Question

78 refer to the following setting. In the language of government statistics, you are "in the labor force" if you are available for work and either working or actively seeking work. The unemployment rate is the proportion of the labor force (not of the entire population) that is unemployed. Here are estimates from the Current Population Survey for the civilian population aged 25 years and over in a recent year. The table entries are counts in thousands of people.

Unemployment Suppose that you randomly select one person 25 years of age or older.

a. What is the probability that a randomly chosen person 25 years of age or older is in the labor force?

b. If you know that a randomly chosen person 25 years of age or older is a college graduate, what is the probability that he or she is in the labor force?

c. Are the events "in the labor force" and "college graduate" independent? Justify your answer.

Short answer

(a) The probability that a randomly chosen person 25 years of age or older is in the labor force $67.035\%$

(b) The probability that he or she is in the labor force$78.30\%$

(c)Are the events "in the labor force" and "college graduate" not independent

Step-by-step solution

Part (a) Step 1: Given Information

Given

Formula used:

$\text{Probability}=\frac{\text{favorable outcomes}}{\text{possible outcomes}}$

Part (a) Step 2: Simplification

The table contains $27,669+59,860+47,556+51,582=186,667$people in total, where $12,470+37,834+34,439+40,390=125,133$of the 186,667 people are in the labour force.

$P\left(Inthelaborforce\right)=\frac{\text{favorable outcomes}}{\text{possible outcomes}}$

$$\begin{gathered} =\frac{125,133}{186,667} \\ =0.67035 \\ =67.035\% \end{gathered}$$

Part (b) Step 1: Given Information

Probability$=\frac{\text{favorable outcomes}}{\text{possible outcomes}}$

Part (b) Step 2: Simplification

In the row "college graduate" and the column "Total population" of the provided, there are 51582 college graduates. In the following table, the row "college graduate" and the column "in labour force" show that 40390 of the 51582 college graduates are employed.

$P\left(Inlaborforcecollegegraduate\right)=\frac{\#\text{of favorable outcomes}}{\#\text{of possible outcomes}}$

$$\begin{gathered} =\frac{40,390}{51,582} \\ =\frac{20,195}{25,791} \\ =0.7830 \\ =78.30\% \end{gathered}$$

Part (c) Step 1: Given Information

Formula used:

Probability$=\frac{\text{favorable outcomes}}{\text{possible outcomes}}$

Part (c) Step 2: Simplification

The table contains $27,669+59,860+47,556+51,582=186,667$people in total, while $12,470+37,834+34,439+40,390=125,133$of the 186,667 people are in the labor force.

$P\left(Inthelaborforce\right)=\frac{\text{\# of favorable outcomes}}{\#\text{of possible outcomes}}$

$$\begin{gathered} =\frac{125,133}{186,667} \\ =0.67035 \\ =67.035\% \end{gathered}$$

In the row "college graduate" and the column "Total population" of the provided table, there are 51582 college graduates. In the row "college graduate" and the column "in labour force" of the following table, 40390 of the 51582 college graduates are employed.

$P\left(Inlaborforcecollegegraduate\right)=\frac{\#\text{of favorable outcomes}}{\#\text{of possible outcomes}}$

$$\begin{gathered} =\frac{40,390}{51,582} \\ =\frac{20,195}{25,791} \\ =0.7830 \\ =78.30\% \end{gathered}$$

Two occurrences are independent if the likelihood of one event occurring has no bearing on the likelihood of the other event occurring.

Because the probabilities P (in the labour force college graduate) are not equal, the probability of the event "in the labour force" is impacted when the event "college graduate" occurs, and the events are thus not independent.