Chapter 9.1, Problem 31E

(a)

To determine

To explain how many of the mathematics degrees given in this year were earned by women.

Answer to Problem 31E

  $7616$ women have earned a degree.

Explanation of Solution

It is given in the question the following information as:

  $\begin{array}{l}P(\text{Bachelor})=0.73\\ P(\text{Master})=0.21\\ P(\text{Doctorate})=0.06\\ P(\text{Women | Bachelor})=0.48\\ P(\text{Women | Master})=0.42\\ P(\text{Women | Doctorate})=0.29\end{array}$

Thus we need to find the probabilities as:

  $\begin{array}{l}P(\text{Women and Bachelor})=P(\text{Women | Bachelor})\times P(\text{Bachelor})\\ =0.48\times 0.73\\ =0.3504\\ P(\text{Women and Master})=P(\text{Women | Master})\times P(\text{Master})\\ =0.42\times 0.21\\ =0.0882\\ P(\text{Women and Doctorate})=P(\text{Women | Doctorate})\times P(\text{Doctorate})\\ =0.29\times 0.06\\ =0.0174\end{array}$

Therefore, add the corresponding probabilities to get:

  $\begin{array}{l}P(\text{Women})=0.3504+0.0882+0.0174\\ =0.4560\\ =45.60\%\end{array}$

In total there are $16701$ women, so, we have,

  $\begin{array}{l}45.60\%\times 16701\\ =7616\end{array}$

Thus, $7616$ women have earned a degree.

(b)

To determine

To explain are the events “degree earned by a woman” and “degree was a master’s degree” independent.

Answer to Problem 31E

They are not independent.

Explanation of Solution

It is given in the question the following information as:

  $\begin{array}{l}P(\text{Bachelor})=0.73\\ P(\text{Master})=0.21\\ P(\text{Doctorate})=0.06\\ P(\text{Women | Bachelor})=0.48\\ P(\text{Women | Master})=0.42\\ P(\text{Women | Doctorate})=0.29\end{array}$

Thus, we need to find the probabilities as:

  $\begin{array}{l}P(\text{Women and Bachelor})=P(\text{Women | Bachelor})\times P(\text{Bachelor})\\ =0.48\times 0.73\\ =0.3504\\ P(\text{Women and Master})=P(\text{Women | Master})\times P(\text{Master})\\ =0.42\times 0.21\\ =0.0882\\ P(\text{Women and Doctorate})=P(\text{Women | Doctorate})\times P(\text{Doctorate})\\ =0.29\times 0.06\\ =0.0174\end{array}$

Therefore, add the corresponding probabilities to get:

  $\begin{array}{l}P(\text{Women})=0.3504+0.0882+0.0174\\ =0.4560\\ =45.60\%\end{array}$

Since the probabilities $P(\text{Women | Master})=0.42$ and $P(\text{Women})=45.60\%$ are not the same, the corresponding events are not independent.

(c)

To determine

To find out what is the probability that at least $1\text{ of the 2}$ degrees was earned by a woman if you choose $\text{2 of the 16701}$ mathematics degrees at random.

Answer to Problem 31E

  $70.4064\%$ .

Explanation of Solution

It is given in the question the following information as:

  $\begin{array}{l}P(\text{Bachelor})=0.73\\ P(\text{Master})=0.21\\ P(\text{Doctorate})=0.06\\ P(\text{Women | Bachelor})=0.48\\ P(\text{Women | Master})=0.42\\ P(\text{Women | Doctorate})=0.29\end{array}$

And $P(\text{Women})=45.60\%$ from part (a). thus using the complement rule we will find out the probability of individuals who earned a degree but are not women are:

  $\begin{array}{l}P(\text{Not women})=1-P(\text{Women})\\ =1-0.4560\\ =0.5440\end{array}$

And from the multiplication rule we have to find out the probability of obtaining two individuals who earned a degree but are not women as:

  $\begin{array}{l}P(2\text{Not women})=P(\text{Not women})\times P(\text{Not women})\\ =0.5440\times 0.5440\\ =0.295936\end{array}$

Now, using the complement rule, we can then find out the probability of at least one degree by women as:

  $\begin{array}{l}P(\text{at least 1 women out of 2})=1-P(2\text{Not women})\\ =1-0.295936\\ =0.704064\\ =70.4064\%\end{array}$