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?

• A. $\frac{1}{720}$
• B. $\frac{1}{80}$
• C. $\frac{1}{10}$
• D. $\frac{1}{9}$
• E. $\frac{1}{5}$

Answer 1

The correct option is E $\frac{1}{5}$

Two probabilities must be calculated here: $\left(1\right)$ the probability of Harry's being chosen for secretary and $\left(2\right)$ the probability of Harry's being chosen for treasurer. For any probability, the probability of an event's occurring $=\frac{\text{number of desired outcomes}}{\text{total number of outcomes that can occur}}$

(1) If Harry is to be secretary, he first CANNOT have been chosen for president, and then he must be chosen for secretary. The probability that he will be chosen for president is $\frac{1}{10}$, so probability of hit NOT being chosen for president is $1-\frac{1}{10}=\frac{9}{10}$. Then, the probability of his being chosen for secretary is $\frac{1}{9}$.

Once he is chosen, the probability that he will be selected for treasurer is $0$, so the probability that he will NOT be selected for treasurer is $1-0=1$. Thus, the probability that Harry will be chosen for secretary is $\left(\frac{9}{10}\right)\left(\frac{1}{9}\right)\left(1\right)=\frac{1}{10}$.

(2) If Harry is to be treasurer, he needs to be NOT chosen for president, then NOT chosen for secretary, and then finally chosen for treasurer.The probability that he will NOT be chosen for president is again $1-\frac{1}{10}=\frac{9}{10}$. The probability of his NOT being chosen for secretary is $1-\frac{1}{9}=\frac{8}{9}$. The probability of his being chosen for treasurer is $\frac{1}{8}$, so the probability that Harry will be chosen for treasurer is $\left(\frac{9}{10}\right)\left(\frac{8}{9}\right)\left(\frac{1}{8}\right)=\frac{1}{10}$

(3) So, finally, the probability of Harry's being chosen as either secretary or treasurer is thus $\frac{1}{10}+\frac{1}{10}=\frac{2}{10}=\frac{1}{5}$.

The correct answer is E.