One survey found that 23% of its members had purchased shares directly through an initial public offering. In a sample of 12 members, what is the probability that three members have purchased in an initial public offering?

To calculate the probability that three out of 12 members have purchased shares directly through an initial public offering (IPO), we can use the binomial probability formula: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k) Where: - P(X = k) is the probability of getting exactly k successes. - n is the number of trials or members in the sample (12 in this case). - k is the number of successes we're interested in (3 members purchasing through IPO). - p is the probability of success on a single trial (the probability that a member has purchased through IPO). In this case, p = 0.23 (23% expressed as a decimal), n = 12, and k = 3. Now, let's calculate the probability: P(X = 3) = (12 choose 3) * (0.23)^3 * (1 - 0.23)^(12 - 3) Using a calculator, you can calculate the binomial coefficient (12 choose 3) as: (12 choose 3) = 220 Now, plug the values into the formula: P(X = 3) = 220 * (0.23)^3 * (0.77)^9 P(X = 3) ≈ 0.2547 So, the probability that three out of 12 members have purchased shares directly through an initial public offering is approximately 0.2547, or about 25.47%.