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Quantifying recube vig

Posted By: Timothy Chow
Date: Tuesday, 12 January 2010, at 3:20 p.m.

In Response To: Quantifying recube vig (Marv Porten)

Marv Porten wrote:

It's very hard to believe that in 15552 trials there wasn't at least one game where, for example, white never had a re-cube to 8, but won the game with the cube at 4 by rolling a doublet on the final roll.

It might seem surprising at first, but I don't think it's all that surprising if you think about it harder. If you roll the position out at DMP, then White wins only about 16.6% of the time. The scenario you envision would seem to fall into this 16.6% (ignoring any differences in checker play between DMP and this score). So right off the bat we're down to .166*15552 = 2582 games. One assumes that the vast majority of these White wins come from hitting a shot. What is the most likely way White can win without ever having a recube?

I claim it will be by White hitting a shot "almost too late" and then never getting lucky until the last roll. For if White hits a shot very late then White has to be very lucky to catch up and win at all; this is not very likely. And if White hits a shot at just the right time, but gets lucky double 6's (say) "too soon," this will again trigger the cube. Therefore a number of things have to go just right: White has to hit a shot at the right moment, and then White has to have normal luck for a while, and then White has to roll the lucky last-roll doublets. The part that I italicized about White having "normal luck" is easy to overlook. We have a tendency to think that normal luck happens with probability close to 1. However, it doesn't take too many rolls before we expect to get at least one joker. So having "normal luck" for a long time is not as likely as our intuition tells us.

None of this proves that the above parlay is unlikely to happen at least once in 15552 trials, but it does show that one has to be careful about jumping to conclusions here based on intuition.

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