Let be the event
"Harry will be either the member chosen to be the secretary or the member chosen to be the treasurer".
By definition the probability of this event
where
- number of desired outcomes;
- total number of outcomes that can occur.
Let's denote people who have chosen for president (P), for secretary (S) and for treasurer (T). So
is equal to the number of different combinations
:
(total number of outcomes that can occur when choosing a president)(the total number of outcomes that can occur when choosing a secretary)(the total number of outcomes that can occur when choosing a treasurer)
Let be the number of desired outcomes when Harry will be the member chosen to be a secretary;
is the number of desired outcomes when Harry will be the member chosen to be a treasurer.
If Harry is to be a secretary then he cannot be chosen for president and is equal to the number of different combinations
:
=(the total number of outcomes that can occur when choosing a president)(total number of outcomes that can occur when choosing a treasurer)
.
If Harry is to be a treasurer then he cannot be chosen for president or secretary and is equal to the number of different combinations
:
=(the total number of outcomes that can occur when choosing a president)(the total number of outcomes that can occur when choosing a secretary)
.
So
Then .
The problem is indifferent whether Harry or the other person is chosen; all 10 people have an equal chance to be a secretery and all 10 people have the same chance to be a treasurer. So, the probability of Harry being chosen as a secretary is and the probability of Harry being chosen as a treasurer is the same,
. Both events are mutualy exclusive, so the probability of Harry being either a secretary or treasurer is
Although the problem is indifferent whether Harry or the other person is chosen, answer C () is wrong because the outcome consists with 2 mutually exclusive events.