List the criteria for a binomial experiment. (Select all that apply:) The trials may have two or more outcomes. The probability of success on each trial is constant. The probability of success can change from trial to trial. A fixed number of trials repeated under identical conditions. The trials are independent. The trials have exactly two outcomes. The trials are mutually exclusive. The trials have exactly three outcomes. What does the random variable of a binomial experiment measure? The random variable measures the number of trials until the first success.
The Correct Answer and Explanation is:
A binomial experiment has several key criteria that must be met to classify it as such. The criteria include:
- A fixed number of trials: There must be a set number of trials that are repeated under identical conditions. For example, if you’re flipping a coin 10 times, the number of trials is fixed at 10.
- Two possible outcomes: Each trial must have exactly two possible outcomes, commonly referred to as “success” and “failure.” For example, flipping a coin gives two outcomes: heads (success) or tails (failure).
- Constant probability of success: The probability of success on each trial must be the same. If you’re flipping a fair coin, the probability of landing heads (success) is always 0.5 for each trial.
- Independence of trials: The trials must be independent, meaning that the outcome of one trial does not affect the outcome of another trial. In coin flipping, the result of one flip does not influence the next flip.
- Mutually exclusive outcomes: The outcomes of the trials must be mutually exclusive, meaning the events do not overlap. For instance, a coin cannot land on both heads and tails simultaneously.
Thus, the correct criteria for a binomial experiment are:
- The trials are independent.
- The trials have exactly two outcomes.
- A fixed number of trials repeated under identical conditions.
- The probability of success on each trial is constant.
- The trials are mutually exclusive.
Regarding the random variable in a binomial experiment, it measures the number of successes in a fixed number of trials. For example, if you’re flipping a coin 10 times, the random variable could represent how many times you get heads (success). This is different from measuring the number of trials until the first success, which would be the scenario for a geometric distribution rather than a binomial distribution. Therefore, in a binomial experiment, the random variable counts how many successes occur across all trials.
