Chapter 9.1, Problem 33E

(a)

To determine

To find: the numbers of the mathematics degrees given that year were earned by women.

Answer to Problem 33E

10258 women

Explanation of Solution

Given:

  $\begin{array}{l}P\left(\text{Bachelor}\right)=70\%=0.70\\ P\left(\text{Master}\right)=24\%=0.24\\ P\left(\text{women/Bachelor}\right)=43\%=0.43\\ P\left(\text{women/Master}\right)=41\%=0.41\\ P\left(\text{women/doctorate}\right)=29\%=0.29\end{array}$

Formula used:

General multiplication rule:

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

Addition rule

  $P\left(A\cup B\right)=P\left(A\text{ or }B\right)=P\left(A\right)+P\left(B\right)$

Calculation:

70% of the mathematics degrees were Bachelor degrees, whereas 24% were master degrees and in total there need to have been 100% degrees. This then implies that 100%-70%-24%=6% of the mathematics degrees were doctorates.

  $P\left(\text{Doctorates}\right)=67\%=0.06$

Using the general multiplication rule:

  $\begin{array}{c}P\left(\text{women and Bachelor}\right)=P\left(\text{Bachelor}\right)\times P\left(\text{women/Bachelor}\right)\\ =0.70\times 0.43=0.301\\ P\left(\text{women and Master}\right)=P\left(\text{Master}\right)\times P\left(\text{women/Master}\right)\\ =0.24\times 0.41=0.0984\\ P\left(\text{women and doctorate}\right)=P\left(\text{Doctorates}\right)\times P\left(\text{women/doctorate}\right)\\ =0.06\times 0.29=0.0174\end{array}$

Using the addition rule

  $\begin{array}{c}P\left(\text{women}\right)=P\left(\text{women/Bachelor}\right)+P\left(\text{women/Master}\right)+\left(\text{women/doctorate}\right)\\ =0.301+0.0984+0.0174\\ =0.4168\end{array}$

Thus a proportion of 0.4168 of the 24611 degrees were earned by the women.

Which are associates with $=0.4168\times 24611=10257.864=10258$ women

(b)

To determine

To Explain: that the events are independent yes or no.

Answer to Problem 33E

Not independent

Explanation of Solution

Given:

  $\begin{array}{l}P\left(\text{Bachelor}\right)=70\%=0.70\\ P\left(\text{Master}\right)=24\%=0.24\\ P\left(\text{women/Bachelor}\right)=43\%=0.43\\ P\left(\text{women/Master}\right)=41\%=0.41\\ P\left(\text{women/doctorate}\right)=29\%=0.29\end{array}$

Formula used:

Multiplication rule:

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

Addition rule

  $P\left(A\cup B\right)=P\left(A\text{ or }B\right)=P\left(A\right)+P\left(B\right)$

Calculation:

70% of the mathematics degrees were Bachelor degrees, whereas 24% were master degrees and in total there need to possess been 100% degrees. This then implies that 100%-70%-24%=6% of the mathematics degrees were doctorates.

  $P\left(\text{Doctorates}\right)=67\%=0.06$

Multiplication rule:

  $\begin{array}{c}P\left(\text{women and Bachelor}\right)=P\left(\text{Bachelor}\right)\times P\left(\text{women/Bachelor}\right)\\ =0.70\times 0.43=0.301\\ P\left(\text{women and Master}\right)=P\left(\text{Master}\right)\times P\left(\text{women/Master}\right)\\ =0.24\times 0.41=0.0984\\ P\left(\text{women and doctorate}\right)=P\left(\text{Doctorates}\right)\times P\left(\text{women/doctorate}\right)\\ =0.06\times 0.29=0.0174\end{array}$

Using the addition rule

  $\begin{array}{c}P\left(\text{women}\right)=P\left(\text{women/Bachelor}\right)+P\left(\text{women/Master}\right)+\left(\text{women/doctorate}\right)\\ =0.301+0.0984+0.0174\\ =0.4168\end{array}$

It is find that the possibilities $P\left(\text{women}\right)=0.4168$ and $P\left(\text{women/Bachelor}\right)=0.43$ are not equal, which suggests that the likelihood of the event “Degree earned by a woman” is have an effect on once grasp that the degree was a Bachelor’s degree.

(c)

To determine

To find: the probability that at least 1 of the 2 degrees was earned by a women.

Answer to Problem 33E

0.65987776=65.987776%

Explanation of Solution

Given:

  $\begin{array}{l}P\left(\text{Bachelor}\right)=70\%=0.70\\ P\left(\text{Master}\right)=24\%=0.24\\ P\left(\text{women/Bachelor}\right)=43\%=0.43\\ P\left(\text{women/Master}\right)=41\%=0.41\\ P\left(\text{women/doctorate}\right)=29\%=0.29\end{array}$

Formula used:

Multiplication rule:

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

Addition rule

  $P\left(A\cup B\right)=P\left(A\text{ or }B\right)=P\left(A\right)+P\left(B\right)$

Complement rule:

  $P\left(A\right)=P\left(\text{not }A\right)=1-P\left(A\right)$

Multiplication rule for independent events:

  $P\left(A\cap B\right)=P\left(A\text{ and }B\right)=P\left(A\right)\times P\left(B\right)$

Calculation:

70% of the mathematics degrees were Bachelor degrees, whereas 24% were master degrees and in total there require to have been 100% degrees. This then implies that 100%-70%-24%=6% of the mathematics degrees were doctorates.

  $P\left(\text{Doctorates}\right)=67\%=0.06$

Use the general multiplication rule:

  $\begin{array}{c}P\left(\text{women and Bachelor}\right)=P\left(\text{Bachelor}\right)\times P\left(\text{women/Bachelor}\right)\\ =0.70\times 0.43=0.301\\ P\left(\text{women and Master}\right)=P\left(\text{Master}\right)\times P\left(\text{women/Master}\right)\\ =0.24\times 0.41=0.0984\\ P\left(\text{women and doctorate}\right)=P\left(\text{Doctorates}\right)\times P\left(\text{women/doctorate}\right)\\ =0.06\times 0.29=0.0174\end{array}$

Use the addition rule for disjoint events:

  $\begin{array}{c}P\left(\text{women}\right)=P\left(\text{women/Bachelor}\right)+P\left(\text{women/Master}\right)+\left(\text{women/doctorate}\right)\\ =0.301+0.0984+0.0174\\ =0.4168\end{array}$

Use the complement rule:

  $\begin{array}{c}P\left(\text{Not woman}\right)=1-P\left(\text{woman}\right)\\ =1-0.4168\\ =0.5832\end{array}$

Use the multiplication rule for independent events (assuming that the two individuals are independent):

  $\begin{array}{c}P\left(\text{Two Not woman}\right)=P\left(\text{Not woman}\right)\times P\left(\text{Not woman}\right)\\ =0.5832\times 0.5832\\ =0.34012224\end{array}$

Use the complement rule:

  $\begin{array}{c}P\left(\text{At least one women}\right)=1-P\left(\text{Two not woman}\right)\\ =1-0.3412224\\ =0.65987776\\ =65.987776\%\end{array}$