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# How many commutative rings with exactly one non-zero zero divisor are there?

I recently rememebered the following theorem by Ganesan:

Let $R$ be a commutative ring with $0<n<\infty$ non-zero zero divisors. Then $\operatorname{card}(R)\leq(n+1)^2.$

The proof proceeds by considering the annihilator of one of the zero divisors. Let $z_1,\ldots,z_n$ be the non-zero zero divisors of $R.$ Let $A_1=\{x\in R\,|\,xz_1=0\}.$ It is a well-known fact that this is an ideal in $R.$ We also have $\operatorname{card}(A_1)<n+1$ because $z_1\neq 0.$

Now let's pick one representative $r_x$ from each coset $x\in R/A_1.$ Consider the map $$x\mapsto r_xz_1.$$ Suppose $r_xz_1=r_yz_1.$ Then $(r_x-r_y)z_1=0,$ so $(r_x-r_y)\in A_1,$ which means $x=y.$ Therefore the map is injective. But for any $r\in R,$ the element $rz_i$ is a (possibly zero) zero-divisor. It follows that there are at most $n+1$ cosets in $R/A_1$ and since each coset has cardinality equal to $\operatorname{card}(A_1)\leq n+1,$ we have $\operatorname{card}(R) \leq (n+1)^2.\square$

So we have that if $R$ has exactly one non-zero zero divisor $x,$ then $R$ has at most $4$ elements. I thought it would be a cool thing to determine which rings (with or without unity, since the theorem doesn't require its existence) have this property. It is possible to do this by checking all rngs with at most $4$ elements. But I found that I can't determine all non-isomorphic commutative rngs with at most $4$ elements. Wikipedia says there are $11$ rngs with four elements counting non-commutative ones. There can't be many commutative ones then, but checking all possible operation tables to find them is tedious since I don't know what the $11$ rings are.

Is there a nice way to do this? The condition that there is exactly one-nonzero zero divisor seems pretty strong and I suspect the only such rng is $\mathbb Z/2\mathbb Z$ with zero multiplication, but this is only a suspicion. I've checked that there are no such rngs with $3$ elements, but what about the four-element ones?

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Do you restrict to unitary rings or is the existence of a multiplicative identity not required? –  Jesko Hüttenhain Mar 28 '12 at 14:58
It's not. But if you have an argument about unitary rings, I would be glad to see it too. –  user23211 Mar 28 '12 at 15:01

Note: The above comments were posted while I was writing this answer - I've assumed $R$ is unitary. In particular, my comment about $\mathbb{Z}/2\mathbb{Z}$ with the zero multiplaction not being a ring no longer stands ("ring" to me includes unitary).

If $|R|=4$, then as the characteristic of $R$ divides $4$, it must be either $1$, $2$ or $4$. If the characteristic is $1$, you have the zero ring, so that's not an example. If the characteristic is $4$, then you have $\mathbb{Z}/4\mathbb{Z}$, which has exactly one non-zero zero divisor.

In the case that the characteristic is $2$, we can write the elements as $0$, $1$, $a$ and $1+a$; if $1+a=0$ then $a=-1=1$, if $1+a=1$ then $a=0$ and if $1+a=a$ then $0=1$, so $1+a$ is different from the first three. Then (as we're assuming $R$ is commutative), the multiplication table is completely determined by the choice of $a^2$.

There are four possible choices, although $a^2=0$ and $a^2=1$ give isomorphic rings by swapping $a$ and $a+1$, so there are three different possibilities. Only the $a^2=0$ choice gives a ring with exactly one non-zero zero divisor (namely $a$). This ring is isomorphic to $\mathbb{F}_2(x)/(x^2)$.

Note that you can't have the zero multiplication on $\mathbb{Z}/2\mathbb{Z}$ and still get a ring, as then you don't have an identity element, so that isn't in fact an example. As the zero ring isn't an example, $\mathbb{Z}/2\mathbb{Z}$ (the only ring of two elements) isn't an example, and as you've checked the ring of $3$ elements isn't an example either, the two order four examples are the only ones.

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$\mathbb{Z}/4\mathbb{Z}$ has only 1 nonzero zero divisor. –  Brandon Carter Mar 28 '12 at 15:19
Yes, you're right. Edited accordingly. –  Matthew Pressland Mar 28 '12 at 16:08
Thank you for the answer. I still have some hope that perhaps someone might come up with the solution for non-unitary rings so will wait a bit with accepting. –  user23211 Mar 29 '12 at 13:24
Sure, that makes sense. I feel somehow that the answer for non-unitary rings could be arrived at along the same lines, although the case analysis will be more unpleasant, and I don't really have the time or the inclination to work through it. If somebody has a really slick answer I'd be interested to see it though. –  Matthew Pressland Mar 29 '12 at 14:25


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Lemma 79.12.3. Denote the common underlying category of $\mathit{Sch}_{fppf}$ and $\mathit{Sch}_{\acute{e}tale}$ by $\mathit{Sch}_\alpha$ (see Topologies, Remark 34.11.1). Let $S$ be an object of $\mathit{Sch}_\alpha$.

$F : (\mathit{Sch}_\alpha /S)^{opp} \longrightarrow \textit{Sets}$

be a presheaf with the following properties:

1. $F$ is a sheaf for the étale topology,

2. the diagonal $\Delta : F \to F \times F$ is representable by algebraic spaces, and

3. there exists $U \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha /S)$ and $U \to F$ which is surjective and smooth.

Then $F$ is an algebraic space in the sense of Algebraic Spaces, Definition 64.6.1.

Proof. The proof mirrors the proof of Lemma 79.12.1. Let $R = U \times _ F U$. By (2) the presheaf $R$ is an algebraic space and by (3) the projections $R \to U$ are smooth and surjective. Denote $(U, R, s, t, c)$ the groupoid associated to the equivalence relation $j : R \to U \times _ S U$ (see Groupoids in Spaces, Lemma 77.11.3). By Theorem 79.10.1 we see that $X = U/R$ (quotient in the fppf-topology) is an algebraic space. Using that the smooth topology and the étale topology have the same sheaves (by More on Morphisms, Lemma 37.38.7) we see the map $U \to F$ identifies $F$ as the quotient of $U$ by $R$ for the smooth topology (details omitted). Thus we have morphisms (transformations of functors)

$U \to F \to X.$

By Lemma 79.11.6 we see that $U \to X$ is surjective, flat and locally of finite presentation. By Groupoids in Spaces, Lemma 77.19.5 (and the fact that $j$ is a monomorphism) we have $R = U \times _ X U$. By Descent on Spaces, Lemma 73.11.26 we conclude that $U \to X$ is smooth and surjective (as the projections $R \to U$ are smooth and surjective and $\{ U \to X\}$ is an fppf covering). Hence for any scheme $T$ and morphism $T \to X$ the fibre product $T \times _ X U$ is an algebraic space surjective and smooth over $T$. Choose a scheme $V$ and a surjective étale morphism $V \to T \times _ X U$. Then $\{ V \to T\}$ is a smooth covering such that $V \to T \to X$ lifts to a morphism $V \to U$. This proves that $U \to X$ is surjective as a map of sheaves in the smooth topology. It follows that $F \to X$ is surjective as a map of sheaves in the smooth topology. On the other hand, the map $F \to X$ is injective (as a map of presheaves) since $R = U \times _ X U$. It follows that $F \to X$ is an isomorphism of smooth ($=$ étale) sheaves, see Sites, Lemma 7.11.2 which concludes the proof. $\square$

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Question

# Why $$5\times 6\times 7\times 11-2\times 3\times 7\times 11$$ is composite number.

Solution

## Multiplication of those number is equal to 1848. Where 1848 comes on the table of 2,3,4...therefore $$5×6×7×11-2×3×7×11$$ is composite numberMathematics

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Examples of mathematical results discovered “late”

What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple that it's surprising no one thought of it sooner.

The example that makes me ask is the 2011 paper John Baez mentioned called "Two semicircles fill half a circle", which proves a fairly simple geometrical fact similar to those that have been pondered for thousands of years.

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I'm likely off, but isn't the posted problem quickly solved by use of coordinate geometry? –  Draksis Mar 4 at 23:00
It took a surprisingly long time (17th century) for even very basic concepts of probability theory to be developed, considering that those concepts would have been immensely valuable in real life, given that gambling has been popular forever. –  dfan May 1 at 21:57

This proof of the irrationality of $\sqrt 2$ appears to have been discovered in 1892 by A.P. Kiselev:

If $\sqrt2$ is rational, let $\triangle ABO$ be the smallest possible isosceles right triangle whose sides are integers.

Construct $CD$ perpendicular to $AO$ with $AC = AB$. $\triangle OCD$ is another isosceles right triangle.

$AC=AB$, therefore $AC$ is an integer, therefore $OC = OA - AC$ is an integer. $\triangle OCD$ is isosceles, so $OC=CD$ and CD is an integer. $CD$ and $BD$ are equal because they are tangent to the same circle, so $BD$ and $OD = BO-BD$ are integers. But then $\triangle OCD$ is an isosceles right triangle with integer sides, contradicting the assumption that $\triangle ABO$ was the smallest such.

I am amazed that this wasn't found by the Greeks, because it is so much more in their style than the proof that they did find.

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It is really amazing that the Greeks should not have found this argument while they knew that $\sqrt2$ is irrational. Since that implies that Euclid's algorithm applied to lengths $\sqrt2:1$ cannot terminate (and find a $\gcd$) in a finite number of steps. In this particular case that is so because the ratio of the remaining lengths is periodic with period $2$. To wit, subtract the shorter AB from the longer OA in OA:AB to get AB:OC; then again subtract OC=CD=BC from AB=OB to get OC:OD, which as the figure shows is the same ratio as OA:AB, so one gets periodicity, and irrationality. –  Marc van Leeuwen Mar 4 at 18:50
Its still a wonderful proof! Wish it was more common. –  Sawarnik Mar 4 at 19:08
The Greeks must have known the same proof of irrationality for the golden ratio, where the construction is built into the definition. I would be surprised if the proof for sqrt(2) were first discovered by Kiselyev or not noted by one of the ancient Greek mathematicians. –  zyx Mar 4 at 21:49
@zyx I would be delighted to see a citation. I first encountered this proof in Thomas Apostol "Irrationality of the Square Root of Two - A Geometric Proof" American Mathematical Monthly Nov 2000 841–842. Apparently neither Apostol nor the reviewers at the Monthly were aware of either its earlier publication by Kiselev, or any hypothetical earlier publication. Of course, there is plenty of Greek mathematics that is completely lost, and among it there are no doubt some results that would be much more surprising than this one. –  MJD Mar 4 at 21:52
@MJD Almost surely this proof is much older. There are many variants on such geometrical forms of the standard descent proofs and many of them are very old. I expect the same is true for this variant. –  Bill Dubuque Mar 5 at 2:39

This is not a theorem but a result which was discovered "too late".

The number $e$ hiding inside Pascal's triangle was found it by Harlan Brothers in 2012, in a very simple way!

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John Baez has a post explaining why this works. If we let $u_n = s_n/s_{n+1}$, then it's easy to see that $u_{n-1} = n!/n^n$, so $$\frac{s_{n-1}s_{n+1}}{s_n^2} = \frac{u_{n-1}}{u_n} = \frac{n!}{n^n}\frac{(n+1)^{n+1}}{(n+1)!} = \left(1+\frac1n\right)^n \to e$$ as $n \to \infty$. –  ShreevatsaR Mar 6 at 9:21
Personally, I don't think that this result is singular or imperative enough that it should have been discovered much earlier. –  azimut Mar 10 at 12:25
For what it's worth, Harlan Brothers actually found this no later than November 2009, as that is when he added this piece of information to the OEIS. –  Nathaniel Johnston Mar 30 at 3:09

An example mentioned for Martin Gardner (Martin Gardner's New Mathematical Diversions from Scientific American): Morley's trisector theorem is elementary enough to be proved two millenia ago, but unknown until 1899.

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I think the use of trisectors was problematic for the Greeks, because they cannot be constructed using a compass and a ruler. –  sds Mar 5 at 19:56
@sds: You're absolutely right. Still, between ancient Greece and 1899 lies some room... –  Pete L. Clark Mar 5 at 21:43
And yet the first proof was heavily complicated, I think there was some time before easier geometric proofs were found. –  Sawarnik Mar 7 at 4:00

Apéry's 1975 proof of the irrationality of of $\zeta(3)=\sum_{n=1}^\infty\frac1{n^3}$ took the experts by surprise, and uses tools that had been available for a very long time.

See A Proof that Euler Missed for more details.

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A Dandelin sphere of a conic section will touch the plane of the conic section at a focus of the conic section. This pretty and useful fact was discovered only in 1822, but there is no reason why it could not have been in the treaties of Apollonius of Perga.

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This is not a theorem but a result as you require that was discovered "late" .

As Erdős mentions :

"At that time,mathematicians of the seventeenth and eighteenth centuries found pairs of odd amicable numbers such as $(12.285,14.595)$.
Curiosly, however,the smallest pair after the one known from antiquity $(1184,1210)$ was found only in 1866 by a 16-year old student Niccolo Paganini".

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What was "that time", for comparison? –  Paŭlo Ebermann Mar 5 at 21:44
@PaŭloEbermann "that time" was "the modern renaissance of mathematics" –  Konstantinos Gaitanas Mar 6 at 10:35
When I read this, the first thing that came to mind was "the famous violinist?" –  qwr Mar 8 at 7:55
@qwr I wondered also if it was Niccolò Paganini the famous violinist, but I checked, and it was a different person with the same name. The violinist died in 1840. –  MJD Mar 29 at 20:44

Informally, the historical definition of an Archimedean solid requires that all its faces are regular polygons and that each vertex of the solid locally looks the same. Since the ancient Greeks it was common knowledge that up to scaling, rotating and reflecting, there are 13 types of Archimedean solids.

It was only in the 20th century that it was noticed that there is a 14th solid which complies with the above definition, the Elongated Square Gyrobicupola. It arises from the Rhombicuboctahedron (one of the 13 established Archimedean solids) by twisting one of its octagonal "caps" by 45 degrees. However, those two solids are not identical.

Since this solid doesn't look as symmetric as the "proper" Archimedean solids (its symmetry group doesn't act regularly on its vertices), nowadays the definition typically is sharpened.

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This reminds me of another example: the Johnson solids, which are polyhedra with regular polygonal faces, were first enumerated by Norman Johnson in 1966, although there was nothing stopping their enumeration by Kepler, Archimedes, or anyone else in the past 2,500 years or so. –  MJD Mar 11 at 0:12
Another similar example: There are exactly eight convex polyhedra whose faces are equilateral triangles. Several, such as the icosahedron, have been known since antiquity. But as far as I know the one with twelve faces is first mentioned in a paper of Freudenthal and Van der Waerden from 1947. –  MJD Mar 12 at 16:04
I find it bizarre that it took mathematicians something like 400 years to realize that they could just rotate one of the "caps" of the rhombicuboctahedron by 45 degrees to obtain the elongated square gyrobicupola, thus not changing the size of any of the faces. –  Vortico Mar 30 at 11:53

The theorem $$\text{PRIMES is in P}$$ was proven only in 2002 (via the AKS primality test). The algorithm certainly isn't rocket science, making it quite surprising that this wasn't found way earlier. The point is that it was the common believe that PRIMES is not in P, so there was not a big search for such an algorithm.

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The answer by Marco13 already mentions this. Although it contains two answers which makes voting difficult… –  MvG Mar 7 at 18:28
I think it is a misconception that "it was the common belief that PRIMES is not in P". It was actually known in the 1980s that if the Generalized Riemann Hypothesis is true then PRIMES is in P (via the not-as-well-known Ankeny's algorithm). Since many people believe GRH, it was believed that PRIMES is in P. The new result of AKS in 2002 is a proof that doesn't depend on GRH. –  Yoni Rozenshein Mar 30 at 13:34
Looked up what I wrote in the previous comment, I made a mistake. It's not called "Ankeny's algorithm", it's simply the Euler-Jacobi test. Ankeny's theorem (1950s) guarantees (conditional GRH) that if $n$ is composite then there is a base $a < 2(\log n)^2$ for which $n$ is not an Euler-Jacobi pseudoprime, which yields a trivial polynomial time algorithm (check all bases $a$ up to $2(\log n)^2$...) –  Yoni Rozenshein Mar 30 at 15:23
@YoniRozenshein: Thanks for your interesting remark. So maybe its more precise to write that nobody expected the existence of an easy explicit algorithm, without any mystery like Riemann... –  azimut Mar 31 at 15:36
Yes, this was a big surprise :) I don't know if it was more surprising that it was independent of RH than that it was elementary. –  Yoni Rozenshein Mar 31 at 19:29

I'd say the fact that zero took until sometime after the 600s before it was used in math takes the cake here.

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Using zero does not qualify as a result that was discovered late. –  Marc van Leeuwen Mar 4 at 20:00
And it's not the mathematical concept of zero, that was adopted late, but the written notation for zero in a place-value system. But its late adoption in Europe and elsewhere is not so surprising, because calculation was not done with paper (scarce and expensive) but with abacuses and counting boards. And on an abacus there is a notation for zero. So when you look carefully at just what it was that was adopted late, you find that it is a very specific technical invention that had has much more to do with engineering than with mathematics. –  MJD Mar 4 at 21:19
In line with @MJD 's comment, it's not easy to come up with a situation, prior to positional notation, where having zero or "nothing" as a number makes a big difference, or where treating positive and (what we would now call) negative numbers as a single unified system is critical. Accounting works fine with debit and credit as separate columns. –  zyx Mar 5 at 0:37
@MJD: Actually the mathematical concept of zero as a number ("how can nothing be a number?") was also very late in being adopted. AFAIK no one in the Western world (so discounting the abacus) did arithmetic with zero as a number ("adding zero to five gives five", etc.) for a long time. Even when Fibonacci introduced the place-value system in his Liber Abaci, he wrote "the nine digits of the Hindus" and "these nine figures, and with the sign $0$", instead of "the ten digits" as we might. And you can Google for "is zero a number" to see many people who have trouble with the idea even now. :-) –  ShreevatsaR Mar 6 at 8:36
But why would that be such a big discovery without some form of positional notation? People understood that income and expenses cancel each other, and that one can travel the same distance in two directions (and how to compose trips). Creating a $0$ or minus sign does not seem as revolutionary without the relation to multiplication, or the ability to process larger numbers by positional notation. @ShreevatsaR –  zyx Mar 6 at 20:05

One possible answer would be the discovery of Strassen algorithm in 1969. This is a way to multiply large matrices together in (slightly) faster than standard time.

Although it may not be the fastest algorithm for matrix multiplication the majority of faster variants stem from this one idea.

It may not be a simple proof but it is very late.

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The whole field of computational complexity could have started earlier by at least a few decades. –  zyx Mar 5 at 5:34
in contrast, I seem to remember that the FFT method for multiplication of large integers was first discovered by Gauss. –  MJD Mar 5 at 20:20
@MJD was the FFT known in those days? Because with a naive DFT, using fourier transforms for multiplication is no better than the standard algorithm. –  Tim Seguine Mar 6 at 10:31
@Tim Acording to Heideman, Johnson, and Burrus “Gauss and the History of the Fast Fourier TransformArchive for History of Exact Sciences 34 #3 pp. 265–277 “This [Gauss’ algorithm] is also exactly the FFT algorithm derived by Cooley and Tukey in 1965 … Gauss’ algorithm is as general and powerful as the Cooley-Tukey common-factor algorithm”. –  MJD Mar 12 at 16:23
@MJD I am aware of that; that is not the problem. The problem is that the standard "naive" DFT is $O(n^2)$, but the FFT is $O(n\log{n})$. Since standard multiplication is also $O(n^2)$, this means there is no algorithmic benefit of utilizing the Gauss algorithm for multiplication (it is not faster). Thus, the question of whether the FFT was known at the time would be extremely relevant. The fact that it can be used for multiplication is fairly trivial (the convolution theorem is a basic result which of course Gauss would have known about) –  Tim Seguine Mar 12 at 17:23

Another result that was discovered late is the construction of the Heptadecagon (regular seventeen-sided polygon) by compass and unmarked straightedge, found by Gauß in 1796.

Wikipedia describes it as "...the first progress in regular polygon construction in over 2000 years"

Remarkable enough, considering that is a purely geometrical result, and the Greek already had everything that was necessary for achieving it in their toolbox.

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Generally the JSJ-decomposition of 3-manifolds is considered to be discovered "late" (by $40$ years, but that is a long time in modern maths!).

From Allen Hatcher's Notes on Basic 3-Manifold Topology,

Beyond the prime decomposition, there is a further canonical decomposition of irreducible compact orientable 3 manifolds, splitting along tori rather than spheres [the JSJ-decomposition]. This was discovered only in the mid 1970’s, by Johannson and Jaco-Shalen, though in the simplified geometric version...it could well have been proved in the 1930’s. (A 1967 paper of Waldhausen comes very close to this geometric version.) Perhaps the explanation for this late discovery lies in the subtlety of the uniqueness statement. There are counterexamples to a naive uniqueness statement, involving a class of manifolds studied extensively by Seifert in the 1930’s. The crucial observation, not made until the 1970’s, was that these Seifert manifolds give rise to the only counterexamples.

For context, JSJ-decompositions are the starting point for Perelman's proof of Geometrisation, which implies the Poincare conjecture. His subsequent shunning of a fields medal and $1m prize is oft-told... - add comment Theorem$\big($Sylvester-Gallai$\big):$For any$n$points in$\mathbb{R}^2$, not all collinear, there exists a line passing through exactly two of them.$\rm{Proof}:$Pick any$2$points and draw a line$\ell$through them: suppose a$3^{\rm{rd}}$point lies on the line$($else we are done$)$and pick the closest point$p\not\in\ell$to the line, at a distance$\delta$say. Of our$3$points$\in\ell$a pair lies on one side of$p:$draw a line$\ell'$through$p$and the furthest of the pair from$p$. The distance$\delta'$between$\ell'$and the second point is$<\delta$.$\qquad\qquad\qquad\quad$The statement was proposed first by Sylvester in$1893$and independently by Erdős in$1943\;($rather surprisingly, Erdős could not find a proof$)$. It was proven the following year by Gallai.$\big[\mathbf{Note:}$the statement can fail in other fields: for instance, in$\mathbb{P}^2(\mathbb{C})$see the Hesse configuration$\big]$- ... So if a third point lies in$l'$, repeat to get two points line joining which has the closest point at even lower non-zero$\delta''$. Since there are finitely many lines with 2 points from the set, this has to stop somewhere, there are two points which have no other point on the line joining them. Incredible proof. – KalEl Mar 30 at 5:20 @KalEl Precisely! – Z Z Mar 30 at 10:42 add comment Pascal's theorem was discovered in 1639 or 1640, but its projective dual, Brianchon's theorem, was only discovered in 1806. So it took about 167 years despite the fact that Pascal already realized the projective nature of his theorem. - add comment It took 2000 years for anyone to recognize that Euclid's Parallel Postulate didn't follow from his other four axioms. It has always been surprising to me that people tried for so long to prove the parallel postulate followed from the other axioms and no one seemed to consider the possibility that it was independent of them. The ultimate realization that there exist models of geometry in which the converse holds has blossomed into an enormous area of mathematics rife with deep and interesting results. - Why should this have been discovered earlier? Spherical geometry was known, but (bidirectionally) interpreting Euclidean geometry in spherical or hyperbolic geometry is complicated, and I would imagine some accumulated experience with spherical trigonometry and calculus was important. Saying "curved surfaces exist and have geodesics as the equivalent of lines" doesn't resolve the issue with the parallel axiom. That several people discovered it at the same time suggests that the effectively necessary concepts were not available much earlier than 1800. – zyx Mar 6 at 20:00 @zyx Riemannian geometry isn't necessary to develop a model of the hyperbolic plane. – IBWiglin Mar 7 at 16:12 Something that is neccessary is a clear understanding that the axioms refer to terms with no further meaning than what the axioms say about them. As long as you have Euclidean intuition about what the term “line” refers to, you have trouble imagining anything satisfying the axioms where that role is played by something you usually wouldn't call a line. I'd associate that level of abstraction with Hilbert, though it might be others should be named here as well or even instead. – MvG Mar 7 at 18:18 @MvG I understand your point of view, but counterintuitive results are abundant in mathematics. It seems to me that the story is always that [insert geometer here] went about trying to prove Euclid's parallel postulate followed from the others using a proof by contradiction. [Geometer] proved a bunch of theorems (for example about quadrilaterals), but never found a contradiction. Then he gave up or assumed something was wrong because the theorems were counter-intuitive. It seems like people just discarded the results off-hand because they seemed odd. – IBWiglin Mar 7 at 18:59 add comment The Mason-Stothers theorem (aka the$abc$conjecture for polynomials): if$f(t)$,$g(t)$, and$h(t)$are nonzero polynomials over a field satisfying$f(t) + g(t) = h(t)$with$f(t)$and$g(t)$being relatively prime and the three polynomials are not all constant, then $$\max(\deg f, \deg g, \deg h) \leq \deg({\rm rad}(fgh)) - 1,$$ where${\rm rad}(F)$for a nonzero polynomial$F$is the product of its irreducible factors (set it to be$1$if$F$is constant). Strictly speaking, the version I wrote is for a field of characteristic$0$. In characteristic$p$, the condition that at least one of the polynomials is nonconstant should be replaced with at least one of them having nonzero derivative. This theorem was proved only in the 1980s, independently, by Mason and Stothers. Its proof uses just some elementary calculations with derivatives of polynomials. There was a proof given by Silverman that shows this result is, to use the OP's term, a "straightforward corollary" of the Riemann-Hurwitz formula, so in principle the theorem could have been formulated in the 19th century, but the simple statement of this theorem did not appear until Mason and Stothers as far as I'm aware. On somewhat the same theme, another example of the type being requested is Belyi's theorem. It says that for a smooth projective algebraic curve$C$over the complex numbers, if$C$is defined over the algebraic numbers then$C$admits a covering of the Riemann sphere ramified over at most three points. (The converse was known earlier.) I say this is somewhat the same theme because Belyi's theorem is closely related to when the inequality of the Mason-Stothers theorem is an equality. Like the Mason-Stothers theorem, the proof of Belyi's theorem is surprisingly low-tech compared to what anyone would imagine when hearing the statement of the theorem. - add comment The first thing that came to my mind was the Kepler conjecture. While it's not a straightforward or simple proof, the fact that fruitsellers stacked their apples optimally for thousands of years until it was proven that they did it is somewhat surprising. - @azimut: Sorry, I wasn't aware of the fact that individual answers were preferred in such a case. I've split them up now. – Marco13 Mar 8 at 13:56 I don't think this one qualifies for the question. Which kind of rigorous proof we should have expected much earlier? – azimut Mar 8 at 19:55 Thanks for splitting your answer! – azimut Mar 8 at 20:10 @azimut Strictly referring to the question (asking for "simple" results), this is certainly true. Maybe the surprising thing here is that the fact itself is so simple (in contrast to the proof). But admittedly, this would apply to many other open problems (e.g. Goldbach)... and BTW: most results that have been mentioned so far do not really look "simple" for me (I'm not a mathematician). However, I wonder when things like Goldbach and Riemann will be added as answers here ;-) abstrusegoose.com/210 – Marco13 Mar 8 at 20:44 add comment Brouwer's demonstrations of the difficulties with classical logic and the excluded-middle principle (tertium non datur), which had been used in mathematics and logic for more than 2000 years. The lateness of discovery was due to the increasing frequency and complexity of infinitary constructions in 19th-century analysis (and the relatively more concrete arguments before that time), and the concomitant pressure to develop precise and consistent ways of handling them. The extension of principles like Excluded Middle from finite to infinite situations is where the problems isolated by Brouwer most clearly appear. - Considering that classical logic, including tertium non datur, is alive and kicking, if space allows it, it would IMO be good to present at least one of Brouwer's points in some detail. – Daniel Fischer Mar 5 at 10:29 Your point is not clear. Adding the law of excluded middle cannot generate (additional) contradictions, so there is no possibility of an internal foundational crisis specific to classical logic. On the other hand, many classical proofs work as-is or with insignificant modification without nonconstructive methods, so that no matter which approach one pursues, there is always a growing body of mathematical knowledge exempt from Brouwer's criticism, and one could argue that this (and political inertia) is what keeps things alive and kicking even if things are somehow "wrong" in the framework. – zyx Mar 7 at 4:29 "Can you give an example of the difficulties Brouwer demonstrated?" Since classical logic is still the predominant logic used, I think many people don't know Brouwer's criticism, and it seems the community in the large didn't find it convincing. Having a point explained would allow tentatively judging it by oneself. – Daniel Fischer Mar 7 at 11:18 add comment Grothendieck said that he found it embarassing that it took humanity so long to define a group. - add comment I got three of them for you. Trisecting the angle, doubling the cube, and squaring the circle. They were three famous geometric problems from ancient Greece. They were concerned about whether or not these could be done with compass and straightedge. They all ended up being impossible, but it wasn't proven for 2000 years. Trisecting the angle was disproved in 1837 using Galois theory Doubling the cube was proved impossible in 1839 by showing that$\sqrt[3] 2$is not a constructable number. Squaring the circle was proved impossible in 1882 by showing that$\pi$is transcendental - The tools to prove, and even to state, those results were not available until the era in which the proof was accomplished. The first two results are nearly trivial given the idea of degree of a field extension, but this was an unknown concept before approximately 1800. The proofs that pi is irrational and transcendental used ideas of good rational and polynomial approximation that were unheard-of before those breakthroughs, which gave birth to transcendental number theory. – zyx Mar 5 at 5:28 Yeah but I still think it fitted nicely with the question. – sicklybeans Mar 5 at 5:31 It's also a brilliant example of seemingly unrelated fields of math coming together – sicklybeans Mar 5 at 5:31 @sicklybeans: While this result and the underlying technique is certainly very nice, I agree with zyx that it doesn't qualify for the question. The proofs were not found late, but they were just in pace with the mathematical development. – azimut Mar 7 at 16:00 add comment I'm not sure this is a real answer, but Wikipedia claims that the Möbius strip was invented in 1858, by Möbius. I find it incredible that there is any property of the strip that was still undiscovered by 1858, so to whatever fraction of Wikipedia's claim is true I think qualifies as a surprisingly late discovery. - Could you be a bit more specific? which properties exactly? – Tim Seguine Mar 13 at 10:05 add comment Smale's paradox: in topology, the fact a sphere in R3 can be turned completely inside out without tears or sharp creases (self-intersection is allowed) was only indirectly proved in 1957 and an actual example of the process wasn't found until 1961. See http://en.wikipedia.org/wiki/Smale%27s_paradox and http://www.geom.uiuc.edu/docs/outreach/oi/history.html - add comment How about the Jordan curve theorem? It seems obvious at first sight, but it is quite complicated to prove. - This isn't exactly what the question is asking for. Just because some statement is "obvious", it doesn't necessarily imply that its rigorous proof should have been found early. In this case, the only known proofs are quite advanced and it is hard to imagine that someone should have come up with it much earlier. – azimut Mar 7 at 15:51 add comment Another result that seems to have appeared late is the AKS primality test ( http://en.wikipedia.org/wiki/AKS_primality_test ). Although the field of computational compexity is relatively new, one could have expected that a proof for the existance of a deterministic polynomial-time primality-proving algorithm should have appeared earlier - given the fact that some of the most intelligent people in the world have studied prime numbers intensively for thousands of years. - Given what you mentioned that computational complexity is relatively new, I find the thousands of years comment slightly irrelevant. – Tim Seguine Mar 12 at 18:27 @TimSeguine I just wanted to point out that people have wondered "Is this a prime number or not?" for quite a while, and it is a very important question, and ... yet they did not find an efficient (deterministic, polynomial time) method for answering this question. (BTW: As you can see here, I don't hesitate to write irrelevant comments occasionally ;-) ) – Marco13 Mar 12 at 21:44 I was just meaning that it doesn't seem very surprising that people didn't think that much about algorithmic complexity before it was a thing. The traditional method of primality testing was probably just considered good enough (it worked). – Tim Seguine Mar 12 at 21:56 @TimSeguine They certainly did not try to find out whether "PRIMES is in P", but nevertheless, one could imagine that the ability to quickly check whether a number is prime or composite could have been of some interest. (I'm not a mathematician, so apologies if this sounds naive) – Marco13 Mar 12 at 22:05 add comment The binary GCD algorithm was only discovered (re-discovered?) in 1967. - add comment The Futurama Theorem: Regardless of how many mind switches between two bodies have been made, they can still all be restored to their original bodies using only two extra people, provided these two people have not had any mind switches prior (assuming two people cannot switch minds back with each other after their original switch). - add comment The Deduction Meta-Theorem for slews of logical calculi, and slews of conditional proofs in ordinary mathematics which can get said to have an interaction with it. - add comment James A. Garfield (20th President of the United States) found 1875 (~2175 years after Euclid) a quite simple, yet clever trapezoid proof of the Pythagorean theorem. Here is a video explaining it: https://www.youtube.com/watch?v=EINpkcphsPQ - add comment The one that bumps up in my mind is Erdos-Mordell inequality. It states that for a point$ p $in a triangle$ ABC $, with$\alpha ,\beta,\gamma $denoting the distance of$ P $to the 3 sides,respectively, then$ AP+BP+CP\geq 2 (\alpha + \beta +\gamma)$This wasn't thought of until Erdos conjectured it in 1935. P.s.People think Erdos might have been inspired by Euler's theorem for triangles, which itself is a 'shame' of Euclid. P.s.2 I can't easily type the proper Hungarian letter on my phone, so sorry. - add comment An example of this that jumps to my mind is Fermats last theorem, for which was conjectured by him in 1637 but only proved in 1995 despite many years work by countless mathematicians. But this proof (obviously) is not simple nor straightforward. I don't know how important that criteria is on our answers, but it is an example of a result that many believed to be true for a very long time, yet to be unable to be proved. Fermats last theorem is that, For a,b and c being positive intergers$a^n+b^n\neq c^n\$ For n bigger than 2

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The proof is not straightforward, which is the importance condition in the question. If you remove this criteria then many more examples come to mind, like the impossibility of squaring the circle. –  Sawarnik Mar 4 at 18:54
I don't think this satisfies the primary criterion, which is "surprisingly late". The theorem turned out to be surprisingly difficult, but there is no way that the proof we have could have been discovered by, say, Euler. –  MJD Mar 4 at 19:07
Fermat allegedly had a proof, that was too big for him to put in the margin. It would have been much simpler than Wiles' proof. –  M. Knight Mar 5 at 12:47
en.wikipedia.org/wiki/… –  oakad Mar 6 at 5:50


text:
# 1.1: Likelihood

## Introduction

Probability models and techniques permeate many important areas of modern life. A variety of types of random processes, reliability models and techniques, and statistical considerations in experimental work play a significant role in engineering and the physical sciences. The solutions of management decision problems use as aids decision analysis, waiting line theory, inventory theory, time series, cost analysis under uncertainty — all rooted in applied probability theory. Methods of statistical analysis employ probability analysis as an underlying discipline.

Modern probability developments are increasingly sophisticated mathematically. To utilize these, the practitioner needs a sound conceptual basis which, fortunately, can be attained at a moderate level of mathematical sophistication. There is need to develop a feel for the structure of the underlying mathematical model, for the role of various types of assumptions, and for the principal strategies of problem formulation and solution.

Probability has roots that extend far back into antiquity. The notion of “chance” played a central role in the ubiquitous practice of gambling. But chance acts were often related to magic or religion. For example, there are numerous instances in the Hebrew Bible in which decisions were made “by lot” or some other chance mechanism, with the understanding that the outcome was determined by the will of God. In the New Testament, the book of Acts describes the selection of a successor to Judas Iscariot as one of “the Twelve.” Two names, Joseph Barsabbas and Matthias, were put forward. The group prayed, then drew lots, which fell on Matthias.

Early developments of probability as a mathematical discipline, freeing it from its religious and magical overtones, came as a response to questions about games of chance played repeatedly. The mathematical formulation owes much to the work of Pierre de Fermat and Blaise Pascal in the seventeenth century. The game is described in terms of a well defined trial (a play); the result of any trial is one of a specific set of distinguishable outcomes. Although the result of any play is not predictable, certain “statistical regularities” of results are observed. The possible results are described in ways that make each result seem equally likely. If there are N such possible “equally likely” results, each is assigned a probability 1/N.

The developers of mathematical probability also took cues from early work on the analysis of statistical data. The pioneering work of John Graunt in the seventeenth century was directed to the study of “vital statistics,” such as records of births, deaths, and various diseases. Graunt determined the fractions of people in London who died from various diseases during a period in the early seventeenth century. Some thirty years later, in 1693, Edmond Halley (for whom the comet is named) published the first life insurance tables. To apply these results, one considers the selection of a member of the population on a chance basis. One then assigns the probability that such a person will have a given disease. The trial here is the selection of a person, but the interest is in certain characteristics. We may speak of the event that the person selected will die of a certain disease– say “consumption.” Although it is a person who is selected, it is death from consumption which is of interest. Out of this statistical formulation came an interest not only in probabilities as fractions or relative frequencies but also in averages or expectatons. These averages play an essential role in modern probability.

We do not attempt to trace this history, which was long and halting, though marked by flashes of brilliance. Certain concepts and patterns which emerged from experience and intuition called for clarification. We move rather directly to the mathematical formulation (the “mathematical model”) which has most successfully captured these essential ideas. This is the model, rooted in the mathematical system known as measure theory, is called the Kolmogorov model, after the brilliant Russian mathematician A.N. Kolmogorov (1903-1987). Kolmogorov succeeded in bringing together various developments begun at the turn of the century, principally in the work of E. Borel and H. Lebesgue on measure theory. Kolmogorov published his epochal work in German in 1933. It was translated into English and published in 1956 by Chelsea Publishing Company.

## Outcomes and events

Probability applies to situations in which there is a well defined trial whose possible outcomes are found among those in a given basic set. The following are typical.

• A pair of dice is rolled; the outcome is viewed in terms of the numbers of spots appearing on the top faces of the two dice. If the outcome is viewed as an ordered pair, there are thirty six equally likely outcomes. If the outcome is characterized by the total number of spots on the two die, then there are eleven possible outcomes (not equally likely).
• A poll of a voting population is taken. Outcomes are characterized by responses to a question. For example, the responses may be categorized as positive (or favorable), negative (or unfavorable), or uncertain (or no opinion).
• A measurement is made. The outcome is described by a number representing the magnitude of the quantity in appropriate units. In some cases, the possible values fall among a finite set of integers. In other cases, the possible values may be any real number (usually in some specified interval).
• Much more sophisticated notions of outcomes are encountered in modern theory. For example, in communication or control theory, a communication system experiences only one signal stream in its life. But a communication system is not designed for a single signal stream. It is designed for one of an infinite set of possible signals. The likelihood of encountering a certain kind of signal is important in the design. Such signals constitute a subset of the larger set of all possible signals.

These considerations show that our probability model must deal with

• A trial which results in (selects) an outcome from a set of conceptually possible outcomes. The trial is not successfully completed until one of the outcomes is realized.
• Associated with each outcome is a certain characteristic (or combination of characteristics) pertinent to the problem at hand. In polling for political opinions, it is a person who is selected. That person has many features and characteristics (race, age, gender, occupation, religious preference, preferences for food, etc.). But the primary feature, which characterizes the outcome, is the political opinion on the question asked. Of course, some of the other features may be of interest for analysis of the poll.

Inherent in informal thought, as well as in precise analysis, is the notion of an event to which a probability may be assigned as a measure of the likelihood the event will occur on any trial. A successful mathematical model must formulate these notions with precision. An event is identified in terms of the characteristic of the outcome observed. The event “a favorable response” to a polling question occurs if the outcome observed has that characteristic; i.e., iff (if and only if) the respondent replies in the affirmative. A hand of five cards is drawn. The event “one or more aces” occurs iff the hand actually drawn has at least one ace. If that same hand has two cards of the suit of clubs, then the event “two clubs” has occurred. These considerations lead to the following definition.

Definition. The event determined by some characteristic of the possible outcomes is the set of those outcomes having this characteristic. The event occurs iff the outcome of the trial is a member of that set (i.e., has the characteristic determining the event).

• The event of throwing a “seven” with a pair of dice (which we call the event SEVEN) consists of the set of those possible outcomes with a total of seven spots turned up. The event SEVEN occurs iff the outcome is one of those combinations with a total of seven spots (i.e., belongs to the event SEVEN). This could be represented as follows. Suppose the two dice are distinguished (say by color) and a picture is taken of each of the thirty six possible combinations. On the back of each picture, write the number of spots. Now the event SEVEN consists of the set of all those pictures with seven on the back. Throwing the dice is equivalent to selecting randomly one of the thirty six pictures. The event SEVEN occurs iff the picture selected is one of the set of those pictures with seven on the back.
• Observing for a very long (theoretically infinite) time the signal passing through a communication channel is equivalent to selecting one of the conceptually possible signals. Now such signals have many characteristics: the maximum peak value, the frequency spectrum, the degree of differentibility, the average value over a given time period, etc. If the signal has a peak absolute value less than ten volts, a frequency spectrum essentially limited from 60 herz to 10,000 herz, with peak rate of change 10,000 volts per second, then it is one of the set of signals with those characteristics. The event "the signal has these characteristics" has occured. This set (event) consists of an uncountable infinity of such signals.

One of the advantages of this formulation of an event as a subset of the basic set of possible outcomes is that we can use elementary set theory as an aid to formulation. And tools, such as Venn diagrams and indicator functions for studying event combinations, provide powerful aids to establishing and visualizing relationships between events. We formalize these ideas as follows:

• Let $$\Omega$$ be the set of all possible outcomes of the basic trial or experiment. We call this the basic space or the sure event, since if the trial is carried out successfully the outcome will be in $$\Omega$$; hence, the event $$\Omega$$ is sure to occur on any trial. We must specify unambiguously what outcomes are “possible.” In flipping a coin, the only accepted outcomes are “heads” and “tails.” Should the coin stand on its edge, say by leaning against a wall, we would ordinarily consider that to be the result of an improper trial.
• As we note above, each outcome may have several characteristics which are the basis for describing events. Suppose we are drawing a single card from an ordinary deck of playing cards. Each card is characterized by a “face value” (two through ten, jack, queen, king, ace) and a “suit” (clubs, hearts, diamonds, spades). An ace is drawn (the event ACE occurs) iff the outcome (card) belongs to the set (event) of four cards with ace as face value. A heart is drawn iff the card belongs to the set of thirteen cards with heart as suit. Now it may be desirable to specify events which involve various logical combinations of the characteristics. Thus, we may be interested in the event the face value is jack or king and the suit is heart or spade. The set for jack or king is represented by the union $$J \cup K$$ and the set for heart or spade is the union $$H \cup S$$. The occurrence of both conditions means the outcome is in the intersection (common part) designated by $$\cap$$. Thus the event referred to is

$$E = (J \cup K) \cap (H \cup S)$$

The notation of set theory thus makes possible a precise formulation of the event $$E$$.

• Sometimes we are interested in the situation in which the outcome does not have one of the characteristics. Thus the set of cards which does not have suit heart is the set of all those outcomes not in event H . In set theory, this is the complementary set (event) $$H^c$$.
• Events are mutually exclusive iff not more than one can occur on any trial. This is the condition that the sets representing the events are disjoint (i.e., have no members in common).
• The notion of the impossible event is useful. The impossible event is, in set terminology, the empty set $$\emptyset$$. Event $$\emptyset$$ cannot occur, since it has no members (contains no outcomes). One use of $$\emptyset$$ is to provide a simple way of indicating that two sets are mutually exclusive. To say $$AB = \emptyset$$ (here we use the alternate $$AB$$ for $$A \cap B$$) is to assert that events $$A$$ and $$B$$ have no outcome in common, hence cannot both occur on any given trial.
• The language and notaton of sets provide a precise language and notation for events and their combinations. We collect below some useful facts about logical (often called Boolean) combinations of events (as sets). The notion of Boolean combinations may be applied to arbitrary classes of sets. For this reason, it is sometimes useful to use an index set to designate membership. We say the index J is countable if it is finite or countably infinite; otherwise it is uncountable. In the following it may be arbitrary.

$${A_i : i \in J}$$ is the class of sets $$A_i$$, one for each index $$i$$ in the index set $$J$$

For example, if $$J = {1, 2, 3}$$ then $${A_i : i \in J}$$ is the class $${A_1, A_2, A_3}$$, and

$$\bigcup_{i \in J} A_i = A_1 \cup A_2 \cup A_3$$, $$\bigcup_{i \in J} A_i = A_1 \cap A_2 \cap A_3$$,

If $$J = {1, 2, \cdot\cdot\cdot}$$ then $${A_i: i \in J}$$ is the sequence $${A_1: 1 \le i}$$, and

$$\bigcup_{i \in J} A_i = \bigcup_{i = 1}^{\infty} A_i$$, $$\bigcap_{i \in J} A_i = \bigcap_{i = 1}^{\infty} A_i$$

If event E is the union of a class of events, then event E occurs iff at least one event in the class occurs. If F is the intersection of a class of events, then event F occurs iff all events in the class occur on the trial.

The role of disjoint unions is so important in probability that it is useful to have a symbol indicating the union of a disjoint class. We use the big V to indicate that the sets combined in the union are disjoint. Thus, for example, we write

$$A = \bigvee_{i = 1}^{n} A_i$$ to signify $$A = \bigcup_{i = 1}^{n} A_i$$ with the proviso that the $$A_i$$ form a disjoint class

Events derived from a class

Consider the class $${E_1, E_2, E_3}$$ of events. Let $$A_k$$ be the event that exactly $$k$$ occur on a trial and $$B_k$$ be the event that $$k$$ or more occur on a trial. Then

$$A_0 = E_1^c E_2^c E_3^c$$, $$A_1 = E_1 E_2^c E_3^c \bigvee E_1^c E E_3^c \bigvee E_1^c E_2^c E_3$$, $$A_2 = E_1 E_2 E_3^c \bigvee E_1 E_2^c E_3 \bigvee E_1^c E_2 E_3$$, $$A_3 = E_1 E_2 E_3$$

The unions are disjoint since each pair of terms has $$E_i$$ in one and $$E_i^c$$ in the other, for at least one $$i$$. Now the $$B_k$$ can be expressed in terms of the $$A_k\. For example \(V_2 = A_2 \bigvee A_3$$

The union in this expression for $$B_2$$ is disjoint since we cannot have exactly two of the $$E_i$$ occur and exactly three of them occur on the same trial. We may express $$B_2$$ directly in terms of the $$E_i$$ as follows:

$$B_2 = E_1 E_2 \cup E_1 E_3 \cup E_2 E_3$$

Here the union is not disjoint, in general. However, if one pair, say $${E_1, E_3}$$ is disjoint, then $$E_1 E_3 = \emptyset$$ and the pair $${E_1 E_2, E_2 E_3}$$ is disjoint (draw a Venn diagram). Suppose $$C$$ is the event the first two occur or the last two occur but no other combination. Then

$$C = E_1 E_2 E_3^c \bigvee E_1^c E_2 E_3$$

Let $$D$$ be the event that one or three of the events occur,

$$D = A_1 \bigvee A_3 = E_1 E_2^c E_3^c \bigvee E_1^c E_2 E_3^c \bigvee E_1^c E_2^c E_3 \bigvee E_1 E_2 E_3$$

The important patterns in set theory known as DeMorgan's rules are useful in the handing of events. For an arbitrary class $${A_i: i \in J}$$ of events,

$$[\bigcup_{i \in J} A_i]^c = \bigcap_{i \in J} A_i^c$$ and $$[\bigcap_{i \in J} A_i]^c = \bigcup_{i \in J} A_i^c$$

An outcome is not in the union (i.e., not in at least one) of the $$A_i$$ iff it fails to be in all $$A_i$$, and it is not in the intersection (i.e. not in all) iff it fails to be in at least one of the $$A_i$$.

continuation of example

Express the event of no more than one occurrence of the events in $${E_1, E_2, E_3}$$ as $$B_2^c$$.

$$B_2^c = [E_1 E_2 \cup E_1 E_3 \cup E_2 E_3]^c = (E_1^c \cup E_2^c) (E_1^c \cup E_3^c) (E_2^3 \cup E_3^c) = E_1^c E_2^c \cup E_1^c E_3^c \cup E_2^c E_3^c$$

The last expression shows that not more than one of the $$E_i$$ occurs iff at least two of them fail to occur.

• 1.1: Likelihood is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Paul Pfeiffer via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


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1.1: What Is Statistical Thinking?

Statistical thinking is a way of understanding a complex world by describing it in relatively simple terms that nonetheless capture essential aspects of its structure, and that also provide us some idea of how uncertain we are about our knowledge. The foundations of statistical thinking come primarily from mathematics and statistics, but also from computer science, psychology, and other fields of study.

We can distinguish statistical thinking from other forms of thinking that are less likely to describe the world accurately. In particular, human intuition often tries to answer the same questions that we can answer using statistical thinking, but often gets the answer wrong. For example, in recent years most Americans have reported that they think that violent crime was worse compared to the previous year (Pew Research Center). However, a statistical analysis of the actual crime data shows that in fact violent crime has steadily decreased since the 1990’s. Intuition fails us because we rely upon best guesses (which psychologists refer to as heuristics) that can often get it wrong. For example, humans often judge the prevalence of some event (like violent crime) using an availability heuristic – that is, how easily can we think of an example of violent crime. For this reason, our judgments of increasing crime rates may be more reflective of increasing news coverage, in spite of an actual decrease in the rate of crime. Statistical thinking provides us with the tools to more accurately understand the world and overcome the fallibility of human intuition.

This page titled 1.1: What Is Statistical Thinking? is shared under a CC BY-NC 2.0 license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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