![]() To explain, 4-1 means four of one number and one of another. The number of these that lead to the different hand patterns are as follows: Pattern There are 6^5 = 7776 outcomes for the first roll. Recall that the number of ways of picking m things out of n isĪ concrete example we will need in a minute is So this strategy for the three rolls game has a better expected score than the strategy for the two rolls game. (the last term in the sum comes from the fact that, if you re-roll after the first roll, then you are back in the situation of the two rolls game, so your expected score is 4.75, as before). ![]() The expected score using this strategy is: Now for the three rolls game, using the same idea as in the two rolls game, we could use the following strategy: after the first roll: keep a 5 or a 6 and re-roll a 1, 2, 3, or 4.
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