To determine whether this probability experiment represents a binomial experiment, we need to verify if it satisfies the four criteria for a binomial experiment:
1. Fixed number of trials: There should be a fixed number of trials in the experiment.
2. Independence: Each trial should be independent of the others.
3. Two possible outcomes: Each trial has exactly two possible outcomes (success or failure).
4. Constant probability: The probability of success should remain the same for each trial.
Let's evaluate the given experiment based on these criteria:
1. Fixed number of trials: The experiment involves selecting a random sample of 25 professional athletes. Here, the fixed number of trials, or sample size, is 25. Therefore, this criterion is satisfied.
2. Independence: Assuming that the hair length of each selected athlete does not influence the hair length of another athlete, each trial can be considered independent. This criterion is satisfied.
3. Two possible outcomes: The experiment asks each athlete to state their hair length. If we categorize hair length into two distinct categories (for instance, "long" and "short"), this criterion can be considered satisfied. However, if hair length has more than two categories or is treated as a continuous measure, this criterion might not strictly be satisfied.
4. Constant probability: Assuming that the probability of an athlete having a certain hair length remains consistent across the sample, this criterion is also satisfied.
Given the assumptions that we can categorize hair length into exactly two categories ("long" and "short") and that the probabilities remain constant, this experiment can be considered a binomial experiment.
Therefore, the correct choice is:
Yes, because the experiment satisfies all the criteria for a binomial experiment. The number of trials, [tex]\( n \)[/tex], is 25.
