Here’s a probability puzzle to ruin your week.
In the Royal Game of Ur, players advance tokens along a track with 14 spaces. To determine how many spaces to advance, a player rolls 4 dice with 4 sides. Two corners on each die are marked; the other two are not. The total number of marked corners — which is 0, 1, 2, 3, or 4 — is the number of spaces to advance.
For example, if the total on your first roll is 2, you could advance a token to space 2. If you roll a 3 on the next roll, you could advance the same token to space 5.
Suppose you have a token on space 13. How many rolls did it take to get there?
Hint: you might want to start by computing the distribution of k given n, where k is the number of the space and n is the number of rolls. Then think about the prior distribution of n.
I’ll post a solution later this week, but I have to confess: I believe my solution is correct, but there is still part of it I am not satisfied with.
[UPDATE November 1, 2018]
Here’s the thread on Twitter where a few people discuss this problem.
And here’s my solution. As you will see there are still some unresolved questions.
And here’s a solution from vlad, also based on simulation, using WebPPL: