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:

	Male	Female	Total
Blue	$0$	$10$	$10$
Brown	$20$	$20$	$40$
Total	$20$	$30$	$50$

Part b. Two-way Table:

	Male	Female	Total
Blue	$4$	$6$	$10$
Brown	$16$	$24$	$40$
Total	$20$	$30$	$50$

Part c. Two-way Table:

	Male	Female	Total
Blue	$0$	$10$	$10$
Brown	$20$	$20$	$40$
Total	$20$	$30$	$50$

Step by step solution

Part a. Step 1. Given information

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

A: Student is male

B: Student has blue eyes

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 “Male” and row “Blue”.

Also,

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

Thus,

The two – way table becomes:

Gender

		Male	Female	Total
Eye Color	Blue	$0$	$10$	$10$
	Brown	$20$	$20$	$40$
	Total	$20$	$30$	$50$

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 – way table:

Gender

		Male	Female	Total
Eye color	Blue	$\frac{10 \times 20}{50}$	$\frac{10 \times 30}{50}$	$10$
	Brown	$\frac{40 \times 20}{50}$	$\frac{40 \times 30}{50}$	$40$
	Total	$20$	$30$	$50$

Thus,

The two – way table becomes:

	Male	Female	Total
Blue	$4$	$6$	$10$
Brown	$16$	$24$	$40$
Total	$20$	$30$	$50$

Part c. Step 1. Explanation

For two – 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 $10$ for male with blue eyes.

Then

Put $10$ in the column “Male” and the row “Blue”.

And

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

Thus,

The two – way table becomes:

Gender

		Male	Female	Total
Eye color	Blue	$10$	$0$	$10$
	Brown	$10$	$30$	$40$
	Total	$20$	$30$	$50$

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Short Answer

Step by step solution

Part a. Step 1. Given information

Part a. Step 2. Explanation

Part b. Step 1. Explanation

Part c. Step 1. Explanation

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