Chapter 5.2, Problem 50E

(a)

To determine

Probability for U.S senators were either Republican or male.

Answer to Problem 50E

Probability of intersection,

  $P\left(\text{R}\cup \text{M}\right)=0.84$

Explanation of Solution

Given:

  $\text{G}$ : Republican

  $\text{M}$ : Male

Probability that senators were Republicans,

  $P\left(\text{R}\right)=52\%=0.52$

Probability that senators were females,

  $P\left({\text{M}}^{\text{c}}\right)=21\%=0.21$

Probability that senators were male Republicans,

  $P\left(\text{R}\cap \text{M}\right)=47\%=0.47$

Complement rule:

  $P\left(\text{M}\right)=P\left({\left({\text{M}}^{\text{c}}\right)}^{\text{c}}\right)=1-P\left({\text{M}}^{\text{c}}\right)=1-0.21=0.79$

General addition rule:

  $\begin{array}{l}P\left(\text{R}\cup \text{M}\right)=P\left(\text{R}\right)+P\left(\text{M}\right)-P\left(\text{R}\cap \text{M}\right)\\ =0.52+0.79-0.47\\ =0.84\\ =84\%\end{array}$

Therefore,

84% of the U.S senators were either Republican or Male.

(b)

To determine

Symbolic form of the event and its probability.

Answer to Problem 50E

Symbolic form of the event,

  ${\text{R}}^{\text{c}}\cap {\text{M}}^{\text{c}}$

Probability of the event,

  $P\left({\text{R}}^{\text{c}}\cap {\text{M}}^{\text{c}}\right)=0.16$

Explanation of Solution

Given:

  ${\text{R}}^{\text{c}}$ : Democrat or Independent

  ${\text{M}}^{\text{c}}$ : Female

Symbolic representation for the randomly selected U.S senators is both female and either Democrat or Independent,

  ${\text{R}}^{\text{c}}\cap {\text{M}}^{\text{c}}$ .

Also note,

  ${\text{R}}^{\text{c}}\cap {\text{M}}^{\text{c}}$ is equivalent to ${\left(\text{R}\cup \text{M}\right)}^{\text{c}}$ .

Complement rule:

  $\begin{array}{l}P\left({\text{R}}^{\text{c}}\cap {\text{M}}^{\text{c}}\right)=P\left({\left(\text{R}\cup \text{M}\right)}^{\text{c}}\right)\\ =1-P\left(\text{R}\cup \text{M}\right)\\ =1-0.84\\ =0.16\end{array}$