How to Calculate Probability using Multiple Dice

To calculate the probabilities associated with results with rolling multiple dice, one must understand the basic concept of probability with outcomes rolling 1 die and independent events. The possible outcomes when rolling one six sided die is 1,2,3,4,5,6. The sum of all the probabilities of all the outcomes must equal 1. Assuming that you have a fair die (all numbers equally likely to occur), the probability of any of the outcomes on a single die roll is 1/6.

When rolling 2 dice, there are 36 possible outcomes. (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 outcome has a 1/36 chance of occurring, assuming both dice are fair six sided dice. Because what number shows up on the first die has no bearing on what shows up on the second die, the events are said to be independent. With independent events you simply multiply the probability of each even to get the probability of the combined event. For instance, say you want to know the probability of rolling a (5,4). Without knowing there are 36 events, you can say.. The probability of rolling a 5 is 1/6, as is the probability of rolling a 4. Therefore the probability of rolling a 5 and a 4 is (1/6)(1/6) = 1/36.

Now let’s think of rolling a (5,4) again but this time the question is, “What is the probability of having a 4 on the second die given that the first die is a 5?” This is a bit different.. Think of the possible outcomes with 5 as the first number showing up on the dice. (5,1), (5,2), (5,3), (5,4), (5,5), (5,6). Therefore the answer to this question is 1/6 since there are 6 possible outcomes, with (5,4) being the outcome sought after in the question.

Another question could be as follows, what is the probability of having the sum of the numbers on the two dice greater than or equal to 9? To answer, need to check all the possible outcomes.. (5,4), (5,5), (5,6), (4,5), (4,6), (6,3), (6,4), (6,5), (6,6), (3,6).. 10 of the possible 36 outcomes will yield a sum of 9 or greater to the answer is 10/36 or simplified to 5/18.

If there are 3 dice involved, the same rules apply. There are just more possible outcomes, 216 to be exact. Adding one more die multiplies the number of possible outcomes by 6. If the question is, what is the probability of rolling a (1,1,1) on 3 dice? The answer is 1/216, by using the probability of independent events. Probability of getting a 1 is 1/6 on each die, so (1/6)(1/6)(1/6) = 1/216.

There are many more examples and extensions of this concept, but I hope what I have shown you makes you understand the concept of calculating the probabilities when rolling multiple dice.