Chapter 4.3, Problem 44E

a.

To determine

Compute the probability that selected person is female.

Answer to Problem 44E

The probability that selected person is female is 0.4637.

Explanation of Solution

Calculation:

There were a total of 64.5 million women and 74.6 million men. Among the women 25.8 had management job and 25 million men had management job.

Consider, $1\text{million}={\text{10}}^6$ .

The probability of the event A can be obtained by the formula:

$P\left(A\right)=\frac{\text{Number of outcomes in }A}{\text{Number of outcomes in}\text{sample space}}$

Here, event A denotes that selected person is female.

$\begin{array}{c}\text{Total person}=64,500,000+74,600,000\\ =139,100,000\end{array}$

Among the 139.1 million persons 64.5 million were women.

Substitute 64,500,000 for “number of outcomes in E” and 139,100,000 for “Number of outcomes in sample space” in the probability formula.

Therefore,

$\begin{array}{c}P\left(A\right)=\frac{64,500,000}{139,100,000}\\ =0.46369\\ =0.4637\end{array}$

Thus, the probability that selected person is female is 0.4637.

b.

To determine

Compute the probability that selected person has a management job.

Answer to Problem 44E

The probability that selected person has a management job is 0.3652.

Explanation of Solution

Calculation:

Here, event B denotes that selected person has a management job. Total number of students is 139,100,000. Among the women 25.8 had management job and 25 million men had management job.

$\begin{array}{c}\text{Number of persons have management job}=25,800,000+25,000,000\\ =50,800,000\end{array}$

Substitute 50,800,000 for “number of outcomes in B” and 139,100,000 for “Number of outcomes in sample space” in the probability formula.

Therefore,

$\begin{array}{c}P\left(B\right)=\frac{50,800,000}{139,100,000}\\ =0.3652\end{array}$

Thus, the probability that selected person has a management job is 0.3652.

c.

To determine

Compute the probability that selected person is female and has a management job.

Answer to Problem 44E

The probability that selected person is female and has a management job is 0.1855.

Explanation of Solution

Calculation:

Here, event ‘A and B’ denotes that selected person is female and has a management job. Among the 139,100,000 persons 25,800,000 of female has a management job.

Substitute 25,800,000 for “number of outcomes in ‘A and B’” and 139,100,000 for “Number of outcomes in sample space” in the probability formula.

Therefore,

$\begin{array}{c}P\left(A\text{ and }B\right)=\frac{25,800,000}{139,100,000}\\ =0.18547\\ =0.1855\end{array}$

Thus, the probability that selected person is female and has a management job is 0.1855.

d.

To determine

Compute the probability that selected person has a management job given that the person is female.

Answer to Problem 44E

The probability that selected person has a management job given that the person is female is 0.4.

Explanation of Solution

Calculation:

The formula for conditional probability is given by:

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

Event A denotes denote that person is female and event B denotes that selected person has a management job.

From part (a) it is clear that, $P\left(A\right)=0.4637$.

From part (c) it is clear that, $P\left(A\text{and}B\right)=0.1855$.

Substitute these values in the formula for conditional probability.

Therefore,

$\begin{array}{c}P\left(B|A\right)=\frac{0.1855}{0.4637}\\ =0.4000\end{array}$

Thus, the probability that selected person has a management job given that the person is female is 0.4.

e.

To determine

Compute the probability that selected person is female given that person has a management job.

Answer to Problem 44E

The probability that selected person is female given that person has a management job is 0.5079.

Explanation of Solution

Calculation:

The formula for conditional probability is given by:

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

Event A denotes that person is female and event B denotes that selected person has a management job.

From part (b) it is clear that, $P\left(B\right)=0.3652$.

From part (c) it is clear that, $P\left(A\text{and}B\right)=0.1855$.

Substitute these values in the formula for conditional probability.

Therefore,

$\begin{array}{c}P\left(A|B\right)=\frac{0.1855}{0.3652}\\ =0.5079\end{array}$

Thus, the probability that selected person is female given that person has a management job is 0.5079.