Chapter 5.3, Problem 44E

Management: The Bureau of Labor Statistics reported that 64.5 trillion women and 74.6 men were employed. Of the women. 25.8 million had management jobs, and of the men, 25.0 million had management jobs. An employed person is chosen at random.

What is the probability that the person is a female?

What is the probability that the person has a management job?

What is the probability that the person is female and has a management job?

Given that the person is female, what is the probability that she has a management job?

Given that the person has a management job, what is due probability that the person is female?

To determine

The probability that randomly chosen person is female.

Answer to Problem 44E

  $0.4637$

Explanation of Solution

Given:

Total number of female is 64.5 million and male is 74.6 million.

Number of person with management job from women is 25.8 million and that of men is 25.0.

Formula Used:

Probability of an event is given by

  $\frac{\text{Number}\text{of}\text{Favorable}\text{Outcome}}{\text{Number}\text{of}\text{total}\text{outcome}}$

Calculation:

Let $F$ be the event that selectedperson is female.

  $\begin{array}{c}n\left(F\right)=64.5\\ n\left(S\right)=64.5+74.6=139.1\\ P\left(F\right)=\frac{64.5}{139.1}\\ =0.4637\end{array}$

Hence, the probability that randomly chosen person is female,is $0.4637$ .

To determine

The probability that randomly chosen person has management job.

Answer to Problem 44E

  $0.3652$

Explanation of Solution

Given:

Total number of female is 64.5 million and male is 74.6 million.

Number of person with management job from women is 25.8 million and that of men is 25.0.

Formula Used:

Probability of an event is given by

  $\frac{\text{Number}\text{of}\text{Favorable}\text{Outcome}}{\text{Number}\text{of}\text{total}\text{outcome}}$

Calculation:

Let $M$ be the event that selected person has management job.

  $\begin{array}{c}n\left(M\right)=25.8+25=50.8\\ n\left(S\right)=64.5+74.6=139.1\\ P\left(M\right)=\frac{50.8}{139.1}\\ =0.3652\end{array}$

Hence, the probability that randomly chosen person has management job is, $0.3652$ .

To determine

The probability that the chosen one is femle and has management job.

Answer to Problem 44E

  $0.1855$

Explanation of Solution

Given:

Total number of female is 64.5 million and male is 74.6 million.

Number of person with management job from women is 25.8 million and that of men is 25.0.

Formula Used:

For any two t events $A$ and B , probability of occurring both the events is

  $\begin{array}{c}P\left(A\text{and}B\right)=P\left(A\cap \text{}B\right)\\ =\frac{n\left(A\cap \text{}B\right)}{n\left(S\right)}\end{array}$

Calculation:

Let $M$ be the event that selected person has management job.

Let $F$ be the event that selected person is female.

So, $M\text{and}F$ be the event that selected one is femle and has management job.

There are 25.8 millionfemale who has management job.

  $\begin{array}{c}P\left(M\cap F\right)=\frac{n\left(M\cap F\right)}{n\left(S\right)}\\ =\frac{25.8}{139.1}\\ =0.1855\end{array}$

The probability that selected one is femle and has management jobis, $0.1855$ .

To determine

The probability that a person has management job, given that person is female.

Answer to Problem 44E

  $0.4$

Explanation of Solution

Given:

Total number of female is 64.5 million and male is 74.6 million.

Number of person with management job from women is 25.8 million and that of men is 25.0.

Formula Used:

For any two dependent events $A$ and B ,

  $P\left(B/A\right)=\frac{P\left(A\text{and}B\right)}{P\left(A\right)}$

Where

  $P\left(B/A\right)=$ The probability of occurring $B$ under the assumption that $A$ has been occurred.

Calculation:

Let $F$ be the event that selected person is female.

  $\begin{array}{c}n\left(F\right)=64.5\\ n\left(S\right)=64.5+74.6=139.1\\ P\left(F\right)=\frac{64.5}{139.1}\\ =0.4637\end{array}$

Let $M$ be the event that selected person has management job.

So, $M\text{and}F$ be the event that selected one is femle and has management job.

There are 25.8 millionfemale who has management job.

  $\begin{array}{c}P\left(M\cap F\right)=\frac{n\left(M\cap F\right)}{n\left(S\right)}\\ =\frac{25.8}{139.1}\\ =0.1855\end{array}$

Therefore, $M/F$ be the event that a person has management job, given that person is female.

So, the probability of the event “a person has management job, given that person is female” is calculated as follows:

  $\begin{array}{c}P\left(M/F\right)=\frac{P\left(M\text{and}F\right)}{P\left(F\right)}\\ =\frac{0.1855}{0.4637}\\ =0.4\end{array}$

The probability thata person has management job, given that person is femaleis, $0.4$ .

To determine

The probability that a person is female, given that person has management job.

Answer to Problem 44E

  $0.508$

Explanation of Solution

Given:

Total number of female is 64.5 million and male is 74.6 million.

Number of person with management job from women is 25.8 million and that of men is 25.0.

Formula Used:

For any two dependent events $A$ and B ,

  $P\left(B/A\right)=\frac{P\left(A\text{and}B\right)}{P\left(A\right)}$

Where

  $P\left(B/A\right)=$ The probability of occurring $B$ under the assumption that $A$ has been occurred.

Calculation:

Let $M$ be the event that selected person has management job.

  $\begin{array}{c}n\left(M\right)=25.8+25=50.8\\ n\left(S\right)=64.5+74.6=139.1\\ P\left(M\right)=\frac{50.8}{139.1}\\ =0.3652\end{array}$

Let $F$ be the event that selected person is female.

So, $M\text{and}F$ be the event that selected one is femle and has management job.

There are 25.8 millionfemale who has management job.

  $\begin{array}{c}P\left(M\cap F\right)=\frac{n\left(M\cap F\right)}{n\left(S\right)}\\ =\frac{25.8}{139.1}\\ =0.1855\end{array}$

Therefore, $F/M$ be the event that a person is female, given that person has management job.

So, the probability of the event “a person is female, given that person has management job” is calculated as follows:

  $\begin{array}{c}P\left(F/M\right)=\frac{P\left(M\text{and}F\right)}{P\left(M\right)}\\ =\frac{0.1855}{0.3652}\\ =0.508\end{array}$

The probability thata person is female, given that person has management jobis, $0.508$ .