Question

In a game involving a pair of fair dice, a player rolls the dice until they get a sum of [7]. Let [z=] the number of rolls it takes a player to get the sum. Is [z] a binomial variable? Why or why not?

Answer 1

The variable z is not a binomial variable because it does not have a fixed number of trials; it is instead a geometric random variable that counts the number of rolls until the first success (sum of 7) is obtained.

Step-by-step explanation:

The question is asking whether the variable z, which represents the number of rolls it takes a player to get a sum of 7 with a pair of fair dice, is a binomial variable. A binomial variable must satisfy three conditions: a fixed number of trials, only two possible outcomes (success or failure) in each trial, and that the trials are independent with the probability of success remaining the same in each trial.

However, obtaining a sum of 7 does not have a fixed number of trials since we do not know in advance how many rolls will be necessary to achieve a sum of 7. Each roll is independent, and the probability of rolling a sum of 7 remains constant (at approximately 1/6), but the number of trials is not fixed. Instead, the variable z is a geometric random variable, which counts the number of independent trials until the first success.