List the criteria for binomial experiment: (Select all that apply: ) The ” trials are mutually exclusive_ The probability of success on each trial constant. The trials have exactly three outcomes. The trials may have two or more outcomes The trials are independent: A fixed number of trials repeated under identical conditions The probability of success can change from trial to trial. The trials have exactly two outcomes What does the random variable of binomial experiment measure? The random variable measures the number of trials until the Ist success. The random variable measures the number of failures in trials_ The random variable measures the number of successes in n trials The random variable measures the number of trials until the nth success
The Correct Answer and Explanation is:
Criteria for a Binomial Experiment:
A binomial experiment has the following key criteria:
- The trials are independent: The outcome of one trial does not affect the outcome of another. Each trial is independent, and the probability of success remains constant across all trials.
- A fixed number of trials: The experiment consists of a predetermined number of trials (n). This is a crucial component, as it defines the scope of the experiment.
- The trials have exactly two outcomes: Each trial in a binomial experiment must result in one of two possible outcomes, often referred to as “success” or “failure.” For example, in a coin toss, the outcomes are “heads” (success) and “tails” (failure).
- The probability of success is constant: In each trial, the probability of success (denoted as p) is the same. This is essential for the consistency of the binomial experiment.
- The trials are mutually exclusive: Each trial is mutually exclusive, meaning only one of the two possible outcomes can occur in a trial.
What the Random Variable Measures:
The random variable in a binomial experiment specifically measures the number of successes in n trials.
- The random variable measures the number of successes in n trials: This is the most common interpretation in binomial experiments. It tells you how many successes occur over the n trials. For example, if you toss a coin 5 times, the random variable might represent the number of heads (successes) out of those 5 tosses.
Other options you mentioned are incorrect:
- The random variable does not measure the number of trials until the first or nth success. That would be a scenario modeled by a geometric distribution.
- The number of failures is also not measured directly in a binomial experiment but can be derived as the difference between the number of trials and the number of successes.
Example:
If you flip a fair coin 10 times, the number of heads (successes) is a binomial random variable. If p = 0.5 for success (heads), the distribution of the number of heads across 10 trials can be modeled using the binomial distribution, where n=10n = 10n=10 and p=0.5p = 0.5p=0.5.
