Question

Exercise 1.A committee of five people is chosen randomly from four men and six women. Find the probability that:a) Exactly four women are on the committee:b) At least four women are on the committee:c) At most four women are on the committee:

Answer 1

Answer with explanation:

Given : Number of men = 4

Number of women = 6

Total people =
`6+4=10`

Total number of ways to make a committee of five people from 10 persons :-

`^(10)C_(4)=(10!)/(4!(10-4)!)=210`

a) Number of ways to make committee that has exactly four women :

`^6C_4* ^4C_1=(6!)/(4!(6-4)!)*4=60`

The probability that committee has exactly four women :

`(60)/(210)=(2)/(7)`

b) Number of ways to make committee that has at-least four women :

`^6C_4* ^4C_1+^6C_5=(6!)/(4!(6-4)!)*4+6=66`

The probability that committee at-least four women :

`(66)/(210)=(22)/(70)`

c) Number of ways that committee has more than 4 women :-

`^6C_5*^4C_0=6`

The probability that committee has more than 4 women :-

`(6)/(210)`

Now, the probability that committee has at most four women :-

`1-(6)/(210)=(204)/(210)=(102)/(105)`