rolling two dice and take the money

Question

In a dice game, there are 9 piles of money valued from 1 to 9. The game involves rolling two dice multiple times. If the total of the dice is 10, 11, or 12, you re-roll until the total is between 1 and 9. For example, if you roll a 3+4=7, you can either take pile 3 and pile 4, or just pile 7. If two dice have the same result like 3+3=6, you can just take pile 6.If either option is not available, the game ends. The goal is to maximize the amount of money taken.

I think as for the distribution of the outcome of two dice, it has larger probability that the sum lying around 7, so maybe if the sum is near 7 I should take the split piles instead of the sum pile? Still very confused about what's the best strategy.

Answer 1

So, let's look at the space: there are 30 possible combinations we can land on in this game (6 combinations lead to re-rolls, so they can be excluded). The probability that we can choose any given pile is equal to the every result that include that pile as a face, as well as any other roll that equals that face (other than doubles). That leads to the following table of probabilities (note that these probabilities do not equal one, because of the significant overlap) (edit: Went and double-checked my work, apparently deducted a few in the wrong place):

|Pile|P(Pile)|
|----|-------|
|1   |10/30  |
|2   |11/30  |
|3   |12/30  |
|4   |11/30  |
|5   |12/30  |
|6   |11/30  |
|7   | 6/30  |
|8   | 5/30  |
|9   | 4/30  |

I believe that the most reasonable strategy would be to always pick up the available pile(s) with the least probability of appearing again - in theory this should give you the best opportunity to collect as many as possible, making it least likely to be in a situation where no pile is available. So, the priority order would be 9,8,7,1,{6,4,2},{5,3}.

(edit: Because I've redone the work, for those who wish to look a bit further into the distribution, here's the full list:

|d1|d2|sum|Piles Offered|
|--|--|---|-------------|
|1 |1 |2  |2            |
|1 |2 |3  |{1, 2}, 3    |
|1 |3 |4  |{1, 3}, 4    |
|1 |4 |5  |{1, 4}, 5    |
|1 |5 |6  |{1, 5}, 6    |
|1 |6 |7  |{1, 6}, 7    |
|2 |1 |3  |{1, 2}, 3    |
|2 |2 |4  |4            |
|2 |3 |5  |{2, 3}, 5    |
|2 |4 |6  |{2, 4}, 6    |
|2 |5 |7  |{2, 5}, 7    |
|2 |6 |8  |{2, 6}, 8    |
|3 |1 |4  |{1, 3}, 4    |
|3 |2 |5  |{2, 3}, 5    |
|3 |3 |6  |6            |
|3 |4 |7  |{3, 4}, 7    |
|3 |5 |8  |{3, 5}, 8    |
|3 |6 |9  |{3, 6}, 9    |
|4 |1 |5  |{1, 4}, 5    |
|4 |2 |6  |{2, 4}, 6    |
|4 |3 |7  |{3, 4}, 7    |
|4 |4 |8  |8            |
|4 |5 |9  |{4, 5}, 9    |
|5 |1 |6  |{1, 5}, 6    |
|5 |2 |7  |{2, 5}, 7    |
|5 |3 |8  |{3, 5}, 8    |
|5 |4 |9  |{4, 5}, 9    |
|6 |1 |7  |{1, 6}, 7    |
|6 |2 |8  |{2, 6}, 8    |
|6 |3 |9  |{3, 6}, 9    |