Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. A football player who completes 41​% of his passes is asked to throw passes until he misses. The number of passes attempted is recorded. Does the probability experiment represent a binomial​ experiment? A. ​No, because the experiment is not performed a fixed number of times. B. ​No, because the trials of the experiment are not independent and the probability of success differs from trial to trial. C. ​Yes, because the experiment satisfies all the criteria for a binomial experiment. D. ​No, because there are more than two mutually exclusive outcomes for each trial.

A. No, because the experiment is not performed a fixed number of times.

In a binomial experiment, several criteria must be met:

1. The number of trials is fixed in advance. That is, before the experiment is started, we should know exactly how many times the event will be repeated. 2. Each trial can result in just two possible outcomes, commonly referred to as 'success' and 'failure'. 3. The trials must be independent, meaning the outcome of one trial does not affect the outcome of another. 4. The probability of success must remain constant from trial to trial.

In the given scenario, the football player throws passes until he misses, which means that there is no pre-determined number of trials. The experiment continues until a 'failure' occurs, and thus it does not meet the fixed number of trials criterion for a binomial experiment.

Extra: Binomial experiments are a fundamental concept in probability theory, often associated with situations where you're conducting the same type of trial multiple times. For example, flipping a coin a fixed number of times, or manufacturing a set number of items and checking them for defects are scenarios that meet the criteria for a binomial experiment. Numerous statistical methods have been developed to analyze binomial experiments because of their clear structure and the power of the binomial distribution, which mathematically describes the probability of having a fixed number of successes in a given number of trials. In contrast, the experiment described in the question is a geometric experiment, as it continues until the first failure occurs. The geometric distribution is used for modeling the number of trials until the first success or first failure.