Question

a student randomly guesses on a 25-question true-false quiz. determine whether the experiment is a binomial experiment. if not, explain what property of a binomial experiment fails.

select all that apply.

a. it is not a binomial experiment because the trials are not independent of each other.

b. it is not a binomial experiment because the experiment has other possible outcomes, that can not be categorized as \success\ and \failure.\

c. it is not a binomial experiment because the experiment is not performed for a fixed number of trials.

d. it is not a binomial experiment because the probability of success varies from trial to trial.

e. the experiment is a binomial experiment.

Explanation:

To determine if the experiment (guessing on a 25 - question true - false quiz) is a binomial experiment, we recall the properties of a binomial experiment:

1. There are a fixed number of trials. Here, the number of questions (trials) is 25, so the "fixed number of trials" property is satisfied.
2. Each trial has only two possible outcomes, which we can classify as "success" (e.g., guessing correctly) and "failure" (e.g., guessing incorrectly). In a true - false quiz, each question has only two outcomes: true or false, so we can categorize them as success and failure.
3. The probability of success is the same for each trial. When guessing on a true - false question, the probability of guessing correctly is \(p=\frac{1}{2}\) for each question, and this probability does not change from question to question.
4. The trials are independent. The result of guessing on one question does not affect the result of guessing on another question.

Now let's analyze each option:

• Option A: The trials (guessing on each question) are independent. The outcome of one question does not influence the outcome of another, so this option is incorrect.
• Option B: Each trial (question) has two outcomes (true or false), which can be categorized as success and failure. So this option is incorrect.
• Option C: The number of trials is 25, which is fixed. So this option is incorrect.
• Option D: The probability of success (guessing correctly) is \(\frac{1}{2}\) for each question, so the probability of success does not vary from trial to trial. This option is incorrect.
• Option E: The experiment satisfies all the properties of a binomial experiment: fixed number of trials (25), two outcomes per trial (correct/incorrect), constant probability of success (\(\frac{1}{2}\) per question), and independent trials.

Answer:

E. The experiment is a binomial experiment.