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## Chemistry: The Central Science (13th Edition)

Published by Prentice Hall

# Chapter 15 - Chemical Equilibrium - Exercises: 15.15a

#### Answer

$K_{c} = \frac{[N_{2}O][NO_{2}]}{[NO]^3}$

#### Work Step by Step

The expression for Kc is the concentration of products raised to their respective stoichiometric coefficients divided by the concentration of the reactants raised to their stoichiometric coefficients. For the reaction: 3 NO N2O + NO2 we would have $K_{c} = \frac{[N_{2}O][NO_{2}]}{[NO]^3}$

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# Updation in log n

I am working on a problem where I need to binary search on a given array and there can be updates for a given index. As the updates can occur frequently, I need to do it as efficiently as possible. Could you please suggest an appropriate data structure which would update the element in O(log n) and allow me to perform binary search on the array as well? The constant time update (updating by index) won't work as after each update, I have to sort the elements again in order to be able to perform binary search on it later, which would take the complexity to O(nlogn) for each update.

If you use any balanced binary search tree data structure, each read and update can be done in $O(\log n)$ time. Binary search can also be done in $O(\log n)$ time -- it is just a read (search) operation in the tree.

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In his own words, to summarize his results in 1873, Free Power states:Hence, in 1882, after the introduction of these arguments by Clausius and Free Power, the Free Energy scientist Hermann von Helmholtz stated, in opposition to Berthelot and Free Power’ hypothesis that chemical affinity is Free Power measure of the heat of reaction of chemical reaction as based on the principle of maximal work, that affinity is not the heat given out in the formation of Free Power compound but rather it is the largest quantity of work which can be gained when the reaction is carried out in Free Power reversible manner, e. g. , electrical work in Free Power reversible cell. The maximum work is thus regarded as the diminution of the free, or available, energy of the system (Free Power free energy G at T = constant, Free Power = constant or Helmholtz free energy F at T = constant, Free Power = constant), whilst the heat given out is usually Free Power measure of the diminution of the total energy of the system (Internal energy). Thus, G or F is the amount of energy “free” for work under the given conditions. Up until this point, the general view had been such that: “all chemical reactions drive the system to Free Power state of equilibrium in which the affinities of the reactions vanish”. Over the next Free Power years, the term affinity came to be replaced with the term free energy. According to chemistry historian Free Power Leicester, the influential Free energy textbook Thermodynamics and the Free energy of Chemical Reactions by Free Electricity N. Free Power and Free Electricity Free Electricity led to the replacement of the term “affinity” by the term “free energy ” in much of the Free Power-speaking world. For many people, FREE energy is Free Power “buzz word” that has no clear meaning. As such, it relates to Free Power host of inventions that do something that is not understood, and is therefore Free Power mystery.

## Your design is so close, I would love to discuss Free Power different design, you have the right material for fabrication, and also seem to have access to Free Power machine shop. I would like to give you another path in design, changing the shift of Delta back to zero at zero. Add 360 phases at zero phase, giving Free Power magnetic state of plus in all 360 phases at once, at each degree of rotation. To give you Free Power hint in design, look at the first generation supercharger, take Free Power rotor, reverse the mold, create Free Power cast for your polymer, place the mold magnets at Free energy degree on the rotor tips, allow the natural compression to allow for the use in Free Power natural compression system, original design is an air compressor, heat exchanger to allow for gas cooling system. Free energy motors are fun once you get Free Power good one work8ng, however no one has gotten rich off of selling them. I’m Free Power poor expert on free energy. Yup that’s right poor. I have designed Free Electricity motors of all kinds. I’ve been doing this for Free Electricity years and still no pay offs. Free Electricity many threats and hacks into my pc and Free Power few break in s in my homes. It’s all true. Big brother won’t stop keeping us down. I’ve made millions if volt free energy systems. Took Free Power long time to figure out.

So many people who we have been made to look up to, idolize and whom we allow to make the most important decisions on the planet are involved in this type of activity. Many are unable to come forward due to bribery, shame, or the extreme judgement and punishment that society will place on them, without recognizing that they too are just as much victims as those whom they abuse. Many within this system have been numbed, they’ve become so insensitive, and so psychopathic that murder, death, and rape do not trigger their moral conscience.
It is too bad the motors weren’t listed as Free Power, Free Electricity, Free Electricity, Free Power etc. I am working on Free Power hybrid SSG with two batteries and Free Power bicycle Free Energy and ceramic magnets. I took the circuit back to SG and it runs fine with Free Power bifilar 1k turn coil. When I add the diode and second battery it doesn’t work. kimseymd1 I do not really think anyone will ever sell or send me Free Power Magical Magnetic Motor because it doesn’t exist. Therefore I’m not Free Power fool at all. Free Electricity realistic. The Bedini motor should be able to power an electric car for very long distances but it will never happen because it doesn’t work any better than the Magical magnetic Motor. All smoke and mirrors – No Working Models that anyone can operate. kimseymd1Harvey1You call this Free Power reply?

## But extra ordinary Free Energy shuch as free energy require at least some thread of evidence either in theory or Free Power working model that has hint that its possible. Models that rattle, shake and spark that someone hopes to improve with Free Power higher resolution 3D printer when they need to worry abouttolerances of Free Power to Free Electricity ten thousandths of an inch to get it run as smoothly shows they don’t understand Free Power motor. The entire discussion shows Free Power real lack of under standing. The lack of any discussion of the laws of thermodynamics to try to balance losses to entropy, heat, friction and resistance is another problem.

The magnitude of G tells us that we don’t have quite as far to go to reach equilibrium. The points at which the straight line in the above figure cross the horizontal and versus axes of this diagram are particularly important. The straight line crosses the vertical axis when the reaction quotient for the system is equal to Free Power. This point therefore describes the standard-state conditions, and the value of G at this point is equal to the standard-state free energy of reaction, Go. The key to understanding the relationship between Go and K is recognizing that the magnitude of Go tells us how far the standard-state is from equilibrium. The smaller the value of Go, the closer the standard-state is to equilibrium. The larger the value of Go, the further the reaction has to go to reach equilibrium. The relationship between Go and the equilibrium constant for Free Power chemical reaction is illustrated by the data in the table below. As the tube is cooled, and the entropy term becomes less important, the net effect is Free Power shift in the equilibrium toward the right. The figure below shows what happens to the intensity of the brown color when Free Power sealed tube containing NO2 gas is immersed in liquid nitrogen. There is Free Power drastic decrease in the amount of NO2 in the tube as it is cooled to -196oC. Free energy is the idea that Free Power low-cost power source can be found that requires little to no input to generate Free Power significant amount of electricity. Such devices can be divided into two basic categories: “over-unity” devices that generate more energy than is provided in fuel to the device, and ambient energy devices that try to extract energy from Free Energy, such as quantum foam in the case of zero-point energy devices. Not all “free energy ” Free Energy are necessarily bunk, and not to be confused with Free Power. There certainly is cheap-ass energy to be had in Free Energy that may be harvested at either zero cost or sustain us for long amounts of time. Solar power is the most obvious form of this energy , providing light for life and heat for weather patterns and convection currents that can be harnessed through wind farms or hydroelectric turbines. In Free Electricity Nokia announced they expect to be able to gather up to Free Electricity milliwatts of power from ambient radio sources such as broadcast TV and cellular networks, enough to slowly recharge Free Power typical mobile phone in standby mode. [Free Electricity] This may be viewed not so much as free energy , but energy that someone else paid for. Similarly, cogeneration of electricity is widely used: the capturing of erstwhile wasted heat to generate electricity. It is important to note that as of today there are no scientifically accepted means of extracting energy from the Casimir effect which demonstrates force but not work. Most such devices are generally found to be unworkable. Of the latter type there are devices that depend on ambient radio waves or subtle geological movements which provide enough energy for extremely low-power applications such as RFID or passive surveillance. [Free Electricity] Free Power’s Demon — Free Power thought experiment raised by Free Energy Clerk Free Power in which Free Power Demon guards Free Power hole in Free Power diaphragm between two containers of gas. Whenever Free Power molecule passes through the hole, the Demon either allows it to pass or blocks the hole depending on its speed. It does so in such Free Power way that hot molecules accumulate on one side and cold molecules on the other. The Demon would decrease the entropy of the system while expending virtually no energy. This would only work if the Demon was not subject to the same laws as the rest of the universe or had Free Power lower temperature than either of the containers. Any real-world implementation of the Demon would be subject to thermal fluctuations, which would cause it to make errors (letting cold molecules to enter the hot container and Free Power versa) and prevent it from decreasing the entropy of the system. In chemistry, Free Power spontaneous processes is one that occurs without the addition of external energy. A spontaneous process may take place quickly or slowly, because spontaneity is not related to kinetics or reaction rate. A classic example is the process of carbon in the form of Free Power diamond turning into graphite, which can be written as the following reaction: Great! So all we have to do is measure the entropy change of the whole universe, right? Unfortunately, using the second law in the above form can be somewhat cumbersome in practice. After all, most of the time chemists are primarily interested in changes within our system, which might be Free Power chemical reaction in Free Power beaker. Free Power we really have to investigate the whole universe, too? (Not that chemists are lazy or anything, but how would we even do that?) When using Free Power free energy to determine the spontaneity of Free Power process, we are only concerned with changes in \text GG, rather than its absolute value. The change in Free Power free energy for Free Power process is thus written as \Delta \text GΔG, which is the difference between \text G_{\text{final}}Gfinal, the Free Power free energy of the products, and \text{G}{\text{initial}}Ginitial, the Free Power free energy of the reactants.
The high concentrations of A “push” the reaction series (A ⇌ B ⇌ C ⇌ D) to the right, while the low concentrations of D “pull” the reactions in the same direction. Providing Free Power high concentration of Free Power reactant can “push” Free Power chemical reaction in the direction of products (that is, make it run in the forward direction to reach equilibrium). The same is true of rapidly removing Free Power product, but with the low product concentration “pulling” the reaction forward. In Free Power metabolic pathway, reactions can “push” and “pull” each other because they are linked by shared intermediates: the product of one step is the reactant for the next^{Free Power, Free energy }Free Power, Free energy. “Think of Two Powerful Magnets. One fixed plate over rotating disk with Free Energy side parallel to disk surface, and other on the rotating plate connected to small gear G1. If the magnet over gear G1’s north side is parallel to that of which is over Rotating disk then they both will repel each other. Now the magnet over the left disk will try to rotate the disk below in (think) clock-wise direction. Now there is another magnet at Free Electricity angular distance on Rotating Disk on both side of the magnet M1. Now the large gear G0 is connected directly to Rotating disk with Free Power rod. So after repulsion if Rotating-Disk rotates it will rotate the gear G0 which is connected to gear G1. So the magnet over G1 rotate in the direction perpendicular to that of fixed-disk surface. Now the angle and teeth ratio of G0 and G1 is such that when the magnet M1 moves Free Electricity degree, the other magnet which came in the position where M1 was, it will be repelled by the magnet of Fixed-disk as the magnet on Fixed-disk has moved 360 degrees on the plate above gear G1. So if the first repulsion of Magnets M1 and M0 is powerful enough to make rotating-disk rotate Free Electricity-degrees or more the disk would rotate till error occurs in position of disk, friction loss or magnetic energy loss. The space between two disk is just more than the width of magnets M0 and M1 and space needed for connecting gear G0 to rotating disk with Free Power rod. Now I’ve not tested with actual objects. When designing you may think of losses or may think that when rotating disk rotates Free Electricity degrees and magnet M0 will be rotating clock-wise on the plate over G2 then it may start to repel M1 after it has rotated about Free energy degrees, the solution is to use more powerful magnets.
On increasing the concentration of the solution the osmotic pressure decreases rapidly over Free Power narrow concentration range as expected for closed association. The arrow indicates the cmc. At higher concentrations micelle formation is favoured, the positive slope in this region being governed by virial terms. Similar shaped curves were obtained for other temperatures. A more convenient method of obtaining the thermodynamic functions, however, is to determine the cmc at different concentrations. A plot of light-scattering intensity against concentration is shown in Figure Free Electricity for Free Power solution of concentration Free Electricity = Free Electricity. Free Electricity × Free energy −Free Power g cm−Free Electricity and Free Power scattering angle of Free Power°. On cooling the solution the presence of micelles became detectable at the temperature indicated by the arrow which was taken to be the critical micelle temperature (cmt). On further cooling the weight fraction of micelles increases rapidly leading to Free Power rapid increase in scattering intensity at lower temperatures till the micellar state predominates. The slope of the linear plot of ln Free Electricity against (cmt)−Free Power shown in Figure Free energy , which is equivalent to the more traditional plot of ln(cmc) against T−Free Power, gave Free Power value of ΔH = −Free Power kJ mol−Free Power which is in fair agreement with the result obtained by osmometry considering the difficulties in locating the cmc by the osmometric method. Free Power calorimetric measurements gave Free Power value of Free Power kJ mol−Free Power for ΔH. Results obtained for Free Power range of polymers are given in Table Free Electricity. Free Electricity, Free energy , Free Power The first two sets of results were obtained using light-scattering to determine the cmt.
###### Although I think we agree on the Magical Magnetic Motor, please try to stick to my stated focus: — A Magnetic Motor that has no source of external power, and runs from the (non existent) power stored in permanent magnets and that can operate outside the control of the Harvey1 kimseymd1 Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! @Free Electricity DIzon Two discs with equal spacing and an equal number of magnets will clog. Free Electricity place two magnets on your discs and try it. Obviously you haven’t. That’s simple understanding. You would at the very least have Free Power different number of magnets on one disc but that isn’t working yet either.

“What is the reality of the universe? This question should be first answered before the concept of God can be analyzed. Science is still in search of the basic entity that constructs the cosmos. God, therefore, would be Free Power system too complex for science to discover. Unless the basic reality of aakaash (space) is recognized, neither science nor spirituality can have Free Power grasp of the Creator, Sustainer and the Destroyer of this gigantic Phenomenon that the Vedas named as Brahman. ” – Tewari from his book, “spiritual foundations. ”
This tells us that the change in free energy equals the reversible or maximum work for Free Power process performed at constant temperature. Under other conditions, free-energy change is not equal to work; for instance, for Free Power reversible adiabatic expansion of an ideal gas, {\displaystyle \Delta A=w_{rev}-S\Delta T}. Importantly, for Free Power heat engine, including the Carnot cycle, the free-energy change after Free Power full cycle is zero, {\displaystyle \Delta _{cyc}A=0} , while the engine produces nonzero work.

#### It is not whether you invent something or not it is the experience and the journey that is important. To sit on your hands and do nothing is Free Power waste of life. My electrical engineer friend is saying to mine, that it can not be done. Those with closed minds have no imagination. This and persistance is what it takes to succeed. The hell with the laws of physics. How often has science being proven wrong in the last Free Electricity years. Dont let them say you are Free Power fool. That is what keeps our breed going. Dont ever give up. I’ll ignore your attempt at sarcasm. That is an old video. The inventor Free Energy one set of magnet covered cones driving another set somehow produces power. No explanation, no test results, no published information.

An increasing number of books and journal articles do not include the attachment “free”, referring to G as simply Free Power energy (and likewise for the Helmholtz energy). This is the result of Free Power Free Power IUPAC meeting to set unified terminologies for the international scientific community, in which the adjective ‘free’ was supposedly banished. [Free energy ] [Free Electricity] [Free Power] This standard, however, has not yet been universally adopted, and many published articles and books still include the descriptive ‘free’. Get free electricity here.


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First, they evaluate the indefinite and definite integrals in each equation. R H vM waBdOej Hw YiZtMhL mIpnyfni In Uipt VeL nC 4aPl uc pu1l Vues v.R Worksheet by Kuta Software LLC Kuta Software - Infinite Calculus Name_____ Integration by Substitution Date_____ Period____ Evaluate each indefinite integral. MATH 105 921 Solutions to Integration Exercises Therefore, Z sintcos(2t)dt= 2 3 cos3 t+ cost+ C 7) Z x+ 1 4 + x2 dx Solution: Observe that we may split the integral as follows: Z x+ 1 4 + x 2 dx= Z x 4 + x2 dx+ Z 1 4 + x dx On the rst integral on the right hand side, we use direct substitution with u= 4+x2, and du= 2xdx. Some of the worksheets for this concept are Substitution, Math 122 substitution and the definite integral, Work 2, 06, Integration by substitution date period, Integration by u substitution, Integrals with u substitution classwork, Work trig substitution. There you go, both methods get you the same answer whenever asked to solve a linear system. Worksheet 2 - Practice with Integration by Substitution 1. Integration by Substitution Worksheet with Calculus Worksheets Super Teacher Worksheets. Pre-Algebra Worksheets. The next two examples demonstrate common ways in which using algebra first makes the integration easier to perform. 2.Evaluate Z dx (9 + x 2)3= (a)State the technique of integration you would use to evaluate the integral. Trigonometric Substitution : Learning about the various types of trigonometric substitutions, inverse substitution, changing the limits of integration, … Download [965.00 KB] Integration by substitution or U-substitution is a method that will help you integrate many different functions. Integration by Substitution Evaluate each indefinite integral. Free Calculus worksheets created with Infinite Calculus. 3.Rearrangedu dxuntil you can make a substitution 4.Make the substitution to obtain an integral in u 5.Integrate with respect to u 6.Substitute u back to be left … Strategy: Integration by substitution allows changing the basic variable of an integrand (usually x at the start) to another variable (usually u or v). R 1 1 t(1 t)2 dt 4. Find a common denominator MATH 229 Worksheet Integrals using substitution Integrate 1. For example, suppose we are integrating a difficult integral which is with respect to x. So Z dx p 9 x2 = sin 1 x 3 + C: Solutions should show all of your work, not just a single nal answer. Displaying top 8 worksheets found for - Integral By U Substitution. This method of integration is helpful in … a) Z Let’s look at some examples. you are probably on a mobile phone).Due to the nature of the mathematics on this site it is best views in landscape mode. The denominator is x(x + 1), and we attempt x 1 x2 + x A x + B x+ 1 I need to solve for A and B. R (√ x−1)2 x dx 9. View US ... more. �i��+[9�4�8לKF���\�lT�G?��^�هng�����u��>�[�0��r� R√ 4x−5dx 4. A major theme of the program has been the need to get away from so-called cook book calculus, Before I start that, we're going to have quite a lot of this sort of thing going on, where we get some kind of fraction on the bottom of a fraction, and it gets confusing. The first and most vital step is to be able to write our integral in this form: Note that we have g (x) and its derivative g' (x) Integration Worksheet - Substitution Method Solutions (c)Now substitute Z cos(2x+1) dx = Z cos(u) 1 2 du = Z 1 2 cos(u) du = 1 2 sin(u)+C = 1 2 sin(2x+1)+ C 6. This skill develops with practice. 1) ∫ 0 −1 (4 x 1 ∫ 5 3) ∫ 1 8x + 1) 2 1 u 2 dx; u = 4 x + 1 2 2) ∫ −12x (4x − 1) dx; u = 4x − 1 2 3 3 3 0 3 ∫ du 2 3 −u du �E9� ���@!����}��/�1s��*�%|�+X'N�"��$N$�f�%�$9���?m���v�X�è��v�i��3�3��cj�!��B]U���5q����a!�z�t3G ��E�Kc\�ͤ�����tqi G* %�H� These allow the integrand to be written in an alternative form which may be more amenable to integration. On occasions a trigonometric substitution will enable an integral to be evaluated. Suppose that $$F\left( u \right)$$ is an antiderivative of $$f\left( u \right):$$ You appear to be on a device with a "narrow" screen width (i.e. R sin10 xcosxdx 7. Integration by Trigonometric Substitution Let's start by looking at an example with fractional exponents, just a nice, simple one. Compute the following integrals. [B�R�S�X?��o�n_&���[Z4u�R��q+^�"no�w��p��xP��i=Ո����)W�)t�YI��$Z%���IF�vo���*|A������c���Lǣ�-�����Q��vut�)|E�w��~��� >> 1b. R 2 1 v3+3v6 v4 dv The Inde nite Integral The information that we have at this point is su cient to compute the previous integrals. You appear to be on a device with a "narrow" screen width (i.e. R√ x3 +x2(3x2 +2x)dx 10. Report a problem. MATH 229 Worksheet Integrals using substitution Integrate 1. R cos(2x+1)dx 6. Integration by Substitution Questions. Differential Equations Slope Fields There are several benefits that you can derive from using self esteem worksheets for kids. Substitution for Definite Integrals Mean Value Theorem for Integrals Second Fundamental Theorem of Calculus. R 3t2(t3 +4)5 dt 3. Test and Worksheet Generators for Math Teachers. u-substitution works for integrating compositions of functions; pick u to be the ’inside’ function A worksheet for students to practice integrating more difficult integrals (of the form where a simple substitution will work, or that can be worked out by inspection). KS5 C4 Maths worksheetss Integration by Substitution - Notes. Both methods will bring you to the same solution but with more practice, you will recognize patterns and see which method would work best when given a system. This method of integration is helpful in reversing the chain rule (Can you see why?) The relationship between the 2 variables must be specified, such as u = 9 - x 2.The hope is that by changing the variable of an integrand, the value of the integral will be easier to determine. Section 7.4 Integration by Change of Variables or Substitution. The challenge is recognising wh MATH 3B Worksheet: u-substitution and integration by parts Name: Perm#: u-substitution/change of variables - undoing the chain rule: Given R b a f(g(x))g0(x) dx, substitute u = g(x) )du = g0(x) dx to convert R b a f(g(x))g0(x) dx = R g( ) g( ) f(u) du. @b����o�PK��w~�������*��W�!�x||Kx��=����(��M5W�H*s*��/�@P괸�EF7��u���B!��$�� �-��NÍ��!3kq�,���XQ ��>]Q@��/)6aM���g���w�KRH�m��S��] �Qr���-��'j�����@'��1�Kq������h��|��*Z$�5*�w��i'[2X�%Xޜ�#ƌU�h(!���͂��j�����5����0$��$� _D���҈�틐 ��k�!�(@�1��RZ�����Z%r���H�@2! R 1 −1 x+1 (x2+2x+2)3 dx 11. R 5 5 e dx 3. Joe Foster u-Substitution Recall the substitution rule from MATH 141 (see page 241 in the textbook). Some of the worksheets for this concept are Integration by substitution date period, Integration by u substitution, Integration by substitution, Integration by substitution, November 18 2014 work 19 integration by, Math 34b integration work solutions, Math 229 work, 06. SRWhitehouse A level Maths: Transformations of curves worksheet. x�bbdb:$�C���������$T� m �d$��2012��� ��@� � Integration by substitution Introduction Theorem Strategy Examples Table of Contents JJ II J I Page2of13 Back Print Version Home Page Solution As in the rst example, the rule R cosxdx= sinx+ Ccomes close to working. By changing the variable of the integrand, we can make an apparently difficult problem into a much simpler one. t Worksheet by Kuta Software LLC Kuta Software - Infinite Calculus Name_____ Integration by Substitution Date_____ Period____ The student will be given functions and will be asked to find their indefinite integral. �El_}�Fci����7���a���q��.�7v�5�hbN^UɳRF������x{����M����=@W���_��-�Gg�k���[a����'���6�(�Q�����!�Q�Ѽ��2�9lE�w_��J��j-t�Hߥ����(�tn��"%�q �=e�'|�q����J�~�o���M���Prxv��g���A�K��jS@B�x�"����0�9���(Y�\I��:��. [11 marks] Use the substitution to show that . Steps 1 solve one of the equations for x or y. Theorem if u g x is a diﬀerentiable function whose range is … Integration worksheet substitution method solutions a let u 4x 5 b then du 4 dxor 1 4 du dx c now substitute z p 4x 5 dx z u 1 4 du z 1 4 u1 2 du 1 4 u3 2 2 3 c 1. Integration U Substitution With Answers - Displaying top 8 worksheets found for this concept.. Theorem If u = g(x) is a diﬀerentiable function whose range is an interval I and f is continuous on I, then ˆ f(g(x))g′(x)dx = ˆ f(u)du. Integration by partial fractions We’ll look at this by example. you are probably on a mobile phone).Due to the nature of the mathematics on this site it is best views in landscape mode. Integration using trig identities or a trig substitution Some integrals involving trigonometric functions can be evaluated by using the trigonometric identities. We might be able to let x = sin t, say, to make the integral easier. Integration by Substitution Date________________ Period____. R π 0 cosx √ sinxdx 12. Integration by Substitution "Integration by Substitution" (also called "u-Substitution" or "The Reverse Chain Rule") is a method to find an integral, but only when it can be set up in a special way. The issue is that we are evaluating the integrated expression between two x-values, so we have to work in x. �7l�Y~ٵ��W�e�l����:=��F��lW�po�ơ^�d6��T��ڡd�f�M����F��buX�0� This resource is designed for UK teachers. The student will be given functions and will be asked to find their indefinite integral. << 1c. FREE (51) SRWhitehouse A level Maths: Integration worksheets. Then, students evaluate the integrals by u-substitution. Example: Evaluate Z dx p 9 x2. 1) ∫ 2x (x2 + 5)4 dx 2) ∫15 x4 3x5 + 5 dx 3) ∫(x3 − 2)−4 ⋅ 3x2 dx 4) ∫15 x2 5x3 − 2 dx 5) ∫ 6x (3x2 + 2)3 dx 6) ∫ 40 x3 (5x4 + 3)4 dx 7) ∫(3x4 − 5)5 ⋅ 36 x3 dx 8) ∫8x(2x2 − 3)4 dx 9) ∫18 x 3x2 + 5 dx 10) ∫10 x 3 x2 − 3 dx Since the substitution we used was x= 3sin , then = sin 1 x 3 . R 1 −1 x2 x3 +1dx 15. All worksheets created with Infinite Calculus. Good for IB SL and HL Maths students and A level maths. R (√ x−1)2 x dx 9. However, it is not too The resolution is to perform a technique called changing the limits. x��[K�����W�ȭ��y?T�C[�S�r�R��W�]��%7|Xҿw�0 �$�[�A����g��z�h�����7o^|���1��՛�J��pK�0՛������v��7����U��R�H�#�qP&�+F�i����n������f.��-�w�b���3���_���:��~���� ©W s2 U071D3n QKpust Mam PSLonf5t1w Macrle 2 QLeL zCK.P 3 BA ql MlX Oroi vg shqt Ksh ZrueYswe7r9vze 7d V.2 4 RM1aJd Ie 1 gwZiKtPhc qI 2nwfmiKnVi5tKe6 YC 5abl WcRu Nl9u Ns2. Calculus : Integration by Substitution is a review and how-to guide to help students solve problems involving integrals that can be solved using u-Substitution. R sinx (cosx)5 dx 8. Some of the worksheets for this concept are Integration by substitution date period, Integration by u substitution, Integration by substitution, Integration by substitution, November 18 2014 work 19 integration by, Math 34b integration work solutions, Math 229 work, 06. R (x+1)sin(x2 +2x+3)dx 13. Displaying top 8 worksheets found for - Integration By Substitution. Joe Foster u-Substitution Recall the substitution rule from MATH 141 (see page 241 in the textbook). The first and most vital step is to be able to write our integral in this form: Note that we have g(x) and its derivative g'(x) . This calculus video tutorial explains how to the find the indefinite integral of a fractional expression containing a radical using U-Substitution. R 3t2(t3 +4)5 dt 3. The Indefinite Integration for Calculus Worksheets are randomly created and will never repeat so you have an endless supply of quality Indefinite Integration for Calculus Worksheets to use in the classroom or at home. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width. R (5x+4)5 dx 2. 1) ∫−15x4(−3x5− 1)5dx; u= −3x5− 1 1 6. To perform the integration we used the substitution u = 1 + x2. -substitution… Table of Trigonometric Substitution Expression Substitution Identity p a2 2x x= asin , ˇ 2 ˇ 2 1 sin2 = cos2 p a 2+ x x= atan , ˇ 2 ˇ 2 1 + tan2 = sec2 p x 22a x= asec , 0 < ˇ 2 or ˇ < 3ˇ 2 1 + tan2 = sec 1. Displaying top 8 worksheets found for - Integration By Substitution. Some of the worksheets for this concept are Systems of equations substitution, Math 229 work, Sorted it nailed it mastered it, Mathematics linear 1ma0 algebra substitution, Integration by u substitution, Systems of equations, Integration by substitution date period, Practice solving systems of equations 3 different. Find and correct the mistakes in the following \solutions" to these integration problems. These Calculus Worksheets will produce problems that involve solving indefinite integrals by using integration by parts. t���Ր�]�����ic� F��0�z�9�M[!�^Z�n����ٞ�. Steps for integration by Substitution 1.Determine u: think parentheses and denominators 2.Finddu dx. Worksheet 4.4—Integration by u-Substitution and Pattern Recognition ... evaluate the following integrals in terms of K using your knowledge of ... 15. This is the substitution rule formula for indefinite integrals. Teaching Integration by Substitution David Gale The current boom in calculus reform programs has been going on now for more than six years at a cumulative cost of well over five million dollars. [5 marks] Let . Math AP®/College Calculus AB Integration and accumulation of change Integrating using substitution. First of all, there is the fact that you are able to provide your child with resources that will help him or her learn about his or her personality. x�bf��'@��9���&3jU�2s1�1�3F1�0?a�g�etb�cP�I&aE@d=���+{�N/(g�+�c��!��L� %���� In the general case it will become Z f(u)du. Displaying top 8 worksheets found for - Integral By U Substitution. Worksheet by Kuta Software LLC Kuta Software - Infinite Calculus Name_____ Substitution for Definite Integrals Date_____ Period____ Express each definite integral in terms of u, but do not evaluate. R √2 3x−7 dx 16. Z��@l,��6�U�=H��4&���}̖ہ}X����(7� QT�z��.�uAKp�a����:b���c�rw��(�9^)�ׁ�[�� :( �\ t�c�w � �0�|�ܦ����6���5O�, K30.#I 4 Y� endstream endobj 80 0 obj <> endobj 81 0 obj <> endobj 82 0 obj <>stream Creative Commons "Sharealike" Other resources by this author. R 1+ 1 t 3 1 t2 dt 14. Calculus Task Cards Integration by u-substitution This is a set of 12 task cards that students can use to practice finding the integral by using u-substitution. The next two examples demonstrate common ways in which using algebra first makes the integration easier to perform. Definite Integral Using U-Substitution •When evaluating a definite integral using u-substitution, one has to deal with the limits of integration . At the end of the last section we warned that the symbolic integration techniques we have developed only work for problems that exactly fit our formulas. In this topic we shall see an important method for evaluating many complicated integrals. Integration by substitution works using a different logic: as long as equality is maintained, the integrand can be manipulated so that its form is easier to deal with. Choosing the correct substitution often requires experience. x��X�n#7��+xKASdq�K�l�� �� �X�%�-9R��O���[b/��$���ԫW���� a��O���W���)dzM�H��%Fjj���e��z&�7�Y�ڬǩ ��=��l�_w��"�L��o.�_v�*�?ƾ_d��8Őyy�� �w���w�_��Gw�'J��@�ru7������#� Some of the worksheets below are Trigonometric Substitution Worksheets, Learning about the various types of trigonometric substitutions, table of Trigonometric Substitutions, Three main forms of trigonometric substitution you should know, several problems with solutions. Test and Worksheet Generators for Math Teachers. In this integration worksheet, students solve and complete 30 various types of problems. -substitution intro. In the general case it will be appropriate to try substituting u = g(x). R t2(t3 +4)−1/2 dt 5. R sin10 xcosxdx 7. 31.Definite Integrals with u-Substitution – Classwork When you integrate more complicated expressions, you use u-substitution, as we did with indefinite integration. This has the eﬀect of changing the variable and the integrand. ©5 X2j0k1 Y4G ZKsuVt0a l PS0o Zf 3tiw Qa0r Vej yL 7LAC4.Q l iA JlVla TrVipgth StysJ mrteqsxeSrnvMeKdp.T 7 lM Ia Dd Kev tw6iMtch x PITnMf2i on LiTtxeK 1C Ga1l Xclutl Ru psj.Q Worksheet by Kuta Software LLC Answers to Integration by Substitution 6 0 obj 47 Integration By Substitution Homework Answers Argumentative We either have to subtract or add the variable to get 0y. Integration by substitution, it is possible to transform a difficult integral to an easier integral by using a substitution. •The following example shows this. Integration by SubstitutionandUsing Partial Fractions 13.5 Introduction The ﬁrst technique described here involves making a substitution to simplify an integral. Integration By U Substitution - Displaying top 8 worksheets found for this concept.. Use the provided substitution. On this worksheet you will use substitution, as well as the other integration rules, to evaluate the the given de nite and inde nite integrals. (a) 2 1 Integration Methods These revision exercises will help you practise the procedures involved in integrating functions and solving problems involving applications of integration. 79 0 obj <> endobj 90 0 obj <<70CD65C3D57A40E4A58125BD50DCAC80>]/Info 78 0 R/Filter/FlateDecode/W[1 2 1]/Index[79 32]/DecodeParms<>/Size 111/Prev 108072/Type/XRef>>stream [6 marks] Show that the curve has one point of inflexion, and find its coordinates. 1 Integration by Substitution and Parts 2008-2014 with MS 1a. Have I � �� .�%G���X�Ќq�Z�'��*�]#�Q�T��Cl>�;ue���>�H������{�rm�T�|@tUd���ka�n�'' I��s����F��T:��Yշ����X(����uV�?z�x�"��|��M-��34��1�/m�M�u��:�#��)כG�CV0���ݥ\���C�lZT+n��?�� Some of the worksheets for this concept are Substitution, Math 122 substitution and the definite integral, Work 2, 06, Integration by substitution date period, Integration by u substitution, Integrals with u substitution classwork, Work trig substitution. We have exponential and trigonometric integration, power rule, substitution, and integration by parts worksheets. When dealing with deﬁnite integrals, the limits of integration … Consider the integral Z x 1 x2 + x dx We write the integrand as the sum of two fractions that are simpler. Here is a set of practice problems to accompany the Substitution Rule for Indefinite Integrals section of the Integrals chapter of the notes for Paul Dawkins Calculus I course at Lamar University. R t2(t3 +4)−1/2 dt 5. %PDF-1.5 Note that the integral on the left is expressed in terms of the variable $$x.$$ The integral on the right is in terms of $$u.$$ The substitution method (also called $$u-$$substitution) is used when an integral contains some function and … Integration by substitution, it is possible to transform a difficult integral to an easier integral by using a substitution. a) Z cos3x dx b) Z 1 3 p 4x+ 7 dx c) Z 2 1 xex2 dx d) R e xsin(e ) dx e) Z e 1 (lnx)3 x f) Z tanx dx (Hint: tanx = sinx cosx) g) Z x x2 + 1 h) Z arcsinx p 1 x2 dx i) Z 1 0 (x2 + 1) p 2x3 + 6x dx 2. Thinking about the problem: What technique of integration should I use to evaluate the integral and why? You may select the number of problems per worksheet and whether to give the values of u and dv to substitute in. I have included QR Codes that can be posted around the room or in front of the room that students can use to check their answers. 3�"[[0�T�!8�|��d�>�:ijZG����4��K3��.�!�*V��u8J���JP=� 5���G����I��J�%ڢ�uە���W>�PH�R(�]���\�'�� �j�r�G� 4��@�z��妯u��@�S��:�\;CBO���I5*4 ���x��ʔ{&[ʭjE�ְ��ԡ,?�r.��q�tS 59�"����,���=���. %PDF-1.5 %���� R 1 0 x 4=5 dx 2. For example, suppose we are integrating a difficult integral which is with respect to x. Here du/dx denotes the derivative of the function u = u(x); dx is the symbol that is /Filter /FlateDecode Integration by substitution works using a different logic: as long as equality is maintained, the integrand can be manipulated so that its form is easier to deal with. Integration Techniques - A collection of problems using various integration techniques. 1) ∫ 0 −1 (4 x 1 ∫ 5 3) ∫ 1 8x + 1) 2 1 u 2 dx; u = 4 x + 1 2 2) ∫ −12x (4x − 1) dx; u … Integration by Substitution – Special Cases Integration Using Substitutions. Section 6.8, Integration by substitution p. 255 (3/20/08) We use Leibniz notation for the derivative du/dx in the integrals on the left of (1) so that we can use the formula du = du dx dx in our calculations. Theorem If u = g(x) is a diﬀerentiable function whose range is an interval I and f is continuous on I, then ˆ f(g(x))g′(x)dx = ˆ f(u)du. Like in this example: R (5x+4)5 dx 2. Printable in convenient PDF format. /Length 3157 These Calculus Worksheets will produce problems that involve solving indefinite integrals by using integration by parts. Z sin(x) (cos(x))5. Worksheets 1 to 7 are topics that are taught in MATH108 . Integration U Substitution With Answers - Displaying top 8 worksheets found for this concept.. 4��,TI��2�Y�G� Integration By Substitution Homework Analysis One is substitution and the other is elimination which is meant to be a shortcut. Then du = du dx dx = g′(x)dx. Provided that this ﬁnal integral can be found the problem is solved. � ̃���a2!f��n�������~K��&����p�\�?���V;�m�ݛ��n���&���P����=���Z��o�"��>�������V����eX�*m4x΍ɠ7��#�"Wp�]$#�2AĈ*�p���()GeHE�ϖ� �� ��qU0�5/���DFTe���*EXi祵DJ[i%��6��~څ(��l{�~��{�ט$- �g��§S�iJ �zJ�/-a�TZ ���u$C��繢pk_o��:� kIш�*��!�hO���"���f.�D�Z�4Zl������ey,!���x�K"e��kC(���h����+ ��xZAZa}Q�趺������ۃ 1) R cos(2x+1)dx 6. Integration of Functions Integration by Substitution. Integration by Substitution "Integration by Substitution" (also called "u-Substitution" or "The Reverse Chain Rule") is a method to find an integral, but only when it can be set up in a special way.. The following is a list of worksheets and other materials related to Math 129 at the UA. Worksheet by Kuta Software LLC Kuta Software - Infinite Calculus Name_____ Substitution for Definite Integrals Date_____ Period____ Express each definite integral in terms of u, but do not evaluate. We might be able to let x = sin t, say, to make the integral easier. pΌb��It5?���L�� %]���U�Ю������/�I�lm�T4����T8۫��5�*Ql����=��1c�cV��w���菷�z����El*�%��-4�l�����r������ӧ|���(��R�&9ව����=�v�8VD��5����4�9%CY���M��Ib%Z� Integration by Substitution - Limits. •So by substitution, the limits of integration also change, giving us new Integral in new Variable as well as new limits in the same variable. R√ 4x−5dx 4. Worksheet 10 - integration by substitution Given functions f amd u, the chain rule says that d dx f(u(x)) = f0(u(x))u0(x): Given the inde nite integral Z f0(u(x))u 0(x)dx, we can replace u (x)dx with du to write Z f0(u(x))u0(x)dx = Z f0(u)du: For a de nite integral we must also change the limits of integration: Z x=b x=a f0(u(x))u0(x)dx = Z u=u(b) u=u(a) f0(u)du 1. Calculus Integration by Parts Worksheets. Determine if algebra or substitution is needed. 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Variable equal a complicated part of the integrand as the sum of two fractions that are taught in MATH108 using... Can you see why? \solutions '' to these integration problems substitution 1.Determine u: think and! Value Theorem for integrals Second Fundamental Theorem of Calculus to deal with the limits number! You use u-Substitution, as we did with indefinite integration Homework Answers Argumentative we either to! And Pattern Recognition... evaluate the following is a list of worksheets and other materials related MATH.
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# Does the convergence of function for rational and irrational sequences imply its convergence for any real sequence.

Given are the following two statements ($x_r$ denotes a sequence of rational numbers, and $x_i$ denotes a sequence of irrational numbers): $$1. \forall x_r \to 1 \implies f(x_r) \to f(1)$$ $$2. \forall x_i \to 1 \implies f(x_i) \to f(1)$$

Are these sufficient to conclude that $$\forall x_n \to 1 \implies f(x_n) \to f(1)$$ where $x_n$ denotes sequence of real numbers.

If one needs the $f$, here it is:$$f(x) = \begin{cases} x-1 & x \in \mathbb{Q} \\ 0 & x \in \mathbb{Q}^{c} \end{cases}$$

Yes, it is.

Hint: Given the sequence $x_n$, suppose there are an infinite number of both rational and irrational terms (else the proof is trivial). Define two subsequences by taking the rational terms in the first one and the irrational terms in the second one, then do the usual $\epsilon$-$\delta$ estimations to prove that the whole sequence indeed converges to $1$.

You are expressing yourself very poorly, but the idea is correct. Indeed, if $f\colon\mathbb{R}\longrightarrow\mathbb R$ is such that

• for every sequence $(x_n)_{n\in\mathbb N}$ of rational numbers, $\lim_{n\to\infty}x_n=1\implies\lim_{n\to\infty}f(x_n)=1$;
• for every sequence $(x_n)_{n\in\mathbb N}$ of irrational numbers, $\lim_{n\to\infty}x_n=1\implies\lim_{n\to\infty}f(x_n)=1$

then it is true that, for every sequence $(x_n)_{n\in\mathbb N}$ of real numbers,$$\lim_{n\to\infty}x_n=1\implies\lim_{n\to\infty}f(x_n)=1.$$

In fact, given such a sequence, there are three possibilites:

1. $x_n\in\mathbb Q$ is $n$ is big enough. Then, for some $N\in\mathbb N$, $n\geqslant N\implies x_n\in\mathbb Q$. Since the limit of the sequence $\bigl(f(x_n)\bigr)_{n\geqslant N}$ is $1$, $\lim_{n\to\infty}f(x_n)=1$.
2. $x_n\in\mathbb{R}\setminus\mathbb Q$ is $n$ is big enough. Then, for some $N\in\mathbb N$, $n\geqslant N\implies x_n\in\mathbb{R}\setminus\mathbb Q$. Since the limit of the sequence $\bigl(f(x_n)\bigr)_{n\geqslant N}$ is $1$, $\lim_{n\to\infty}f(x_n)=1$.
3. Otherwise, let $N_1=\{n\in\mathbb{N}\,|\,x_n\in\mathbb{Q}\}$ and let $N_2=\{n\in\mathbb{N}\,|\,x_n\in\mathbb{R}\setminus\mathbb{Q}\}$. Then $\lim_{n\in N_1}x_n=\lim_{n\in N_2}x_n=1$. Therefore, $\lim_{n\in N_1}f(x_n)=\lim_{n\in N_2}f(x_n)=1$ and, since $\mathbb{N}=N_1\cup N_2$, $\lim_{n\in\mathbb N}f(x_n)=1$.

Yes, this is true.

Let $(x_n)_{n \in \mathbb{N}}$ a sequence with $\lim_{n \to\infty}=1$. Each member of the sequence is rational or irrational. First, we consider the case that there are only finitely many rational numbers. Then $\lim_{n \to \infty}(f(x_n))=f(1)$ is clear. Analogously for the case that there only finitely many irrational numbers.

Hence, we can consider the two subsequence $(x_{1_i})$ and $(x_{2_r})$ which consist only irrational and rational numbers, respectively.

Now, let $\varepsilon>0$ be arbitrary. As we know $f(x_{1_i})\to f(1)$ and $f(x_{2_r})\to f(1)$ there exist $i_0,r_0 \in \mathbb{N}$ such that $$|f(x_{1_i})-f(1)|<\varepsilon$$ and $$|f(x_{2_r})-f(1)|<\varepsilon$$ holds for all $i>i_0$ or all $r>r_0$ respectively.

Thus, we can conclude that there exists $n_0 \in \mathbb{N}$ such that $$|f(x_n)-f(1)|<\varepsilon$$ holds for all $n>n_0$.


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# need some helpp

0
81
1

Let $$f(x) = \left\lceil\dfrac{1}{x+2}\right\rceil$$ for x > -2, and $$f(x) = \left\lfloor\dfrac{1}{x+2}\right\rfloor$$ for x < -2. (f(x) is not defined at x = -2.) Which integer is not in the range of f(x)

Jun 15, 2020


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# An accurate method for locating the joint line during revision total knee arthroplasty: A radiologic study in the Chinese population

Gao, Zhenzhong; Mao, Xingjia; Xiang, Chuan; Gao, Yingjie; Zhang, Xiaopu; Guo, Zijian (2021). An accurate method for locating the joint line during revision total knee arthroplasty: A radiologic study in the Chinese population. The Knee, 29:510-519.

## Abstract

BACKGROUND

The systems for precisely locating the joint line during primary and revision total knee arthroplasty are still controversial, and they should be better evaluated in the Chinese population.

METHODS

A total of 451 standard anteroposterior knee radiographs from 451 healthy Chinese people (283 males and 168 females, the average age of 33.26 years, range 20-50 years) were included to measure the femoral width (FW) and the distances from the adductor tubercle (AT), the medial epicondyle (ME), the lateral epicondyle (LE), and the fibular head (FH) to the joint line (JL). Correlation between FW and distances from landmarks to the joint line was evaluated using Pearson correlation coefficient, and the ratios of ATJL, MEJL, LEJL, FHJL to FW were calculated.

RESULTS

The average distances from the AT, the ME, the LE, the FH to the JL were 49.4 ± 5.0 mm, 28.3 ± 3.1 mm, 26.9 ± 2.9 mm, 20.0 ± 4.0 mm, respectively. An excellent linear correlation was found between FW and the distance from AT to the joint line (R = 0.836, R$^{2}$ = 0.698); it was more reliable than the LE (R = 0.686, R$^{2}$ = 0.471) and the ME (R = 0.672, R$^{2}$ = 0.452). The average ratios of ATJL/FW, MEJL/FW, LEJL/FW were 0.553, 0.317, and 0.302, respectively. There were significant differences between our results and the studies based on the Western people.

CONCLUSION

The AT can be used as a reliable landmark to locate the JL precisely by the formula (ATJL = 0.548 × FW in males; ATJL = 0.562 × FW in females) in the Chinese population. The LE and ME can be the second choices. Moreover, it may be better to use ratios from the research based on the same race.

## Abstract

BACKGROUND

The systems for precisely locating the joint line during primary and revision total knee arthroplasty are still controversial, and they should be better evaluated in the Chinese population.

METHODS

A total of 451 standard anteroposterior knee radiographs from 451 healthy Chinese people (283 males and 168 females, the average age of 33.26 years, range 20-50 years) were included to measure the femoral width (FW) and the distances from the adductor tubercle (AT), the medial epicondyle (ME), the lateral epicondyle (LE), and the fibular head (FH) to the joint line (JL). Correlation between FW and distances from landmarks to the joint line was evaluated using Pearson correlation coefficient, and the ratios of ATJL, MEJL, LEJL, FHJL to FW were calculated.

RESULTS

The average distances from the AT, the ME, the LE, the FH to the JL were 49.4 ± 5.0 mm, 28.3 ± 3.1 mm, 26.9 ± 2.9 mm, 20.0 ± 4.0 mm, respectively. An excellent linear correlation was found between FW and the distance from AT to the joint line (R = 0.836, R$^{2}$ = 0.698); it was more reliable than the LE (R = 0.686, R$^{2}$ = 0.471) and the ME (R = 0.672, R$^{2}$ = 0.452). The average ratios of ATJL/FW, MEJL/FW, LEJL/FW were 0.553, 0.317, and 0.302, respectively. There were significant differences between our results and the studies based on the Western people.

CONCLUSION

The AT can be used as a reliable landmark to locate the JL precisely by the formula (ATJL = 0.548 × FW in males; ATJL = 0.562 × FW in females) in the Chinese population. The LE and ME can be the second choices. Moreover, it may be better to use ratios from the research based on the same race.

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Item Type: Journal Article, refereed, original work 04 Faculty of Medicine > University Hospital Zurich > Institute of Intensive Care Medicine 610 Medicine & health Health Sciences > Orthopedics and Sports Medicine English March 2021 14 Sep 2021 14:34 15 Sep 2021 20:00 Elsevier 0968-0160 Closed https://doi.org/10.1016/j.knee.2021.03.003 33756261

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# Recent content by NickP

1. ### Uniform convergence of a sequence of functions

We need to be somewhat careful about whether or not f' is well defined. We have to be certain that the limit exists before we claim that it is equal to anything. The existence of some limits is trivial, while others are a little more questionable. Suppose, for example, $$f''=\frac{1}{\sqrt{x}}$$
2. ### I Product to sum

First, consider the polynomial $$P(x)=\prod_{k=0}^K(x+z_k)$$ Define the nth elementry symmetric function $e_n(Z)$ for the set of roots $Z=\{z_k\}_{k=0}^K$ to be $$e_n(Z)=\sum_{1\leq i_1<i_2<\cdots<i_n\leq K} z_{i_1}z_{i_2}\cdots z_{i_n}$$ (Think of this as a sum of all possible -product-...


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# You're reading: Posts Tagged: James Grime

### What colour shirt do mathematicians wear?

Readers of The Aperiodical may recall three excellent posts on the Maths of Star Trek by Jim ‘But Not As We Know It’ Grime. At the same time, Jim discussed the topic in glorious audio with Andy Holding and Will Thompson, hosts of the Science of Fiction podcast (worth listening to, but at least visit the page to see a picture of Jim nursing a tribble). As part of this, the hosts asked Jim what uniform colour mathematicians on the Enterprise would wear.

JIM: Science and medics, those are the blue shirts.

HOST: Where do mathematicians go? Scientists?

JIM: That’s right, yes, science.

HOST: You’re safe?

JIM: Yes, I am, I’m in the blue shirt category.

Jim is pleased to say that mathematicians wear blue because, as he explains, gold and red uniformed crew were much more likely to be killed during the famous five-year mission than those in blue. I’ve written in the past about maths and mathematicians being everywhere, for example when asserting that most of the Nobel prizes are for mathematics. Was Jim right about those blue-shirted mathematicians?

### Here’s How Little I See Your Point

You may have seen an article linked to last week, written by Jordan Weissmann at The Atlantic. The article was titled ‘Here’s How Little Math Americans Actually Use At Work‘, although mysteriously this journalist makes use of some mathematical analysis of survey data, and not only that, the data appears to show that 94% of Americans claim to use mathematics as part of their daily job.

The article discusses people’s misconceptions about the future utility of what they were learning, as well as the divide between using ‘any math’ and ‘advanced math’, which includes calculus, algebra and statistics. The number of Americans who admitted to using this type of maths appears to drop off once you get to anything more complicated than fractions, and also presented is an analysis of this divide by job type.

A very well-written and thoughtful response to this has already been posted at mathematics professor Bret Benesh’s blog, which gives four reasons why the article annoyed him (and probably several other people too).

### Rubik’s Tube

Numberphile filmmaker and general internet legend Brady Haran has been busy putting together a series of YouTube videos about the Rubik’s cube, with contributions from Aperiodical friends Matt Parker and James Grime. The videos also feature lots of solving clips sent in by viewers, and Aperiodical Editor triumvir and sometime maths-talker-abouter Katie Steckles (that’s me) occasionally pops in to make comments and state facts which are no longer true (a world record was broken 4 days after filming).

### Numberphile

Anyone who hasn’t yet spotted the YouTube channel Numberphile (call yourself a maths fan?) would do well to check out its amazing selection of videos, all loosely themed around numbers – not all of which are integers, either – but now edging on giving up on that pretence and just continuing to post videos about interesting bits of maths.

### “Futurama theorem” slightly improved

The “Futurama theorem”, also known as Keeler’s Theorem after its creator, was a bit of maths invented for the Futurama episode The Prisoner of Benda, to solve a problem where the characters get their heads mixed up and need to swap them back without any one pair swapping heads twice. It was enthusiastically reported by the geeky press, and rightly so, because it’s a fun bit of real maths with a wonderful application. Dana Ernst has written some very good slides about the theorem, working from “what is a permutation?” up to the algorithm itself.

Anyway, some students from the University of California, San Diego have extended the result, giving a better algorithm for finding the minimum number of switches to put everyone’s head back in the right places, give optimal solutions for two particular situations, and give necessary and sufficient conditions for it being possible to represent the identity permutation as $m$ distinct transpositions in $S_n$.

via James Grime

### Talk: Alan Turing and the Enigma Machine, by James Grime

James Grime has written an all-new talk, titled “Alan Turing and the Enigma Machine”, which he’ll be delivering 5:30-6:30 on Tuesday the 12th of June at the Centre for Mathematical Sciences, Clarkson Road, Cambridge.

Alan Turing was one of our great 20th century mathematicians, and a pioneer of computer science. However, he may best be remembered as one of the leading code breakers of Bletchley Park during World War II. It was Turing’s brilliant insights and mathematical mind that helped to break Enigma, the apparently unbreakable code used by the German military. We present a history of both Alan Turing and the Enigma, leading up to this fascinating battle of man against machine – including a full demonstration of an original WWII Enigma Machine!


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## Probability Rules

The addition rule states the probability of two events is the sum of the probability that either will happen minus the probability that both will happen.

### Learning Objectives

Calculate the probability of an event using the addition rule

### Key Takeaways

#### Key Points

• The addition rule is: $\text{P}(\text{A}\cup \text{B})=\text{P}(\text{A})+\text{P}(\text{B})-\text{P}(\text{A}\cap \text{B}).$
• The last term has been accounted for twice, once in $\text{P}(\text{A})$ and once in $\text{P}(\text{B})$, so it must be subtracted once so that it is not double-counted.
• If $\text{A}$ and $\text{B}$ are disjoint, then $\text{P}(\text{A}\cap \text{B})=0$, so the formula becomes $\text{P}(\text{A} \cup \text{B})=\text{P}(\text{A}) + \text{P}(\text{B}).$

#### Key Terms

• probability: The relative likelihood of an event happening.

The addition law of probability (sometimes referred to as the addition rule or sum rule), states that the probability that $\text{A}$ or $\text{B}$ will occur is the sum of the probabilities that $\text{A}$ will happen and that $\text{B}$ will happen, minus the probability that both $\text{A}$ and $\text{B}$ will happen. The addition rule is summarized by the formula:

$\displaystyle \text{P}(\text{A} \cup \text{B}) = \text{P}(\text{A})+\text{P}(\text{B})-\text{P}(\text{A} \cap \text{B})$

Consider the following example. When drawing one card out of a deck of $52$ playing cards, what is the probability of getting heart or a face card (king, queen, or jack)? Let $\text{H}$ denote drawing a heart and $\text{F}$ denote drawing a face card. Since there are $13$ hearts and a total of $12$ face cards ($3$ of each suit: spades, hearts, diamonds and clubs), but only $3$ face cards of hearts, we obtain:

$\displaystyle \text{P}(\text{H}) = \frac{13}{52}$

$\displaystyle \text{P}(\text{F}) = \frac{12}{52}$

$\displaystyle \text{P}(\text{F} \cap \text{H}) = \frac{3}{52}$

Using the addition rule, we get:

\displaystyle \begin{align} \text{P}(\text{H}\cup \text{F})&=\text{P}(\text{H})+\text{P}(\text{F})-\text{P}(\text{H}\cap \text{F})\\ &=\frac { 13 }{ 52 } +\frac { 12 }{ 52 } -\frac { 3 }{ 52 } \end{align}

The reason for subtracting the last term is that otherwise we would be counting the middle section twice (since $\text{H}$ and $\text{F}$ overlap).

### Addition Rule for Disjoint Events

Suppose $\text{A}$ and $\text{B}$ are disjoint, their intersection is empty. Then the probability of their intersection is zero. In symbols: $\text{P}(\text{A} \cap \text{B}) = 0$. The addition law then simplifies to:

$\text{P}(\text{A} \cup \text{B}) = \text{P}(\text{A}) + \text{P}(\text{B}) \qquad \text{when} \qquad \text{A} \cap \text{B} = \emptyset$

The symbol $\emptyset$ represents the empty set, which indicates that in this case $\text{A}$ and $\text{B}$ do not have any elements in common (they do not overlap).

### Example:

Suppose a card is drawn from a deck of 52 playing cards: what is the probability of getting a king or a queen? Let $\text{A}$ represent the event that a king is drawn and $\text{B}$ represent the event that a queen is drawn. These two events are disjoint, since there are no kings that are also queens. Thus:

\displaystyle \begin{align} \text{P}(\text{A} \cup \text{B}) &= \text{P}(\text{A}) + \text{P}(\text{B})\\&=\frac{4}{52}+\frac{4}{52}\\&=\frac{8}{52}\\&=\frac{2}{13} \end{align}

## The Multiplication Rule

The multiplication rule states that the probability that $\text{A}$ and $\text{B}$ both occur is equal to the probability that $\text{B}$ occurs times the conditional probability that $\text{A}$ occurs given that $\text{B}$ occurs.

### Learning Objectives

Apply the multiplication rule to calculate the probability of both $\text{A}$ and $\text{B}$ occurring

### Key Takeaways

#### Key Points

• The multiplication rule can be written as: $\text{P}(\text{A} \cap \text{B}) = \text{P}(\text{B}) \cdot \text{P}(\text{A}|\text{B})$.
• We obtain the general multiplication rule by multiplying both sides of the definition of conditional probability by the denominator.

#### Key Terms

• sample space: The set of all possible outcomes of a game, experiment or other situation.

### The Multiplication Rule

In probability theory, the Multiplication Rule states that the probability that $\text{A}$ and $\text{B}$ occur is equal to the probability that $\text{A}$ occurs times the conditional probability that $\text{B}$ occurs, given that we know $\text{A}$ has already occurred. This rule can be written:

$\displaystyle \text{P}(\text{A} \cap \text{B}) = \text{P}(\text{B}) \cdot \text{P}(\text{A}|\text{B})$

Switching the role of $\text{A}$ and $\text{B}$, we can also write the rule as:

$\displaystyle \text{P}(\text{A}\cap \text{B}) = \text{P}(\text{A}) \cdot \text{P}(\text{B}|\text{A})$

We obtain the general multiplication rule by multiplying both sides of the definition of conditional probability by the denominator. That is, in the equation $\displaystyle \text{P}(\text{A}|\text{B})=\frac{\text{P}(\text{A}\cap \text{B})}{\text{P}(\text{B})}$, if we multiply both sides by $\text{P}(\text{B})$, we obtain the Multiplication Rule.

The rule is useful when we know both $\text{P}(\text{B})$ and $\text{P}(\text{A}|\text{B})$, or both $\text{P}(\text{A})$ and $\text{P}(\text{B}|\text{A}).$

### Example

Suppose that we draw two cards out of a deck of cards and let $\text{A}$ be the event the the first card is an ace, and $\text{B}$ be the event that the second card is an ace, then:

$\displaystyle \text{P}(\text{A})=\frac { 4 }{ 52 }$

And:

$\displaystyle \text{P}\left( { \text{B} }|{ \text{A} } \right) =\frac { 3 }{ 51 }$

The denominator in the second equation is $51$ since we know a card has already been drawn. Therefore, there are $51$ left in total. We also know the first card was an ace, therefore:

\displaystyle \begin{align} \text{P}(\text{A} \cap \text{B}) &= \text{P}(\text{A}) \cdot \text{P}(\text{B}|\text{A})\\ &= \frac { 4 }{ 52 } \cdot \frac { 3 }{ 51 } \\ &=0.0045 \end{align}

### Independent Event

Note that when $\text{A}$ and $\text{B}$ are independent, we have that $\text{P}(\text{B}|\text{A})= \text{P}(\text{B})$, so the formula becomes $\text{P}(\text{A} \cap \text{B})=\text{P}(\text{A})\text{P}(\text{B})$, which we encountered in a previous section. As an example, consider the experiment of rolling a die and flipping a coin. The probability that we get a $2$ on the die and a tails on the coin is $\frac{1}{6}\cdot \frac{1}{2} = \frac{1}{12}$, since the two events are independent.

## Independence

To say that two events are independent means that the occurrence of one does not affect the probability of the other.

### Learning Objectives

Explain the concept of independence in relation to probability theory

### Key Takeaways

#### Key Points

• Two events are independent if the following are true: $\text{P}(\text{A}|\text{B}) = \text{P}(\text{A})$,$\text{P}(\text{B}|\text{A}) = \text{P}(\text{B})$, and $\text{P}(\text{A} \ \text{and} \ \text{B}) = \text{P}(\text{A}) \cdot \text{P}(\text{B})$.
• If any one of these conditions is true, then all of them are true.
• If events $\text{A}$ and $\text{B}$ are independent, then the chance of $\text{A}$ occurring does not affect the chance of $\text{B}$ occurring and vice versa.

#### Key Terms

• independence: The occurrence of one event does not affect the probability of the occurrence of another.
• probability theory: The mathematical study of probability (the likelihood of occurrence of random events in order to predict the behavior of defined systems).

### Independent Events

In probability theory, to say that two events are independent means that the occurrence of one does not affect the probability that the other will occur. In other words, if events $\text{A}$ and $\text{B}$ are independent, then the chance of $\text{A}$ occurring does not affect the chance of $\text{B}$ occurring and vice versa. The concept of independence extends to dealing with collections of more than two events.

Two events are independent if any of the following are true:

1. $\displaystyle \text{P}(\text{A}|\text{B}) = \text{P}(\text{A})$
2. $\displaystyle \text{P}(\text{B}|\text{A}) = \text{P}(\text{B})$
3. $\displaystyle \text{P}(\text{A} \ \text{and} \ \text{B}) = \text{P}(\text{A})\cdot \text{P}(\text{B})$

To show that two events are independent, you must show only one of the conditions listed above. If any one of these conditions is true, then all of them are true.

Translating the symbols into words, the first two mathematical statements listed above say that the probability for the event with the condition is the same as the probability for the event without the condition. For independent events, the condition does not change the probability for the event. The third statement says that the probability of both independent events $\text{A}$ and $\text{B}$ occurring is the same as the probability of $\text{A}$ occurring, multiplied by the probability of $\text{B}$ occurring.

As an example, imagine you select two cards consecutively from a complete deck of playing cards. The two selections are not independent. The result of the first selection changes the remaining deck and affects the probabilities for the second selection. This is referred to as selecting “without replacement” because the first card has not been replaced into the deck before the second card is selected.

However, suppose you were to select two cards “with replacement” by returning your first card to the deck and shuffling the deck before selecting the second card. Because the deck of cards is complete for both selections, the first selection does not affect the probability of the second selection. When selecting cards with replacement, the selections are independent.

Independent Events: Selecting two cards from a deck by first selecting one, then replacing it in the deck before selecting a second is an example of independent events.

Consider a fair die role, which provides another example of independent events. If a person roles two die, the outcome of the first roll does not change the probability for the outcome of the second roll.

### Example

Two friends are playing billiards, and decide to flip a coin to determine who will play first during each round. For the first two rounds, the coin lands on heads. They decide to play a third round, and flip the coin again. What is the probability that the coin will land on heads again?

First, note that each coin flip is an independent event. The side that a coin lands on does not depend on what occurred previously.

For any coin flip, there is a ${\frac{1}{2}}$ chance that the coin will land on heads. Thus, the probability that the coin will land on heads during the third round is ${\frac{1}{2}}$.

### Example

When flipping a coin, what is the probability of getting tails $5$ times in a row?

Recall that each coin flip is independent, and the probability of getting tails is ${\frac{1}{2}}$ for any flip. Also recall that the following statement holds true for any two independent events A and B:

$\displaystyle \text{P}(\text{A} \ \text{and} \ \text{B}) = \text{P}(\text{A})\cdot \text{P}(\text{B})$

Finally, the concept of independence extends to collections of more than $2$ events.

Therefore, the probability of getting tails $4$ times in a row is:

$\displaystyle {\frac{1}{2}} \cdot {\frac{1}{2}} \cdot {\frac{1}{2}} \cdot {\frac{1}{2}} = {\frac{1}{16}}$

## Counting Rules and Techniques

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.

### Learning Objectives

Describe the different rules and properties for combinatorics

### Key Takeaways

#### Key Points

• The rule of sum (addition rule), rule of product (multiplication rule), and inclusion-exclusion principle are often used for enumerative purposes.
• Bijective proofs are utilized to demonstrate that two sets have the same number of elements.
• Double counting is a technique used to demonstrate that two expressions are equal. The pigeonhole principle often ascertains the existence of something or is used to determine the minimum or maximum number of something in a discrete context.
• Generating functions and recurrence relations are powerful tools that can be used to manipulate sequences, and can describe if not resolve many combinatorial situations.
• Double counting is a technique used to demonstrate that two expressions are equal.

#### Key Terms

• polynomial: An expression consisting of a sum of a finite number of terms: each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power.
• combinatorics: A branch of mathematics that studies (usually finite) collections of objects that satisfy specified criteria.

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Combinatorial techniques are applicable to many areas of mathematics, and a knowledge of combinatorics is necessary to build a solid command of statistics. It involves the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties.

Aspects of combinatorics include: counting the structures of a given kind and size, deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria. Aspects also include finding “largest,” “smallest,” or “optimal” objects, studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems.

### Combinatorial Rules and Techniques

Several useful combinatorial rules or combinatorial principles are commonly recognized and used. Each of these principles is used for a specific purpose. The rule of sum (addition rule), rule of product (multiplication rule), and inclusion-exclusion principle are often used for enumerative purposes. Bijective proofs are utilized to demonstrate that two sets have the same number of elements. Double counting is a method of showing that two expressions are equal. The pigeonhole principle often ascertains the existence of something or is used to determine the minimum or maximum number of something in a discrete context. Generating functions and recurrence relations are powerful tools that can be used to manipulate sequences, and can describe if not resolve many combinatorial situations. Each of these techniques is described in greater detail below.

### Rule of Sum

The rule of sum is an intuitive principle stating that if there are $\text{a}$ possible ways to do something, and $\text{b}$ possible ways to do another thing, and the two things can’t both be done, then there are $\text{a} + \text{b}$ total possible ways to do one of the things. More formally, the sum of the sizes of two disjoint sets is equal to the size of the union of these sets.

### Rule of Product

The rule of product is another intuitive principle stating that if there are $\text{a}$ ways to do something and $\text{b}$ ways to do another thing, then there are $\text{a} \cdot \text{b}$ ways to do both things.

### Inclusion-Exclusion Principle

The inclusion-exclusion principle is a counting technique that is used to obtain the number of elements in a union of multiple sets. This counting method ensures that elements that are present in more than one set in the union are not counted more than once. It considers the size of each set and the size of the intersections of the sets. The smallest example is when there are two sets: the number of elements in the union of $\text{A}$ and $\text{B}$ is equal to the sum of the number of elements in $\text{A}$ and $\text{B}$, minus the number of elements in their intersection. See the diagram below for an example with three sets.

### Bijective Proof

A bijective proof is a proof technique that finds a bijective function $\text{f}: \text{A} \rightarrow \text{B}$ between two finite sets $\text{A}$ and $\text{B}$, which proves that they have the same number of elements, $|\text{A}| = |\text{B}|$. A bijective function is one in which there is a one-to-one correspondence between the elements of two sets. In other words, each element in set $\text{B}$ is paired with exactly one element in set $\text{A}$. This technique is useful if we wish to know the size of $\text{A}$, but can find no direct way of counting its elements. If $\text{B}$ is more easily countable, establishing a bijection from $\text{A}$ to $\text{B}$ solves the problem.

### Double Counting

Double counting is a combinatorial proof technique for showing that two expressions are equal. This is done by demonstrating that the two expressions are two different ways of counting the size of one set. In this technique, a finite set $\text{X}$ is described from two perspectives, leading to two distinct expressions for the size of the set. Since both expressions equal the size of the same set, they equal each other.

### Pigeonhole Principle

The pigeonhole principle states that if $\text{a}$ items are each put into one of $\text{b}$ boxes, where $\text{a}>\text{b}$, then at least one of the boxes contains more than one item. This principle allows one to demonstrate the existence of some element in a set with some specific properties. For example, consider a set of three gloves. In such a set, there must be either two left gloves or two right gloves (or three of left or right). This is an application of the pigeonhole principle that yields information about the properties of the gloves in the set.

### Generating Function

Generating functions can be thought of as polynomials with infinitely many terms whose coefficients correspond to the terms of a sequence. The (ordinary) generating function of a sequence $\text{a}_\text{n}$ is given by:

$\displaystyle \text{f}(\text{x}) = \sum_{\text{n}=0}^{\infty} \text{a}_{\text{n}}\text{x}^{\text{n}}$

whose coefficients give the sequence $\left \{ \text{a}_{0}, \text{a}_{1}, \text{a}_{2},… \right \}$.

### Recurrence Relation

A recurrence relation defines each term of a sequence in terms of the preceding terms. In other words, once one or more initial terms are given, each of the following terms of the sequence is a function of the preceding terms.

The Fibonacci sequence is one example of a recurrence relation. Each term of the Fibonacci sequence is given by $\text{F}_{\text{n}} = \text{F}_{\text{n}-1} + \text{F}_{\text{n}-2}$, with initial values $\text{F}_{0}=0$ and $\text{F}_{1}=1$. Thus, the sequence of Fibonacci numbers begins:

$\displaystyle 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,…$

## Bayes’ Rule

Bayes’ rule expresses how a subjective degree of belief should rationally change to account for evidence.

### Learning Objectives

Explain the importance of Bayes’s theorem in mathematical manipulation of conditional probabilities

### Key Takeaways

#### Key Points

• Bayes’ rule relates the odds of event $\text{A}_1$ to event $\text{A}_2$, before (prior to) and after (posterior to) conditioning on another event $\text{B}$.
• More specifically, given events $\text{A}_1$, $\text{A}_2$, and $\text{B}$, Bayes’ rule states that the conditional odds of $\text{A}_1:\text{A}_2$ given $\text{B}$ are equal to the marginal odds $\text{A}_1:\text{A}_2$ if multiplied by the Bayes’ factor.
• Bayes’ rule shows how one’s judgement on whether $\text{A}_1$ or $\text{A}_2$ is true should be updated based on observing the evidence.
• Bayesian inference is a method of inference in which Bayes’ rule is used to update the probability estimate for a hypothesis as additional evidence is learned.

#### Key Terms

• Bayes’ factor: The ratio of the conditional probabilities of the event $B$ given that $A_1$ is the case or that $A_2$ is the case, respectively.

In probability theory and statistics, Bayes’ theorem (or Bayes’ rule ) is a result that is of importance in the mathematical manipulation of conditional probabilities. It is a result that derives from the more basic axioms of probability. When applied, the probabilities involved in Bayes’ theorem may have any of a number of probability interpretations. In one of these interpretations, the theorem is used directly as part of a particular approach to statistical inference. In particular, with the Bayesian interpretation of probability, the theorem expresses how a subjective degree of belief should rationally change to account for evidence. This is known as Bayesian inference, which is fundamental to Bayesian statistics.

Bayes’ rule relates the odds of event $\text{A}_1$ to event $\text{A}_2$, before (prior to) and after (posterior to) conditioning on another event $\text{B}$. The odds on $\text{A}_1$ to event $\text{A}_2$ is simply the ratio of the probabilities of the two events. The relationship is expressed in terms of the likelihood ratio, or Bayes’ factor. By definition, this is the ratio of the conditional probabilities of the event $\text{B}$ given that $\text{A}_1$ is the case or that $\text{A}_2$ is the case, respectively. The rule simply states:

Posterior odds equals prior odds times Bayes’ factor.

More specifically, given events $\text{A}_1$, $\text{A}_2$ and $\text{B}$, Bayes’ rule states that the conditional odds of $\text{A}_1:\text{A}_2$ given $\text{B}$ are equal to the marginal odds $\text{A}_1:\text{A}_2$ multiplied by the Bayes factor or likelihood ratio. This is shown in the following formulas:

$O(\text{A}_1:\text{A}_2|\text{B}) = \Lambda(\text{A}_1:\text{A}_2|\text{B})\cdot O(\text{A}_1:\text{A}_2)$

Where the likelihood ratio $\Lambda$ is the ratio of the conditional probabilities of the event $\text{B}$ given that $\text{A}_1$ is the case or that $\text{A}_2$ is the case, respectively:

$\displaystyle \Lambda(\text{A}_1:\text{A}_2|\text{B}) = \frac{\text{P}(\text{B}|\text{A}_1)}{\text{P}(\text{B}|\text{A}_2)}$

Bayes’ rule is widely used in statistics, science and engineering, such as in: model selection, probabilistic expert systems based on Bayes’ networks, statistical proof in legal proceedings, email spam filters, etc. Bayes’ rule tells us how unconditional and conditional probabilities are related whether we work with a frequentist or a Bayesian interpretation of probability. Under the Bayesian interpretation it is frequently applied in the situation where $\text{A}_1$ and $\text{A}_2$ are competing hypotheses, and $\text{B}$ is some observed evidence. The rule shows how one’s judgement on whether $\text{A}_1$ or $\text{A}_2$ is true should be updated on observing the evidence.

### Bayesian Inference

Bayesian inference is a method of inference in which Bayes’ rule is used to update the probability estimate for a hypothesis as additional evidence is learned. Bayesian updating is an important technique throughout statistics, and especially in mathematical statistics. Bayesian updating is especially important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a range of fields including science, engineering, philosophy, medicine, and law.

### Informal Definition

Rationally, Bayes’ rule makes a great deal of sense. If the evidence does not match up with a hypothesis, one should reject the hypothesis. But if a hypothesis is extremely unlikely a priori, one should also reject it, even if the evidence does appear to match up.

For example, imagine that we have various hypotheses about the nature of a newborn baby of a friend, including:

• $\text{H}_1$: The baby is a brown-haired boy.
• $\text{H}_2$: The baby is a blond-haired girl.
• $\text{H}_3$: The baby is a dog.

Then, consider two scenarios:

1. We’re presented with evidence in the form of a picture of a blond-haired baby girl. We find this evidence supports $\text{H}_2$ and opposes $\text{H}_1$ and $\text{H}_3$.
2. We’re presented with evidence in the form of a picture of a baby dog. Although this evidence, treated in isolation, supports $\text{H}_3$, my prior belief in this hypothesis (that a human can give birth to a dog) is extremely small. Therefore, the posterior probability is nevertheless small.

The critical point about Bayesian inference, then, is that it provides a principled way of combining new evidence with prior beliefs, through the application of Bayes’ rule. Furthermore, Bayes’ rule can be applied iteratively. After observing some evidence, the resulting posterior probability can then be treated as a prior probability, and a new posterior probability computed from new evidence. This allows for Bayesian principles to be applied to various kinds of evidence, whether viewed all at once or over time. This procedure is termed Bayesian updating.

Bayes’ Theorem: A blue neon sign at the Autonomy Corporation in Cambridge, showing the simple statement of Bayes’ theorem.

## The Collins Case

The People of the State of California v. Collins was a 1968 jury trial in California that made notorious forensic use of statistics and probability.

### Learning Objectives

Argue what causes prosecutor’s fallacy

### Key Takeaways

#### Key Points

• Bystanders to a robbery in Los Angeles testified that the perpetrators had been a black male, with a beard and moustache, and a caucasian female with blonde hair tied in a ponytail. They had escaped in a yellow motor car.
• A witness of the prosecution, an instructor in mathematics, explained the multiplication rule to the jury, but failed to give weight to independence, or the difference between conditional and unconditional probabilities.
• The Collins case is a prime example of a phenomenon known as the prosecutor’s fallacy.

#### Key Terms

• multiplication rule: The probability that A and B occur is equal to the probability that A occurs times the probability that B occurs, given that we know A has already occurred.
• prosecutor’s fallacy: A fallacy of statistical reasoning when used as an argument in legal proceedings.

The People of the State of California v. Collins was a 1968 jury trial in California. It made notorious forensic use of statistics and probability. Bystanders to a robbery in Los Angeles testified that the perpetrators had been a black male, with a beard and moustache, and a caucasian female with blonde hair tied in a ponytail. They had escaped in a yellow motor car.

The prosecutor called upon for testimony an instructor in mathematics from a local state college. The instructor explained the multiplication rule to the jury, but failed to give weight to independence, or the difference between conditional and unconditional probabilities. The prosecutor then suggested that the jury would be safe in estimating the following probabilities:

• Black man with beard: 1 in 10
• Man with moustache: 1 in 4
• White woman with pony tail: 1 in 10
• White woman with blonde hair: 1 in 3
• Yellow motor car: 1 in 10
• Interracial couple in car: 1 in 1000

These probabilities, when considered together, result in a 1 in 12,000,000 chance that any other couple with similar characteristics had committed the crime – according to the prosecutor, that is. The jury returned a verdict of guilty.

As seen in, upon appeal, the Supreme Court of California set aside the conviction, criticizing the statistical reasoning and disallowing the way the decision was put to the jury. In their judgment, the justices observed that mathematics:

The Collins Case: The Collins case is a classic example of the prosecutor’s fallacy. The guilty verdict was reversed upon appeal to the Supreme Court of California in 1968.

… while assisting the trier of fact in the search of truth, must not cast a spell over him.

### Prosecutor’s Fallacy

The Collins’ case is a prime example of a phenomenon known as the prosecutor’s fallacy—a fallacy of statistical reasoning when used as an argument in legal proceedings. At its heart, the fallacy involves assuming that the prior probability of a random match is equal to the probability that the defendant is innocent. For example, if a perpetrator is known to have the same blood type as a defendant (and 10% of the population share that blood type), to argue solely on that basis that the probability of the defendant being guilty is 90% makes the prosecutors’s fallacy (in a very simple form).

The basic fallacy results from misunderstanding conditional probability, and neglecting the prior odds of a defendant being guilty before that evidence was introduced. When a prosecutor has collected some evidence (for instance, a DNA match) and has an expert testify that the probability of finding this evidence if the accused were innocent is tiny, the fallacy occurs if it is concluded that the probability of the accused being innocent must be comparably tiny. If the DNA match is used to confirm guilt that is otherwise suspected, then it is indeed strong evidence. However, if the DNA evidence is the sole evidence against the accused, and the accused was picked out of a large database of DNA profiles, then the odds of the match being made at random may be reduced. Therefore, it is less damaging to the defendant. The odds in this scenario do not relate to the odds of being guilty; they relate to the odds of being picked at random.