Mathematics

Probability with Multiple Dice



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When asking about probability with multiple dice, the key things to remember are that there are multiple dice and that the roll of each die is independent of the others.  That is, what you roll with one die has no effect on what you roll on the other dice. 

There are several questions that can be asked about multiple dice.  But perhaps the most common questions are about the sum of the dice.  If you have two dice, you can get a total from 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.  But are they each equally likely?  Rolling dice for a while will convince you they are not, and looking at the odds in the game of craps will also convince you.  But exactly how likely is each sum?  Remember that there are two dice.  There are 6 ways to roll the first die, and 6 ways to roll the second die, so there are 36 rolls in total.  In particular, they are:

(1 1), (1 2), (1 3), (1 4), (1, 5), (1, 6)

(2 1), (2 2), (2 3), (2 4), (2, 5), (2, 6)

(3 1), (3 2), (3 3), (3 4), (3, 5), (3, 6)

(4 1), (4 2), (4 3), (4 4), (4, 5), (4, 6)

(5 1), (5 2), (5 3), (5 4), (5, 5), (5, 6)

(6 1), (6 2), (6 3), (6 4), (6, 5), (6, 6)

Each of these 36 is equally likely (because the roll of one die does not affect the other), so each has probability of 1/36.

What about sums of the dice?  If you go through the table of 36 rolls, you will see that only 1 combination sums to 2, and 1 sums to 12.  But more sum to the intermediate numbers.  Specifically:

2, 12 – 1 combination  1/36 probability

3, 11 – 2 combinations 1/18 probability

4, 10 – 3 combinations, 1/12 probability

5, 9 –  4 combinations, 1/9 probability

6, 8 – 5 comninations 5/36 probability

7 – 6 combinations, 1/6 probability.

Any other question you might ask about the rolls of two dice can be answered by looking at this table, and similar tables can be drawn for more dice, although they get big quickly: With 3 dice, there are 216 different combinations.

Suppose, for example, you ask “How likely is a roll of 6 on exactly one die?”  If you count them up, you will find there are 10 ways to get exactly one 6, so the probability is 5/18 or about 1 in 5.  But there are 11 ways to get at least one 6.

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