91Ó°ÊÓ

A survey classified a large number of adults according to whether they were judged to need eyeglasses to correct their reading vision and whether they used eyeglasses when reading. The proportions falling into the four categories are shown in the table. (Note that a small proportion, .02, of adults used eyeglasses when in fact they were judged not to need them.) $$\begin{array}{lcc} & \begin{array}{l}\text { Used Eyeglasses } \\\\\text { for Reading }\end{array} \\\\\hline \text { Judged to Need } & & \\ \text { Eyeglasses } & \text { Yes } & \text { No } \\\\\hline \text { Yes } & .44 & .14 \\\\\text { No } & .02 & .40\end{array}$$ If a single adult is selected from this large group, find the probability of each event: a. The adult is judged to need eyeglasses. b. The adult needs eyeglasses for reading but does not use them. c. The adult uses eyeglasses for reading whether he or she needs them or not.

Step by step solution

01

Identify cells corresponding to Event A

From the table, the cells corresponding to adults that are judged to need eyeglasses are: "Yes" in the "Judged to Need Eyeglasses" row, and "Yes" and "No" in the "Used Eyeglasses for Reading" column.

02

Calculate the probability of Event A

The probability of Event A can be calculated by adding the proportions in the identified cells: P(A) = 0.44 + 0.14 = 0.58 #b. Probability of the adult needing eyeglasses but not using them# Event B: The adult needs eyeglasses for reading but does not use them.

03

Identify cells corresponding to Event B

From the table, the cells corresponding to adults that need eyeglasses but do not use them are: "Yes" in the "Judged to Need Eyeglasses" row, and "No" in the "Used Eyeglasses for Reading" column.

04

Calculate the probability of Event B

The probability of Event B can be calculated by adding the proportions in the identified cells: P(B) = 0.14 #c. Probability of the adult using eyeglasses for reading whether he or she needs them or not# Event C: The adult uses eyeglasses for reading whether he or she needs them or not.

05

Identify cells corresponding to Event C

From the table, the cells corresponding to adults that use eyeglasses for reading are: "Yes" in the "Used Eyeglasses for Reading" column, and "Yes" and "No" in the "Judged to Need Eyeglasses" row.

06

Calculate the probability of Event C

The probability of Event C can be calculated by adding the proportions in the identified cells: P(C) = 0.44 + 0.02 = 0.46

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