The City Transit Authority plans to hire 4 new bus drivers. From a group of 100
ID: 3302093 • Letter: T
Question
The City Transit Authority plans to hire 4 new bus drivers. From a group of 100 qualified applicants, of whom 65 are men and 35 are women, 4names are to be selected by lot. Suppose that Mary and John Lewis are among the 100 qualified applicants.
(a) What is the probability that Mary's name will be selected?
What is the probability that both Mary's and John's names will be selected? (Round your answer to four decimal places.)
(b) If it is stipulated that an equal number of men and women are to be selected (2 men from the group of 65 men and 2 women from the group of 35 women), what is the probability that Mary's name will be selected? (Round your answer to four decimal places.)
What is the probability that Mary's and John's names will be selected? (Round your answer to four decimal places.)
Explanation / Answer
(a) There are 100 applicants and 4 are to be selected.
Total number of ways to select 4 drivers = 100C4
If Mary is selected, total number of ways to select 3 other drivers = 99C3
=> Probability of Mary being selected
= 99C3 / 100C4
= 156849 / 3921225
= 0.04
If both Mary and John are selected, the number of ways to select other 2 drivers = 98C2
=> Probability of both Mary and John being selected
= 98C2 / 100C4
= 4753 / 3921225
= 0.0012
(b) Total number of ways to select 2 men among 65 and 2 women among 35
= 65C2 * 35C2
= 2080 * 595
= 1237600
Since Mary's name is selected, the other woman ccan come in 34 ways.
The two men can come in 65C2 = 2080 ways
=> Number of ways to select two men and two women including Mary = 34 * 2080 = 70720
=> Probability of Mary being selected = 70720 / 1237600 = 0.0571
If Mary and John are both selected, the other male and female can be selected in 64 and 34 ways respectively
=> Number of ways of selecting the four people = 64*34 = 2176
=> Probability of Mary and John both being selected = 2176 / 1237600
= 0.0018