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Tables to Improve Your Game

Posted By: Bob Koca
Date: Monday, 15 March 2010, at 5:28 a.m.

In Response To: Tables to Improve Your Game (Phil Simborg)

Here are a couple ways to calculate it.

1) Suppose the question was just how long to get an ace. An ace occurs 11/36 of the time and the expected number of rolls is the reciprocal 36/11. For more on why let me direct you to read about geometric distributions. Similarly it will take 36/11 turns on average to get the 6. So that is a total of 72/11. However you might get lucky and get the 6 with the same roll as the 1. That will happen 2/11 of the time (2 of the 11 rolls with an ace also have the six). This will save the 36/11 rolls needed for the 6. Thus, overall we have 72/11 - (2/11)(36/11) which is about 5.95

2) Another way is to set up a recursive equation. Let's call E the answer to our question. There will be definitely at least one roll. 2/36 of the time we got a 16 and need 0 more rolls. 9/36 of the time we got a 1 without a 6 and thus need the 36/11 more on average to get the 6. 25/36 of the time we get no ace and are back where we started, remember that we gave a name of E to this amount.

Thus we have the equation E = 1 + (2/36)(0) + (9/36)(36/11) + (25/36)E Multiplying out the fraction, adding the 1, and subtracting (25/36)E from each side results in (11/36)E = 20/11. Dividing each side by (11/36) we obtain E = (20 * 36)/ (11 *11) which is about 5.95 (exact value is 5 and 115/121)

If you are instead on the ace and the prime is from 4 to 8 then the change is that we have 36/12 + 36/11 - (2/36)(36/11). This is 3/11 rolls less.

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