How to Calculate Multiple Dice Probabilities
Note the number of dice, their sides, and the desired sum. , Enumerate all the ways that sum can be reached., Not all partitions listed in the previous step are equally likely., Add the total number of ways to get the desired sum. , Divide by the...
Step-by-Step Guide
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Step 1: Note the number of dice
This can be tedious for large numbers of dice, but is fairly straightforward.
This is equivalent to the finding all partitions of k into exactly n parts with no part larger than r.
An example for n=5, r=6, and k=12 is shown as an example.
In order to ensure that the count is both exhaustive and that no partition is counted twice, the partitions are presented in lexicographic order and the dice in each partition in non-decreasing order. , This is why they must be listed, not simply counted.
In a smaller 3 die example, the partition 123 covers 6 possibilities (123, 132, 213, 231, 312, 321) while the partition 114 covers only 3 (114, 141, 411) and 222 only includes itself.
Use the multinomial formula to compute the number of ways to permute the digits in each partition.
This information has been added to the table from the previous section. ,, Since each die has r equally probable faces, this is simply rn. -
Step 2: their sides
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Step 3: and the desired sum.
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Step 4: Enumerate all the ways that sum can be reached.
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Step 5: Not all partitions listed in the previous step are equally likely.
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Step 6: Add the total number of ways to get the desired sum.
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Step 7: Divide by the total number of outcomes.
Detailed Guide
This can be tedious for large numbers of dice, but is fairly straightforward.
This is equivalent to the finding all partitions of k into exactly n parts with no part larger than r.
An example for n=5, r=6, and k=12 is shown as an example.
In order to ensure that the count is both exhaustive and that no partition is counted twice, the partitions are presented in lexicographic order and the dice in each partition in non-decreasing order. , This is why they must be listed, not simply counted.
In a smaller 3 die example, the partition 123 covers 6 possibilities (123, 132, 213, 231, 312, 321) while the partition 114 covers only 3 (114, 141, 411) and 222 only includes itself.
Use the multinomial formula to compute the number of ways to permute the digits in each partition.
This information has been added to the table from the previous section. ,, Since each die has r equally probable faces, this is simply rn.
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