Assume that a procedure yields a $binomial$ distribution. Which of the following is a necessary condition for the $binomial$ distribution to apply?

• A. The probability of success must increase after each trial.
• B. The number of trials must be $infinite$.
• C. Each trial must have only two possible outcomes, commonly referred to as success and failure.
• D. Each trial must have a different probability of success.

Answer: (C)

Verified step by step guidance

1. Recall the definition of a binomial distribution: it models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure).
2. Identify the key conditions for a binomial distribution: (1) fixed number of trials, (2) trials are independent, (3) each trial has exactly two possible outcomes, and (4) the probability of success remains constant across trials.
3. Analyze each option given in the problem to see if it matches these conditions. For example, the probability of success must NOT increase or change after each trial; it must remain constant.
4. Recognize that the number of trials must be fixed and finite, not infinite, for the binomial distribution to apply.
5. Conclude that the necessary condition among the options is that each trial must have only two possible outcomes, commonly called success and failure.