Which distribution is characterized by the number of failures before a specified number of successes?

Which distribution is characterized by the number of failures before a specified number of successes?

• A. Binomial distribution
• B. Negative binomial distribution
• C. Geometric distribution
• D. Hypergeometric distribution

Answer: (B)

The negative binomial distribution is the correct choice because it specifically models the number of failures that occur before achieving a fixed number of successes in a series of Bernoulli trials. This distribution is especially useful in scenarios where you want to predict how many times an event will fail before you successfully achieve a predetermined number of successes. For example, if you are interested in how many times a coin must be flipped before you get three heads, the negative binomial distribution allows you to calculate probabilities based on the number of tails (failures) encountered before those three heads (successes) are achieved. In contrast, the binomial distribution focuses on the number of successes in a fixed number of trials rather than failures before success. The geometric distribution specifically deals with the count of failures before the first success, and the hypergeometric distribution involves sampling without replacement, which does not fit the scenario of counting failures before a certain number of successes in independent trials.

The negative binomial distribution is the correct choice because it specifically models the number of failures that occur before achieving a fixed number of successes in a series of Bernoulli trials. This distribution is especially useful in scenarios where you want to predict how many times an event will fail before you successfully achieve a predetermined number of successes.

For example, if you are interested in how many times a coin must be flipped before you get three heads, the negative binomial distribution allows you to calculate probabilities based on the number of tails (failures) encountered before those three heads (successes) are achieved. In contrast, the binomial distribution focuses on the number of successes in a fixed number of trials rather than failures before success. The geometric distribution specifically deals with the count of failures before the first success, and the hypergeometric distribution involves sampling without replacement, which does not fit the scenario of counting failures before a certain number of successes in independent trials.