National surveys indicate that 36% of people have been in a car accident in the last 5 years. If you randomly sample 10 people, how likely is that exactly 4 have had an accident in the last 5 years?

• A. 0.0767
• B. 0.242
• C. 0.00527
• D. 0.00129

Answer: (B)

Verified step by step guidance

1. Identify the type of probability distribution involved. Since we are dealing with a fixed number of trials (10 people) and each trial has two possible outcomes (accident or no accident), this is a binomial distribution problem.
2. Define the parameters of the binomial distribution: n = 10 (number of trials), p = 0.36 (probability of success, i.e., having an accident), and k = 4 (number of successes we are interested in).
3. Use the binomial probability formula to calculate the probability of exactly k successes: P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) is the binomial coefficient, calculated as n! / (k! * (n-k)!).
4. Calculate the binomial coefficient C(10, 4) using the formula: C(10, 4) = 10! / (4! * (10-4)!).
5. Substitute the values into the binomial probability formula: P(X = 4) = C(10, 4) * (0.36)^4 * (0.64)^6, and simplify the expression to find the probability.