# Understanding Probability with Multiple Dice

The probability of dice outcomes uses the regular six-sided dice that are used in board games. The six sides have one, two, three, four, five, and six dots respectively. The sides will have values of the number of dots each side-one, two, three, four, five, and six. The smallest value possible if two dice are used is therefore two (two ones), and the highest value possible is twelve (two sixes). All the possible values are two, three, four, five, six, seven, eight, nine, ten, eleven, and twelve.

The possible outcomes if two dice are thrown 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.

There are thirty-six possible outcomes for the values of the two dice. The possibility of rolling each of the thirty-six possible outcomes is equal. The 1,1 and the 6,6 only occur one time each. This means the probability of rolling a 1,1 (2) or a 6,6 (12) is one in thirty-six on one roll of the dice. The number of ways a value of three can be rolled is either 1,2 or 2,1. The difference between 1,2 and 2,1 is 1,2 means a one is rolled on the first die and a two is rolled on the second die, while 2,1 is the opposite case: a two is rolled on the first die and a one on the second die.

So, the probability of rolling a three is two in thirty-six, or one in eighteen in lowest terms. A four is the outcome if a 1,3; 2,2; or 3,1 is rolled. The probability is therefore three in thirty-six or one in twelve. A five is rolled if the dice show 1,4; 2,3; 3,2; or 4,1 is rolled. This makes the probability four in thirty-six or one in nine. A six is rolled if a 1,5; 2,4; 3,3; 4,2; or 5,1 is the outcome. The odds of a six being rolled are five in thirty-six. A seven is the outcome if a 1,6; 2,5; 3,4; 4,3; 5,2; or 6,1 is rolled. The odds are therefore six of thirty-six or one in six. Eight is the result if a 2,6; 3,5; 4,4; 5,3; or 6,2 is the outcome. The odds of rolling an eight is therefore five in thirty-six. A nine is rolled if a 3,6; 4,5; 5,4; or 6,3 is the result.

This makes the odds of a nine four in thirty-six or one in nine. Ten is the outcome if a 4,6; 5,5; or 6,4 is rolled. The odds are three in thirty-six or one in twelve. An eleven is rolled if a 5,6 or 6,5 is the result. The odds are therefore two in thirty-six or one in eighteen.

The odds are symmetrical because the odds of throwing a two or twelve are one in thirtysix, odds of throwing a three or eleven are one in eighteen, odds of throwing a four or ten are one in twelve, odds of throwing a five or nine are one in nine, odds of throwing a six or eight are five in thirty-six, and the odds of throwing a seven are one in six.

The same idea is used for the rolling of three, four, five, six, or more dice at the same time. For three dice rolled simultaneously, the possible outcomes increase to 6 times 6 times 6, or 216. If four dice are rolled at the same time, then the number of combinations are 6 times 6 times 6 times 6, or 1296.

If five dice are rolled, then the number of possible outcomes are 6 times 6 times 6 times 6, times 6, or 7776. If n dice are used, then the number of combinations is 6^n, which is 6 times 6 times 6 n number of times. The minimum value for the dice changes to the number of dice times one. So, the minium value is three, four, five, six, seven, etc. for three, four, five, six, seven, etc. dice rolled at the same time respectively. For n number of dice rolled simultaneously, the minimum value is n times 1 or n. The maximum value is the number of dice times six. This makes it 6*3=18 if three dice are rolled. The maximum value for four, five, six, dice are 4*6=24, 5*6=30, 6*6=36 respectively. For n number of dice rolled at the same time, the maximum value is n times six. The method of figuring out the odds for rolling more than two dice is analogous to the method for two dice.