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Real Backgammon
Posted By: Timothy Chow
Date: Wednesday, 28 April 2010, at 4:52 a.m.
In Response To: Real Backgammon (Daniel Murphy)
Daniel Murphy wrote:
If "probably observe" is introducing a new point that one large error and 10 small "equivalent" errors should somehow have different effects on results, then I would not agree with that.
Let me try another way to explain it. Suppose you play 10 money games against a perfect player. Let's imagine three cases:
1. You play perfectly, too.
2. You make exactly one error of size 0.100.
3. You make exactly ten errors, each of size exactly 0.010.
In each case, let us ask, what is the probability that you will be behind at the end of the session?
Clearly, in case 1, the probability is 1/2. In cases 2 and 3, we can't answer the question exactly because it depends on the details. Knowing the sizes of your errors doesn't tell us the probability that you'll come out behind. In particular, cases 2 and 3 will not, in general, yield the same probabilities.
Bill, I think, is going further and conjecturing that a small number of large errors is more likely to manifest itself in the short run (i.e., having a larger effect on the probability that your win/loss record will be negative after a small number of games) than a large number of small errors. I'm uncertain about that claim, but it seems somewhat plausible to me. In the very long run it will average out, but the number of games might have to be enormous.

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