Identifying and Understanding Binomial Experiments In Exercises 15–18, determine whether the experiment is a binomial experiment. If it is, identify a success; specify the values of n, p, and q; and list the possible values of the random variable x. If it is not a binomial experiment, explain why.


Basketball A’ja Wilson, the 2020 WNBA Most Valuable Player, makes a free throw shot about 78% of the time. The random variable represents the number of free throws that she makes on eight attempts. (Source: Women’s National Basketball Association)

Verified step by step guidance

1. Step 1: Recall the criteria for a binomial experiment. A binomial experiment must satisfy the following conditions: (1) The experiment consists of a fixed number of trials (n). (2) Each trial has only two possible outcomes: success or failure. (3) The probability of success (p) is the same for each trial. (4) The trials are independent of each other.
2. Step 2: Analyze the problem to determine if it meets the criteria for a binomial experiment. In this case, the experiment involves A’ja Wilson attempting 8 free throws. Each attempt can result in either a success (making the shot) or a failure (missing the shot). The probability of success (making a free throw) is 78% (p = 0.78), and the trials are independent because the outcome of one free throw does not affect the others.
3. Step 3: Identify the success, n, p, and q. A success is defined as making a free throw. The number of trials (n) is 8. The probability of success (p) is 0.78, and the probability of failure (q) is 1 - p = 0.22.
4. Step 4: List the possible values of the random variable x. The random variable x represents the number of free throws made. Since there are 8 trials, x can take on any integer value from 0 to 8 (inclusive).
5. Step 5: Conclude whether this is a binomial experiment. Since the problem satisfies all four criteria for a binomial experiment, it is indeed a binomial experiment. Summarize the key parameters: n = 8, p = 0.78, q = 0.22, and x can range from 0 to 8.

Key Concepts

Binomial Experiment

A binomial experiment is a statistical experiment that meets four criteria: it consists of a fixed number of trials, each trial has two possible outcomes (success or failure), the trials are independent, and the probability of success remains constant across trials. In this context, the experiment involves A’ja Wilson attempting free throws, where each shot can be classified as either a success (making the shot) or a failure (missing the shot).

Parameters n, p, and q

In a binomial experiment, 'n' represents the number of trials, 'p' is the probability of success on each trial, and 'q' is the probability of failure, which can be calculated as 1 - p. For A’ja Wilson's free throw attempts, n is 8 (the number of shots), p is 0.78 (the probability of making a shot), and q is 0.22 (the probability of missing a shot).

Random Variable x

The random variable x in a binomial experiment represents the number of successes in n trials. In this scenario, x would denote the number of successful free throws made by A’ja Wilson out of her 8 attempts. The possible values of x range from 0 to 8, indicating that she could make anywhere from none to all of her free throws.