Question

A certain club has 10 members, including Harry. One of the 10 members is to be chosen at random to be the president, one of the remaining 9 members is to be chosen at random to be the secretary, and one of the remaining 8 members is to be chosen at random to be the treasurer. What is the probability that Harry will be either the member chosen to be the secretary or the member chosen to be the treasurer

Answer 1

The probability that Harry will be chosen as either as Secretary or as treasurer but not as president out of those 10 members will be
`(1)/(5)`.

Explanation:

We have 10 members out of which one is Harry, we need to choose one of the 10 members as President, one to be as secretary out of the remaining 9, one as the treasurer of the remaining 8 members.

We want harry to be as either as secretary or as treasurer. so we will consider both the cases

Probability is given by = No of favorable outcomes/ Total no of outcomes

(1) Harry as secretary : If harry is to be chosen as secretary then he can't be a president or he cant be a Treasurer

As president =
`(1)/(10)`

Not as president =
`1 - (1)/(10)` =
`(9)/(10)`

Now he is selected for the secretary so he can't be selected as treasurer.

Not as treasurer = 1

Probability for chosen as secretary =
`(9)/(10)*(1)/(9)*1` =
`(1)/(10)`

(2) Harry as treasurer : If harry is to be chosen as the treasurer so he can't be chosen as a President or as a Secretary.

Probability for not chosen as president =
`(9)/(10)`

Probability for not chosen as secretary =
`1-(1)/(9) = (8)/(9)`

Probability to be chosen as treasurer =
`(1)/(8)`

Probability to be chosen as Treasurer =
`(9)/(10)*(8)/(9)*(1)/(8)` =
`(1)/(10)`

Total probability for Harry to be chosen as either secretary or as treasurer will be = P(Secretary) +P(Treasurer)

=
`(1)/(10) +(1)/(10) = (1)/(5)`

Therefore the probability that Harry will be chosen as either as Secretary or as treasurer but not as president out of those 10 members will be
`(1)/(5)`.