91Ó°ÊÓ

The probability that any randomly selected individual from the U.S.population uses some type of vision correction is 0.75.

The complement of an event, say A, is the situation when event A does not occur.

The probability of the complement of an event is equal to the probability of the event deduced from 1.

a.

Let A be the probability that the selected individual uses vision correction. Consequently, will be the event that the randomly selected individual does not use vision correction.

As given, $P\left(A\right)=0.75$.

The probability that the selected individual does not use vision correction is:

$$\begin{gathered} P\left(\stackrel{¯}{A}\right)=1-P\left(A\right) \\ =1-0.75 \\ =0.25 \end{gathered}$$

Thus, the probability that a selected individual would not use vision correction is 0.25.

For n events such as$A_1,A_2,...,A{}_n$,the probability that the events co-occur is computed as shown below.

$P\left(A_{1}and{A}_{2}and...and{A}_n\right)=P\left(A_1\right)×P\left(A_2\right)×...×P\left(A_n\right)$

b.

Define $A_1,A_2,A_3,A_4$ as the event that four randomly selected people use vision correction.

Using the multiplication rule, the probability that four randomly selected people use vision correction is shown below:

$$\begin{gathered} P\left(A_1\right)×P\left(A_2\right)×P\left(A_3\right)×P\left(A_4\right)=P\left(A\right)×P\left(A\right)×P\left(A\right)×P\left(A\right) \\ ={\left(P\left(A\right)\right)}^4 \\ ={\left(0.75\right)}^4 \\ =0.316 \end{gathered}$$

Thus, the probability that all randomly selected people use vision correction is 0.316.

c.

An unlikely or rare event is one thathas a probability of occurrence of 0.05 or less.

The probability that all randomly selected individuals use vision correction is larger than 0.05, which implies the event is not unlikely.