Chapter 5.2, Problem 49E

(a)

To determine

Probability that the students were either graduates or primarily used Macs.

Answer to Problem 49E

Probability that the students were graduates or primarily used Macs,

  $P\left(\text{G}\cup \text{M}\right)=0.7$

Explanation of Solution

Given:

  $\text{G}$ : Graduate student

  $\text{M}$ : Primarily uses a PC

Probability that students mainly used Macs,

  $P\left(\text{M}\right)=60\%=0.60$

Probability that school’s students were undergraduates,

  $P\left({\text{G}}^{\text{c}}\right)=67\%=0.67$

Probability respondents were graduate students and used a Mac as their main computer,

  $P\left(\text{G}\cap \text{M}\right)=23\%=0.23$

Complement rule:

  $P\left(\text{G}\right)=P\left({\left({\text{G}}^{\text{c}}\right)}^{\text{c}}\right)=1-P\left({\text{G}}^{\text{c}}\right)=1-0.67=0.33$

General addition rule:

  $\begin{array}{l}P\left(\text{G}\cup \text{M}\right)=P\left(\text{G}\right)+P\left(\text{M}\right)-P\left(\text{G}\cap \text{M}\right)\\ =0.33+0.60-0.23\\ =0.70\\ =70\%\end{array}$

Therefore,

70% of the students were either graduates or primarily the Mac users.

(b)

To determine

Symbolic form of the event and its probability.

Answer to Problem 49E

Symbolic form of the event,

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

Probability of the event,

  $P\left({\text{G}}^{\text{c}}\cap {\text{M}}^{\text{c}}\right)=0.30$

Explanation of Solution

Given:

  ${\text{G}}^{\text{c}}$ : Undergraduate student

  ${\text{M}}^{\text{c}}$ : Primarily uses a PC

Symbolic representation for the randomly selected student is an undergraduate student and primarily uses a PC,

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

Also note,

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

Complement rule:

  $\begin{array}{l}P\left({\text{G}}^{\text{c}}\cap {\text{M}}^{\text{c}}\right)=P\left({\left(\text{G}\cap \text{M}\right)}^{\text{c}}\right)\\ =1-P\left(\text{G}\cap \text{M}\right)\\ =1-0.70\\ =0.30\end{array}$