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.


Video Games A survey found that 29% of gamers own a virtual reality (VR) device. Ten gamers are randomly selected. The random variable represents the number who own a VR device. (Source: Entertainment Software 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 given problem. The experiment involves selecting 10 gamers (fixed number of trials, n = 10). Each gamer either owns a VR device (success) or does not own a VR device (failure), satisfying the two-outcome condition. The probability of success (owning a VR device) is given as p = 0.29, and the probability of failure is q = 1 - p = 0.71. The trials are independent because the ownership of a VR device by one gamer does not affect another.
3. Step 3: Conclude that this is a binomial experiment because it satisfies all four conditions of a binomial experiment: fixed number of trials, two outcomes per trial, constant probability of success, and independence of trials.
4. Step 4: Identify the success, n, p, q, and the possible values of the random variable x. Success is defined as a gamer owning a VR device. The number of trials is n = 10. The probability of success is p = 0.29, and the probability of failure is q = 0.71. The random variable x represents the number of gamers who own a VR device, and its possible values are x = 0, 1, 2, ..., 10.
5. Step 5: Summarize the findings. This is a binomial experiment with n = 10, p = 0.29, q = 0.71, and the possible values of x ranging from 0 to 10. If the experiment did not meet the binomial criteria, you would explain which condition(s) were violated.

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 selecting a fixed number of gamers and determining how many own a VR device.

Random Variable

A random variable is a numerical outcome of a random phenomenon. In binomial experiments, the random variable typically represents the number of successes in the fixed number of trials. For the given question, the random variable x represents the number of gamers out of ten who own a VR device, which can take on values from 0 to 10.

Probability Parameters (n, p, 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, where q = 1 - p. For the survey of gamers, n is 10 (the number of gamers selected), p is 0.29 (the probability that a gamer owns a VR device), and q is 0.71 (the probability that a gamer does not own a VR device).