Which of the following is not an example of a binomial distribution?

• A. Counting the number of students who pass an exam out of $30$ students, where each has the same probability of passing
• B. Counting the number of heads in $10$ flips of a fair coin
• C. Counting the number of defective bulbs in a sample of $20$ bulbs from a large production line
• D. Measuring the time until the first success in a sequence of independent Bernoulli trials

Answer: (D)

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 Bernoulli trials, each with the same probability of success.
2. Examine each example to see if it fits the binomial criteria: fixed number of trials, independent trials, two possible outcomes (success/failure), and constant probability of success.
3. For 'counting the number of students who pass an exam out of 30 students,' this fits the binomial model because there are 30 fixed trials, each student either passes or fails, and the probability of passing is the same for each student.
4. For 'counting the number of heads in 10 flips of a fair coin,' this is a classic binomial example with 10 fixed trials, two outcomes (head or tail), and constant probability of success (head).
5. For 'counting the number of defective bulbs in a sample of 20 bulbs from a large production line,' assuming the sample is random and the probability of defect is constant, this also fits the binomial model.
6. For 'measuring the time until the first success in a sequence of independent Bernoulli trials,' this does not fit the binomial distribution because the number of trials is not fixed in advance; instead, it follows a geometric distribution.