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Chapter 5: Q. 97 (page 353)

The two-way table summarizes data on the gender and eye color of students in a college statistics class. Imagine choosing a student from the class at random. Define event A: student is male, and event B: student has blue eyes.

a. Copy and complete the two-way table so that events A and B are mutually exclusive.

b. Copy and complete the two-way table so that events A and B are independent.

c. Copy and complete the two-way table so that events A and B are not mutually exclusive and not independent.

Short Answer

Expert verified

Part a. Two-way Table:


MaleFemaleTotal
Blue0
10
10
Brown20
20
40
Total20
30
50

Part b. Two-way Table:


MaleFemaleTotal
Blue4
6
10
Brown16
24
40
Total20
30
50

Part c. Two-way Table:


MaleFemaleTotal
Blue0
1010
Brown20
20
40
Total20
30
50

Step by step solution

01

Part a. Step 1. Given information

Data on gender and eye color of the students summarized in two 鈥� way table:

A: Student is male

B: Student has blue eyes

02

Part a. Step 2. Explanation

Two events are disjoint or mutually exclusive when both events cannot occur at same time.

In this part,

Events A and B are mutually exclusive.

This implies

No male student has blue eyes.

Thus,

In the table, put 0 in the column 鈥淢ale鈥� and row 鈥淏lue鈥�.

Also,

Put the remaining counts according to the total counts of the rows and columns.

Thus,

The two 鈥� way table becomes:

Gender



MaleFemaleTotal
Eye ColorBlue0

10
10

Brown20
2040

Total20
30
50
03

Part b. Step 1. Explanation

Two events are independent, when the probability of occurrence of one event does not affect the probability of occurrence of other event.

Then

The counts will be the product of the row total and the column total, divided by the table total provided in the bottom left corner of the table.

Calculate the counts in the two 鈥� way table:

Gender



MaleFemaleTotal
Eye colorBlue10脳2050
10脳305010

Brown40脳2050
40脳305040

Total20
30
50

Thus,

The two 鈥� way table becomes:


MaleFemaleTotal
Blue4
610
Brown16
2440
Total20
30
50
04

Part c. Step 1. Explanation

For two 鈥� way table, where A and B are not mutually exclusive and not independent as well.

In this part, the count for male with blue eyes should be different from the other two parts (Part (a) and Part (b)).

Suppose, if we choose the count 10for male with blue eyes.

Then

Put 10in the column 鈥淢ale鈥� and the row 鈥淏lue鈥�.

And

Put the remaining counts according to the total counts of the rows and columns.

Thus,

The two 鈥� way table becomes:

Gender



MaleFemaleTotal
Eye colorBlue10
010

Brown10
3040

Total20
30
50

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