Find the indicated probability. Round your answer to 6 decimal places when necessary

Question

Among the contestants in a competition are 40 women and 20 men. If 5 winners are randomly selected, what is the probability that they are all men?

Answer 1

The probability that all 5 winners are men can be calculated by the following formula:

P(all men) = (number of ways to choose 5 men) / (total number of ways to choose 5 winners)

The number of ways to choose 5 men out of 20 is given by the combination formula:

C(20, 5) = 20! / (5!(20-5)!) = 20! / (5!15!) = (20 * 19 * 18 * 17 * 16) / (5 * 4 * 3 * 2 * 1) = 15504

The total number of ways to choose 5 winners out of 60 (40 women + 20 men) is given by:

C(60, 5) = 60! / (5!(60-5)!) = 60! / (5!55!) = 216,156

Therefore, the probability that all 5 winners are men is:

P(all men) = 15504 / 216156 ≈ 0.071679

Rounded to 6 decimal places, the probability that all 5 winners are men is approximately 0.071679.