Question

Among the contestants in a competition are 38 women and 22 men. if 5 winners are randomly selected, what is the probability that they are all men

Answer 1

There are two ways to solve this problem:

(1) The basic way:
Total number of contestants = 38 + 22 = 60 contestant
Probability that 1 winner is a man = 22/60
Probability that 2 winners are men = (22/60) * (21/59)
Probability that 3 winners are men = (22/60) * (21/59) * (20/58)
Probability that 4 winners are men = (22/60) * (21/59) * (20/58) * (19/57)
Probability that 5 winners are men = (22/60)*(21/59)*(20/58)*(19/57)*(18/56)
Doing the calculation, we will find that:
probability that 5 winners are men = 0.0048217

(2) The combination way:
Total number of contestants = 38 + 22 = 60
probability that 5 winners are men = C(22, 5) / C(60, 5)
where:
C(n, r) = n! / ((n - r)! * r!)
(The exclamation mark refers to the factorial)
Now, we will substitute:
C (22,5) = (22)! / ((22-5)!*(5)!) = 26334
C (60,5) = (60)! / ((60-5)!*(5)!) = 5461512
Substitute to get the probability as follows:
probability that 5 winners are men = C(22, 5) / C(60, 5)
= (26334) / (5461512)
= 0.0048217