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Answer and comments
Posted By: Timothy Chow In Response To: Roulette puzzle (Timothy Chow)
Date: Wednesday, 28 April 2010, at 6:05 p.m.
Several people have solved the problem correctly; the answer is E.
The first instinct that many people (including myself) have is that since roulette has negative expected value, you're going to lose in the long run. And 108 bets seems like a lot of trials, doesn't it? By this reasoning the probability of losing money should be high, so the answer should be A, B, or C.
However, if you think about it a little, then you'll realize that to break even you need to win only three times. You have a 1/38 chance of winning so after 108 trials you expect to win about three times. This reasoning suggests that your chances of coming out even or ahead are going to be 50% or maybe even better than that. And this reasoning is more sound than the previous reasoning.
To get the exact answer you need to run through all the different ways you could lose: winning exactly 0, 1, or 2 times. Others have given this calculation so I won't repeat it here. I'll just mention for the really hardcore math geeks that you can just barely compute the answer in your head if you approximate your wins as having a Poisson distribution with mean 3; then your probability of coming out behind is about (1 + 3 + 4.5)/e3, and if you know, or can work out, that e3 is about 20, then you'll come very close to the actual answer.
The relevance of this for backgammon is to illustrate the distinction between equity and probability of coming out ahead. Choosing a play with higher equity means that you'll come out ahead in the long run, but the long run can be much longer than you might think at first. It's possible that a lower-equity play will give you a higher probability of coming out ahead in the short run. We all know this fact in at least one guise: we know that checker play in matches isn't the same as checker play in money games. But even if we restrict ourselves to money games, even the worst addicts among us play only a finite number of money games, so the effect is potentially at work even in money games.
In a recent thread, I've been discussing with Daniel Murphy whether ten plays that each lose equity 0.010 are "equivalent" to one play that loses equity 0.100. They're equivalent as far as equity goes, but it's not clear whether they are the same as far as probability of coming out ahead goes. Hopefully the roulette puzzle illustrates some of the subtleties at play here.
Finally, let me also point out that if in the roulette game, you were to lose every single time—an unlikely, but not outrageously unlikely, event—you would conclude that your average loss per game was $1 and your confidence would be 100%. Of course, this conclusion is totally wrong. This is another illustration of the dangers of relying on short rollouts that claim to give you high confidence.
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