Question

determine if the following probability experiment represents a binomial experiment. if not, explain why. if the probability experiment is a binomial experiment, state the number of trials, n.
four cards are selected from a standard 52 - card deck without replacement. the number of fives selected is recorded.
select the correct choice below and, if necessary, fill in the answer box to complete your answer
○ a. yes, because the experiment satisfies all the criteria for a binomial experiment, n=
○ b. no, because the trials of the experiment are not independent since the probability of success differs from trial to trial
○ c. no, because the experiment is not performed a fixed number of times
○ d. no, because there are more than two mutually exclusive outcomes for each trial

Explanation:

A binomial experiment has the following properties:

• A fixed number of trials (\(n\)).
• Each trial has only two possible outcomes (success or failure).
• The probability of success (\(p\)) is the same for each trial.
• The trials are independent.

In this case, four cards are selected without replacement. Since the cards are not replaced, the probability of selecting a five changes from one trial to the next. For example, the probability of selecting a five on the first draw is \(\frac{4}{52}\), on the second draw (if a five was not selected first) it is \(\frac{4}{51}\), and so on. This means the trials are not independent, which violates a key property of a binomial experiment.

Answer:

B. No, because the trials of the experiment are not independent since the probability of success differs from trial to trial.