The dice game craps is played by repeatedly rolling two six-sided dice, and making various wagers on the outcomes of the sequence of rolls. The details of the wagers don’t concern us here– instead, let’s consider just one particular example scenario involving a common wager called the “pass line bet”:
Suppose that we have just rolled a 4 (which, by the way, occurs with probability 3/36 on any single roll). Having thus established 4 as “the point,” our objective now is to roll a 4 again, rolling repeatedly if necessary… but if we ever roll a 7 (which, by the way, occurs with probability 6/36 on any single roll), then we lose. If and once we roll a 4, we win.
To summarize: we are trying to roll a 4 (with probability 3/36). If we roll anything else except a 7 (where “rolling anything else except 7” has probability 27/36), we continue rolling.
So, here is a puzzle, let’s call it Problem 1: how long does it take on average to win? More precisely, what is the expected length of a winning sequence of rolls, i.e., where we never roll a 7?
This problem is not new, nor is it even particularly sophisticated. But it is very similar to a problem that circulated a few years ago, and that generated some interesting discussion. Here is that earlier problem, let’s call it Problem Zero:
Roll a single fair six-sided die repeatedly until you roll a 6. What is the expected number of rolls required, given that we observe that all rolls are even?
My motivation for this post is two-fold. First, this is a sort of pedagogical thought experiment. Problem Zero has already been shown in the wild to be dangerously non-intuitive. Problem 1 is the same problem— that is, it is essentially equivalent to Problem Zero, just dressed up with different values for the single-trial probabilities. But is Problem 1 inherently easier, less “tricky?” And if so, why? Is it because the numbers are different, or is it that the problem is cast in a more concrete setting as a casino game, etc.?
I don’t know if these problems are actually equally “tricky” or not. But at least in the case of the known trickiness of Problem Zero, I have a theory. Both of these problems are like many others that I have enjoyed discussing here in the past, in that they are mathematical problems… but of a sort that a student may approach not just with pencil and paper, but also with a computer, even if only as an initial exploratory tool.
Which brings me to my theory, based on observation of past debate of Problem Zero. We can begin tackling this problem with pencil and paper, or by writing a simulation… and I suspect that, in this case, starting with a simulation makes it much harder to come up with a (the?) wrong answer.