Exercise 21 Page 962

We know that of young workers, $71\%$ are paid by the hour. We randomly choose two workers out of a group of $100.$ We can define two following events.

• $A$ — the first person is paid by the hour
• $B$ — the second chosen person is paid by the hour

We are interested in finding the probability that exactly one of them is paid by the hour. This means there are two possible situations.

• Only the first chosen person is paid by the hour.
• Only the second person is paid by the hour.

Notice that these two situation correspond to finding the following probabilities.

• $P(A\text{ and }(\text{ not }B))$
• $P((\text{not }A)\text{ and }B)$

Let's first find the first probability $P(A\text{ and }(\text{not }B)).$ The events $A$ and $\text{not }B$ are two independent events, since the outcome of the first event does not affect the probability of the outcome of the second one. Let's recall the rule for finding the probability of independent events.

Probability of Independent Events

If two events $E$ and $F$ are independent, then $P(E\text{ and }F)=P(E)\cdot P(F).$

We already know that the probability that a young worker is paid by the hour is equal to $71\%=0.71.$ By the probability of the complement we get that the probability that a young worker is not paid by the hour is equal to $1-0.71=0.29.$ We have enough information to find $P(A\text{ and }(\text{not }B)).$ Based on the previously obtained information, we conclude that $P((\text{not }A)=0.29$ and $P(B)=0.71.$ Therefore, we can state the second probability we are looking for. Finally, to find the probability that exactly one of two chosen young workers is paid by the hour, we need to add $P(A\text{ and }(\text{not }B))$ and $P((\text{not }A)\text{ and }B).$ This is because they are mutually exclusive events, which do not share common outcomes. The probability that exactly one of two chosen young workers is paid by the hour is about $0.42,$ or $42\%.$