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Puzzle about rolling a die

Posted By: Timothy Chow
Date: Monday, 8 January 2024, at 12:54 a.m.

Bob Koca mentioned being excited about puzzles. He may know this one, but even so, I think other readers will enjoy it.

Suppose I roll a fair die repeatedly until I roll a 6. On average, how many rolls will it take? Your instinct is probably that the answer is 6 rolls on average, and you would be correct.

Now for the real puzzle. Suppose again that I roll a fair die repeatedly until I roll a 6, except that if I roll an odd number, then I abort and start over again. The aborted sequence of rolls is ignored.

Question: In this new game, on average, how many rolls will it take for me to roll a 6?

Just to make sure that the question is clear, let's do an example. I might accumulate the following data:

4 6 success!

2 2 1 abort!

6 success!

4 2 2 4 6 success!

In this example, I completed 3 experiments, and the number of rolls it took me to roll a 6 was 2, 1, and 5 respectively. Based on this limited data, I can estimate that the average number of rolls is (2+1+5)/3 ≈ 2.67.

To get the most enjoyment out of this puzzle, I recommend that you try to work it out in your head, or on a piece of paper, before simulating it.

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