Question

A party planning committee is being selected randomly in an office. Jim and Pam are married with a child and only one of them will be on the committee. Dwight does not like Jim and will not be on the committee if Jim is on the committee. The committee must contain 2 males and 3 females. There are a total of 7 males and 8 females in the office. a) In how many ways can the committee be formed? b)What is the probability that Dwight is on the committee?

Answer 1

Number of Males in the office =7

Number of Females in the office = 8

The committee must contain 2 males and 3 females.

a) In how many ways can the committee be formed

Option 1 : Jim is on the committee;

Then only another male could be in the committee and Dwight will not be in the committee; So number males left to choose is 7-2(Jim already on the committee and  Dwight will not be in the committee)=5 ; Therefore number of ways of selecting 1 male member (as jim is already on the committee) from 5 members = 5

Since Jim is the committee ; pam can not be on the committee; so number of females left to choose are 8-1 = 7

Number of ways of selecting 3 females from 7 females = = 35

Number of ways of choosing committee if Jim on the committee = 35 x 5 = 175

Option 2 Jim is not on the committee;

Number of males left to choose =6; Number of ways of selecting 2 male members from 6 males =  

As Jim is not on the committee ; no restrictions for female members i.e number of ways of selecting 3 female members from 8 females =

Number of ways of choosing committee if Jim not on the committee = 15 x 56 = 840

Committee can be formed with jim on the committee or Jim not on the committee i.e

Number of ways the committee be formed = 175 + 840 = 1015

b)What is the probability that Dwight is on the committee

Number of ways a committee be formed if Dwight is on the committee;

If Dwight is on the committee then Jim can not be on the committee Then only another male could be in the committee as Dwight is already in the committee; So number males left to choose is 7-2(Dwight already on the committee and Jim will not be in the committee)=5 ; Therefore number of ways of selecting 1 male member from 5 members = 5

No restrictions on the Female members selection; Number of ways of selecting 3 female members from 8 females =

Number of ways a committee be formed if Dwight is on the committee =5 x 56 = 280

probability that Dwight is on the committee

= Number of ways a committee be formed if Dwight is on the committee / Number of ways the committee be formed

= 280/ 1015 = 0.27586

Probability that Dwight is on the committee = 0.27586