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.
A random sample of 50 professional athletes is obtained, and the individuals selected are asked to state their ages.
Select the correct choice below:
A. Yes, because the experiment satisfies all the criteria for a binomial experiment, n =
B. No, because the experiment is not performed a fixed number of times.
C. No, because the trials of the experiment are not independent since the probability of success differs from trial to trial.
D. No, because there are more than two mutually exclusive outcomes for each trial.
Answer (2)
The experiment is not a binomial experiment because there are more than two mutually exclusive outcomes for each trial (each athlete can state a different age).
Explanation
Analyze the experiment characteristics Let's analyze the given probability experiment to determine if it represents a binomial experiment. A binomial experiment has the following characteristics:
The experiment is performed a fixed number of times (fixed number of trials).
The trials are independent.
There are only two mutually exclusive outcomes for each trial, typically labeled as "success" or "failure".
The probability of success remains constant for each trial.
Check binomial experiment conditions In this experiment, we have a random sample of 50 professional athletes, and each athlete is asked to state their age. Let's check if the binomial experiment conditions are met:
The experiment is performed a fixed number of times: Yes, the experiment is performed 50 times (50 athletes are sampled).
The trials are independent: Yes, each athlete's age is independent of the other athletes' ages.
There are only two mutually exclusive outcomes for each trial: No, this condition is not met. When an athlete states their age, the outcome can be any number within a reasonable range (e.g., 18 to 50). There are more than two possible outcomes. The outcomes are not simply "success" or "failure".
Conclusion Since the condition of having only two mutually exclusive outcomes is not met, the given probability experiment is not a binomial experiment.
Examples
Consider a quality control process where you inspect items and record their defects. If you sampled 50 items and recorded the number of defects for each, this would not be a binomial experiment because the number of defects can be any non-negative integer, not just success or failure. However, if you only recorded whether each item was defective or not defective, then it would be a binomial experiment with n=50.
The given probability experiment does not represent a binomial experiment because it doesn't meet the condition of having only two mutually exclusive outcomes. Each athlete can state a different age, resulting in multiple possible outcomes. The correct answer is option D.
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