Notice that those two events cannot happen at the same time, therefore they are mutually exclusive events.
D
Practice makes perfect
Let's look at the given table with the type and number of prizes.
Prize
Number
manicure
10
pedicure
6
massage
3
facial
1
We want to find the probability that the first customer wins manicure or a massage. We can define the following events.
A — the customer wins a manicure
B — the customer wins a massageThese two events cannot happen at the same time, therefore they are mutually exclusive events. Let's recall the formula for finding it.
Probability of Mutually Exclusive Events
If two events $A$ and $B$ are mutually exclusive, then the probability that $A$ or $B$ occurs is the sum of the probabilities of each individual event.
$P(A \text{ or }B) = P(A) + P(B)$
In order to find $P(A)$ and $P(B),$ we will use theoretical probability.
\begin{gathered}
P = \dfrac{\text{Favorable Outcomes}}{\text{Possible Outcomes}}
\end{gathered}
The number of possible outcomes will be the sum of numbers of all possible prizes a customer can win.
\begin{gathered}
10+6+3+1 = \colIV{20}
\end{gathered}
There are $10$ manicures to win, so the number of favorable outcomes for the event $A$ is equal to $\colIII{10}.$ While for the event $B,$ the number of favorable outcomes is $\colV{3}.$ We are ready to calculate $P(A)$ and $P(B).$
\begin{aligned}
P(A) &= \dfrac{\colIII{10}}{\colIV{20}} \\[0.5em]
P(B) &= \dfrac{\colV{3}}{\colIV{20}}
\end{aligned}
We are ready to use the formula for $P(A \text{or} B)$ and find the probability that the first customer wins a manicure or a massage.
