The likelihood of getting a sum between 9 and 16 is 72/144 = 1/2 = 0.5. The total probability of rolling a sum of 10, 11, or 12 is also 30/144.

For example, the probability of rolling a sum between 5 and 9 (inclusive) is 30/144 ? 0.2083 = 20.83%. The total probability of rolling a sum between 17 and 24 (inclusive) is 28/144 ? 0.1944 = 19.44%.

With two 12-sided dice, there is no range that gives a probability of exactly 20% since the probability is always a fraction over 144, and 144 is not divisible by 5.

Practical Example

Suppose you are designing a board game that uses two dodecahedral dice. You want to find a range of sums whose total probability of occurring is about 1/5 = 20%, so that roughly one fifth of the time a certain outcome happens during the game.

Other Probabilities

The chances of rolling an even sum are exactly 72/144 = 1/2 = 0.5, as are the chances of rolling an odd sum. The probability of throwing a sum that is a multiple of 3 is 48/144 = 1/3 or 0.33333.. If you look at the list of probabilities you will find many possible solutions to this problem.

When you roll two 12-sided dice, the likelihood of getting a sum of 10, 12, 13, 14, or 15 is 54/144 = 3/8 = 0.375

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