Chapter 4.2, Problem 1CYU

a.

To determine

Compute the probability that a randomly selected worker is a college graduate.

Answer to Problem 1CYU

The probability that a randomly selected worker is a college graduate is 0.342.

Explanation of Solution

Calculation:

The U.S workers are classified based on type of occupation and education level. The table provides the frequency in thousands for each category.

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

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

Here, event V denotes that selected worker is a college graduate. Event V consist all the college graduates with any type of occupation.

Therefore,

$\begin{array}{c}\text{Number of college graduate}=\left(\begin{array}{l}31,103,000+2,385,000+7,352,000\\ +1,033,000+1,308,000\end{array}\right)\\ =43,181,000\end{array}$

$\begin{array}{c}\text{Number of workers}=\left[\begin{array}{l}31,103,000+2,385,000+7,352,000+1,033,000+\\ 1,308,000+17,564,000+15,967,000+22,352,000+\\ 12,511,000+14,597,000\end{array}\right]\\ =126,172,000\end{array}$

Substitute 43,181,000 for “number of outcomes in V” and 126,172,000 for “Number of outcomes in sample space” in the probability formula.

Therefore,

$\begin{array}{c}P\left(V\right)=\frac{43,181,000}{126,172,000}\\ =0.342\end{array}$

Thus, the probability that a randomly selected worker is a college graduate is 0.342.

b.

To determine

Compute the probability that the occupation of randomly selected worker is either sales and office or production and transportation.

Answer to Problem 1CYU

The probability that the occupation of randomly selected worker is either sales and office or production and transportation is 0.361.

Explanation of Solution

Calculation:

Event V denotes that selected worker’s occupation is sales and office and event W denote that selected worker’s occupation is production and transportation. Event V and W are mutually exclusive since the worker can choose any one of the occupation. Event V contains workers who are college graduates and his or her occupation is sales and office and workers who are non-college graduates and his or her occupation is sales and office. Event W contains workers who are college graduates and his or her occupation is production and transportation and workers who are non-college graduates and his or her occupation is production and transportation.

The probability that the occupation of selected worker is either sales and office or production and transportation can be expressed as, $P\left(V\text{or }W\right)$.

Addition rule for mutually exclusive events:

For two mutually exclusive events V and W the addition rule states that

$P\left(V\text{or }W\right)=P\left(V\right)+P\left(W\right)$

From table, total number of workers of Sales and office is,$22,353,000+7,352,000=29,704,000$ and total number of workers of Production and transportation is $14,597,000+1,308,000=15,905,000$ .

Therefore,

$P\left(V\right)=\frac{29,704,000}{126,172,000}\text{, and }P\left(W\right)=\frac{15,905,000}{126,172,000}$.

Substitute these values in the addition rule.

Therefore,

$\begin{array}{c}P\left(V\text{or }W\right)=\frac{29,704,000}{126,172,000}+\frac{15,905,000}{126,172,000}\\ =\frac{45,609,000}{126,172,000}\\ =0.361\end{array}$

Thus, the probability that the occupation of selected worker is either sales and office or production and transportation is 0.361.

c.

To determine

Compute the probability that the occupation of selected worker is either a college graduate or has a service occupation.

Answer to Problem 1CYU

The probability that the occupation of selected worker is either a college graduate or has a service occupation is 0.469.

Explanation of Solution

Calculation:

Event V denotes that selected worker is a college graduate and event W denotes that selected worker has a service occupation. Event W contains workers who are college graduates and has a service occupation and workers who are non-college graduates and has a service occupation.

The probability that the occupation of selected worker is either a college graduate or has a service occupation can be expressed as, $P\left(V\text{or }W\right)$.

From table, total number of workers of service occupation is,$15,967,000+2,385,000=18,352,000$

General Addition rule:

For any two events V and W the general addition rule states that

$P\left(V\text{or }W\right)=P\left(V\right)+P\left(W\right)-P\left(V\text{ and }W\right)$

From part (a), $P\left(V\right)=\frac{43,181,000}{126,172,000}$.

From table $P\left(V\text{ and }W\right)=\frac{2,385,000}{126,172,000}$

$P\left(W\right)=\frac{18,352,000}{126,172,000}$

Substitute these values in the general addition rule.

Therefore,

$\begin{array}{c}P\left(V\text{or }W\right)=\frac{43,181,000}{126,172,000}+\frac{18,352,000}{126,172,000}-\frac{2,385,000}{126,172,000}\\ =\frac{59,148,000}{126,172,000}\\ =0.4687\\ =0.469\end{array}$

Thus, the probability that the occupation of selected worker is either a college graduate or has a service occupation is 0.469.