The missing probabilities in the table.

Question

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Answer 1

Question

Chapter 4.5, Problem 109E

(a)

To determine

To find: The missing probabilities in the table.

(a)

Expert Solution

Answer to Problem 109E

Solution: The complete table is obtained as follows.

		Gender		Total
		Men	Women	Total
Institution	4-year institution	0.2684	0.3416	0.61
Institution	2-year institution	0.1599	0.2301	0.39
Total		0.4283	0.5717	1

Explanation of Solution

Calculation: Let M and F denote the gender of the students as “Male” and “Female.” Let 4Y denote that the student attends a 4-year institution and 2Y denote that the student attends a 2-year institution. The conditional probability, which is defined as the occurrence of one event provided the other event has already occurred, is used in this exercise. The probability that a male student attends a 4-year institution is denoted as P(M|4Y) and the probability that a male student attends a 2-year institution is denoted as P(M|2Y).

It is provided in the question that the 4-year institutions constitute 44% males; 61% of students attend the 4-year institution and the rest 2-year institution.

The probability that a student attends a 4-year institution is as follows:

P(4-year institution)=P(4Y)=0.61

The probability that a male student attends a 4-year institution is as follows:

P(Males|4-year instituion)=P(M|4Y)=44%=0.44

The probability that a male student attends a 2-year institution is as follows:

P(Males|2−year instituion)=P(M|2Y)=41%=0.41

The provided probabilities are represented in the following table:

		Gender		Total
		Men	Women
Institution	4-year institution			0.61
	2-year institution
Total				1

The remaining or missing probabilities can be ascertained as follows.

The probability that a student attends a 4-year institution is calculated as

P(2-year institution)=P(2Y)=1−P(4-year institution)=1−P(4Y)=1−0.61

=0.39

The concept of conditional probability is the probability of an event (A) when the other event has already occurred (B), and it is calculated as

P(A|B)=P(A and B)P(B)=P(A∩B)P(B)

The concept of conditional probability is used to ascertain the following probabilities.

The probability that a student is a male and also that he attends a 4-year institution is calculated as follows:

P(Males|4-year institution)=P(Males who attend 4-year institution)P(4-year institution)P(M|4Y)=P(M∩4Y)P(4Y)0.44=P(M∩4Y)0.61P(M∩4Y)=0.44×0.61

=0.2684

The probability that a student is a male and also that he attends a 2-year institution is calculated as follows:

P(Males|2-year institution)=P(Males who attend 2-year institution)P(2-year institution)P(M|2Y)=P(M∩2Y)P(2Y)0.41=P(M∩2Y)0.39P(M∩2Y)=0.41×0.39

=0.1599

The probability that the student is a female and also that she attends a 4-year institution is calculated as follows:

P(Female | 4-year institution)=(P(4-year institution)−P(Males who attend 4-year institution))P(F∩4Y)=P(4Y)−P(M∩4Y)=0.61−0.2684=0.3416

The probability that the student is a female and also that she attends a 2-year institution is calculated as follows:

P(Female|2-year institution)=(P(2-year institution)−P(Males who attend 2-year institution))P(F∩2Y)=P(2Y)−P(M∩2Y)=0.39−0.1599=0.2301

Therefore, the probability that a student is a male and belongs to the 4-year institution {P(M∩4Y)} is 0.2684.

The probability that a student is a male and belongs to the 2-year institution {P(M∩2Y)} is 0.1599.

The probability that a student is a female and belongs to the 4-year institution {P(F∩4Y)} is 0.3416.

The probability that a student is a female and belongs to the 2-year institution {P(F∩2Y)} is 0.2301.

The probabilities P(4Y) and P(2Y) sum to 1. Also, the P(M) and P(W) sum to 1.

Therefore, the complete probability table is shown as below:

		Gender		Total
		Men	Women	Total
Institution	4-year institution	0.2684	0.3416	0.61
Institution	2-year institution	0.1599	0.2301	0.39
Total		0.4283	0.5717	1

(b)

To determine

To find: The probability that a randomly selected female student attends the 4-year institution.

(b)

Expert Solution

Answer to Problem 109E

Solution: The required probability is 0.5975.

Explanation of Solution

Calculation: The concept of conditional probability is the probability of an event (A) when the other event has already occurred (B), and it is calculated as

P(A|B)=P(A and B)P(B)=P(A∩B)P(B)

The conditional probability rule is used to ascertain the required probability. It is calculated as follows.

The probability that the randomly selected student attends a 4-year institution provided the fact that the student is a female is calculated as follows:

P(4-year institution|female)=P(Female who attend 4-year institution)P(Female)=P(F∩4Y)P(F)=0.34160.5717=0.5975

Hence, the required probability is 0.5975.

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Answer 2

Question

Chapter 4.5, Problem 109E

(a)

To determine

To find: The missing probabilities in the table.

(a)

Expert Solution

Answer to Problem 109E

Solution: The complete table is obtained as follows.

		Gender		Total
		Men	Women	Total
Institution	4-year institution	0.2684	0.3416	0.61
Institution	2-year institution	0.1599	0.2301	0.39
Total		0.4283	0.5717	1

Explanation of Solution

Calculation: Let M and F denote the gender of the students as “Male” and “Female.” Let 4Y denote that the student attends a 4-year institution and 2Y denote that the student attends a 2-year institution. The conditional probability, which is defined as the occurrence of one event provided the other event has already occurred, is used in this exercise. The probability that a male student attends a 4-year institution is denoted as P(M|4Y) and the probability that a male student attends a 2-year institution is denoted as P(M|2Y).

It is provided in the question that the 4-year institutions constitute 44% males; 61% of students attend the 4-year institution and the rest 2-year institution.

The probability that a student attends a 4-year institution is as follows:

P(4-year institution)=P(4Y)=0.61

The probability that a male student attends a 4-year institution is as follows:

P(Males|4-year instituion)=P(M|4Y)=44%=0.44

The probability that a male student attends a 2-year institution is as follows:

P(Males|2−year instituion)=P(M|2Y)=41%=0.41

The provided probabilities are represented in the following table:

		Gender		Total
		Men	Women
Institution	4-year institution			0.61
	2-year institution
Total				1