Question

A state lottery randomly chooses 6 balls numbered from 1 through 39 without replacement. You choose 6 numbers and purchase a lottery ticket. The random variable represents the number of matches on your ticket to the numbers drawn in the lottery. Determine whether this experiment is binomial. If so, identify a success, specify the values n, p, and q and list the possible values of the random variable x.

Answer 1

The state lottery does not satisfy the criteria for a binomial experiment since the probability changes with each draw and the trials are not independent.

Step-by-step explanation:

To determine if the state lottery is a binomial experiment, we need to consider if it meets the three criteria for a binomial distribution:

1. There is a fixed number of trials.
2. There are only two possible outcomes per trial: success or failure.
3. The trials are independent and the probabilities remain constant.

In the case of the state lottery where 6 balls are drawn from a set of 39 without replacement, this scenario does not meet the criteria for a binomial experiment. This is because the probability of success changes with each draw since the balls are not replaced. Thus:

• A 'success' would be matching a number on your ticket with one drawn from the lottery.
• The variable n would represent the number of balls drawn.
• The probabilities p (success) and q (failure) are not constant.
• The possible values of the random variable x would be 0 to 6, representing the number of matches.

Since the probability of choosing a correct number changes after each ball is drawn, the draws are not independent, and p and q are not constant, this cannot be considered a binomial experiment.