Suppose you have a fair six-sided die, and repeatedly toss it. What is the probability that you throw exactly one 4 before your first 6, or you throw exactly one 5 before your first 6 (inclusive or)?
(STEP I Q13, 2015)
There are several ways to approach this problem. There’s a “conventional” method that involves careful use of binomial series (and that was the approach taken in the mark scheme). However, the approach I used for this was somewhat different; some of it may be attributable to my computer science (or perhaps even logic/verification) background.
We can model this event using a finite state machine, where each state has a transition distribution that indicates the probability of transitioning to each other state in the state machine. For example, we can consider a simpler problem: what is the probability that you throw exactly one 4 before your first 6?
Well, we can be in multiple “states”:
- We could not have rolled anything of significance yet. In this case, we have a 4 in 6 probability of staying in this state (1, 2, 3 or 5); a 1 in 6 probability of immediate failure (rolling a 6), and a 1 in 6 probability of registering a 4.
- We could have already registered a 4. In this case, we again have a 4 in 6 probability of staying in this state (1, 2, 3 or 5), a 1 in 6 probability of success (rolling a 6), and a 1 in 6 probability of failure (rolling a 4).
- We could already have succeeded or failed. Additional rolls won’t change anything in these cases.
This can be modelled as the following state machine:
We can then compute a success probability for each state – that is, “given I’m in this state, what’s the probability I succeed”?
- The success probability PS of the state Pass is 1, and that of Fail is 0; we write this as PS(Pass) = 1 and PS(Fail) = 0.
- PS(4) is a bit more interesting; it is and solving this yields half. This is intuitively expected; at this point, we’re counting on rolling a 6 before rolling a 4, and on a fair die the probability of that is half.
- We can now determine PS(Start); this is . Solving the equation yields the answer of .
Note that we could only use this approach because, with the way we’ve defined the states, the process is memoryless; that is, the transition probabilities depend only on the current state. We could try to directly construct a state machine for the original question (either exactly one 4 or exactly one 5 before the first 6, or both), though it seems that the requirement of memorylessness makes things somewhat complex; we would need states tracking whether we experienced zero, one, or two-plus of each number.
We can use a well-known probability axiom here, though; – defining rolling exactly one 4 before one 6 as event A and respectively with a 5 for event B. Furthermore, our initial exploration already yielded values for as a quarter, and would also be a quarter by symmetry. We thus, instead, construct a state machine for the intersection, where both need to be satisfied.
- Throughout, a 1, 2 or 3 does nothing and simply puts us back in the state we were in.
- As we start, we can also roll a 4 or 5, which mean we’ve got the one 4 or 5 we want; or we can roll a 6, which is an immediate failure.
- Once we have a 4 registered, we want to roll a 5 which gets us into the (4, 5) state. 6 remains an immediate failure, but 4 also now becomes one (remember that we want to have exactly one 4 and exactly one 5 before our 6).
- The above logic also holds for 5s, except with 4 and 5 swapped.
- Finally, in the (4, 5) state, rolling a 6 yields success, but either a 4 or a 5 would become immediate failures.
Here’s the result:
We could use the same method of success probabilities used earlier; I’ll take a little shortcut.
- PS(4,5) must be as we’re counting on a 6 being rolled before either a 4 or a 5, and with a fair die each of these has equal probability of being the first of the set to be rolled.
- PS(5) is , which gives us . By symmetry, we can be sure that too.
- PS(Start) is . Solving that yields .
We’re not done yet, though, as this isn’t what we wanted. We wanted ; but now we can apply the identity and get
which is the same answer obtained by the STEP problem-setters, though they used binomial series instead.
The above approach can of course be generalised to handle expected values (on the basis that there can be multiple distinct terminal states, where each state is assigned a value). For example, if a player won a consolation prize of $21 if he ‘bust out’ (rolled a 6 before rolling either a 4 or a 5), and a full $60 if he won, and we wanted to determine the value of the game, we could draw the graph with three terminals instead, perhaps something like this:
We would then perform our computations on the expected value (EV) of each state, with terminals being exactly their value.
Also, though for our examples we didn’t have cycles in our graphs, it’s possible that in general there could be cycles. Of course, these can be handled by solving the multiple PS or EV-equations simultaneously.