Question

Vision Correction About 75% of the U.S. population uses some type of vision correction (such as glasses or contact lenses).

a. If someone is randomly selected, what is the probability that he or she does not use vision

correction?

b. If four different people are randomly selected, what is the probability that they all use vision

correction?

c. Would it be unlikely to randomly select four people and find that they all use vision correction? Why or why not?

Short answer

a. The probability that a randomly selected person does not use vision correction is 0.25.

b. The probability that all use vision correction is 0.316.

c. As the probability is greater than 0.05, the event of finding all four people using vision correction is not unlikely.

Step-by-step solution

Given information

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

Define complement of an event 

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.

Compute the probability of the complement of an event

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:

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

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

Describe the multiplication rule

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

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

Step 5:Compute the probability of the individual event

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:

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

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

Describe an unusual event

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.