# Rolling Multiple Dice

This Demonstration lets you simulate rolling multiple six-sided dice. You can roll up to 50 fair dice at once. The values of two random variables are recorded, the sum of the dice and the number of sixes that appear. You can repeat the experiment 1, 100, or 1000 times with a single mouse click. The number of times each outcome has been observed is displayed in a histogram. Experimental probabilities are compared to the theoretical distribution.

### DETAILS

When the number of rolls is increased, the results of a random experiment are seen to approach the theoretical distribution. Theoretical probabilities for obtaining a given number of sixes when multiple dice are rolled are given by a binomial distribution with parameters and 1/6, where is the number of fair dice. The maximum of this distribution is at , which is the most likely number of sixes. The probability of a particular sum of dice is somewhat more cumbersome to compute. According to the central limit theorem, as the number of dice per roll is increased, the theoretical probabilities approach the normal distribution. In this Demonstration, however, instead of using the normal approximation, the theoretical probability of obtaining as the sum of the roll of dice is computed exactly as the coefficient of in , divided by . The expected value of the total of dice is and the distribution is symmetric about the expectation.

### PERMANENT CITATION

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