Question

A freshman English class at a certain college has 49 women and 20 men. If 5 students are randomly selected, what is the probability that they are all men? Round to five decimal places.

a) 0.01133
b) 0.18061
c) 0.00138
d) 0.00813

Answer 1

The probability that all 5 randomly selected students are men is calculated using the hypergeometric probability formula, resulting in approximately 0.00138, which is option c).

Step-by-step explanation:

To calculate the probability that all 5 randomly selected students are men, we will use the hypergeometric probability formula since we are dealing with two distinct groups (men and women), and we are selecting without replacement. The total number of ways to select 5 students out of 69 (49 women + 20 men) is the combination of 69 taken 5 at a time, denoted as C(69, 5).

The number of ways to select 5 men out of the 20 available is the combination of 20 taken 5 at a time, denoted as C(20, 5). Thus, the probability is calculated as:

P(5 men) = C(20, 5) / C(69, 5)

Filling in the values from the combination formula n! / (r!(n-r)!) for C(20, 5) and C(69, 5), we calculate:

P(5 men) = (20! / (5!(20-5)!)) / (69! / (5!(69-5)!))

After computing, we find the probability to be approximately P(5 men) = 0.00138, which corresponds to option c).