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Calculation

Posted By: Tom Keith
Date: Thursday, 9 March 2017, at 7:03 p.m.

In Response To: Calculation (Casper Van der Tak)

Casper wrote: "[-3*cube, +3*cube]"

In money-play backgammon, there is no limit to the cube, so aren't these bounds essentially plus-or-minus infinity?

Let's say that two players decide, for one reason or another, that it is to their advantage to reach a "problem" position. And let's say there is a one-in-a-quadrillion* chance of them rolling the sequence of rolls needed to let them get there. (* The actual probability doesn't matter much to this example.)

Once they have reached the problem position, there is a .482 chance of both sides being able to double/take and get back to the same position they were in two rolls earlier but with the cube now 4 times higher. The chance of that happening, say, 250 times in a row is 5.78e+80. The value of the cube at that point will be 3.27e+150. The chance of all this (reaching the position, then being able to double/take 500 times in a row) is 1 in (1e+15 * 5.78+80), or about 1 in 5.78e+95.

The value of the position at that point, times the probability of reaching it, is about 5.66e+54. That's going to have a pretty big effect on the value of the starting position!

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