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BGonline.org Forums
SOLUTION
Posted By: Timothy Chow In Response To: Puzzle about rolling a die (Timothy Chow)
Date: Wednesday, 10 January 2024, at 2:49 a.m.
The answer is 1.5 rolls on average.
Most people, myself included, find this answer surprising. It's tempting to reason that, since we're excluding odd numbers, the game is equivalent to rolling a three-sided die (with 2, 4, 6 on the faces) until we roll a 6. If this were the case, then on average, it would take 3 rolls to roll a 6. But, as you can verify yourself by simulation, this is the wrong answer!
Here's one way to see that the game is not equivalent to rolling a three-sided die. If it were, then 1/3 of all (non-aborted) sequences would start with a 6, 1/3 would start with a 2, and 1/3 would start with a 4. At first glance, this seems correct; we abort any sequence that starts with a 1, 3, or 5, and we're equally likely to roll a 2, 4, or 6 first, right? The catch is, if your first roll is a 6, then you're guaranteed not to abort, because you're already done; but if your first roll is a 2 or a 4, then there's a chance you'll eventually abort. So in fact, the first roll of a non-aborted sequence is more likely to be a 6 than it is to be a 2 (or a 4).
There are various ways to arrive at the correct answer of 1.5, but here's an argument that uses a minimum amount of explicit calculation. Consider what I'll call The Modified Game, in which we never abort; instead, we keep rolling until we roll a 1, 3, 5, or 6, and then we declare success and stop. What is the average number of rolls per trial in The Modified Game? Well, at each step, we have a 2/3 probability of rolling one of the four lucky numbers, so the average number of rolls per trial is the reciprocal of 2/3, or 3/2 = 1.5.
Now comes the key step of the reasoning. Imagine we're playing The Modified Game over and over, and keeping a record of all our trials. We use four separate sheets of paper. On the first sheet, we record the trials that end with rolling a 1; on the second sheet, we record the trials that end with rolling a 3; etc. The key observation is that the average number of rolls per trial on each sheet is also 1.5. This is because by symmetry, the average number of rolls per trial on each sheet has to be the same as the average number of rolls per trial on every other sheet, and overall, the average number of rolls per trial is 1.5. So the average number of rolls per trial on any particular sheet must be 1.5.
The final observation is that the sheet with trials ending in 6 is the same as the original game. For the only difference between the original game and The Modified Game is that in the original game, we throw away the three other sheets of paper instead of keeping them. So the average number of rolls per trial in the original game is 1.5.
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