Suppose you are counting the number of trials needed to get the first success in a sequence of independent $Bernoulli$ trials, where each trial has the same probability of success. Is it appropriate to use the $geometric$ distribution to calculate probabilities in this situation?

• A. No, because the $geometric$ distribution is only used for continuous random variables.
• B. Yes, because the $geometric$ distribution models the number of trials until the first success in independent trials with constant probability.
• C. No, because the $geometric$ distribution is used when the probability of success changes after each trial.
• D. Yes, but only if the number of trials is fixed in advance.

Answer: (B)

Verified step by step guidance

1. Understand the scenario: We have a sequence of independent Bernoulli trials, each with the same probability of success, and we want to find the probability related to the number of trials until the first success occurs.
2. Recall the definition of the geometric distribution: It models the number of trials needed to get the first success in a sequence of independent Bernoulli trials with a constant probability of success \(p\).
3. Check the conditions for using the geometric distribution: trials must be independent, the probability of success \(p\) must remain constant for each trial, and the variable of interest is the count of trials until the first success.
4. Evaluate the given options based on these conditions: The geometric distribution is not for continuous variables, it does not require a fixed number of trials in advance, and it assumes constant probability of success, so options contradicting these facts are incorrect.
5. Conclude that the geometric distribution is appropriate here because it exactly fits the scenario of counting trials until the first success with constant success probability in independent trials.