Question

Mac or PC? A recent census at a major university revealed that$60\%$of its students mainly used Macs. The rest mainly used PCs. At the time of the census, $67\%$of the school’s students were undergraduates. The rest were graduate students. In the census, $23\%$of respondents were graduate students and used a Mac as their main computer. Suppose we select a student at random from among those who were part of the census. Define events G: is a graduate student and M: primarily uses a Mac.

a. Find P(G ∪ M). Interpret this value in context.

b. Consider the event that the randomly selected student is an undergraduate student and

primarily uses a PC. Write this event in symbolic form and find its probability.

Short answer

(a) The value of P$\left(G\cup M\right)$is$0.70$.

(b) The symbolic form is $P\left(U\cap P\right)$and the probability is$0.30$.

Step-by-step solution

Part (a) Step 1: Given Information

We are given the values of the students who are undergraduate and who use Macs and also those who are both graduate and use Macs, and we have to find out the probability of those who are either graduate or use Macs.

Part (a) Step 2: Explanation

Applying the union rule of probability

which is$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)+P(A\cap B)$,

Here, P (G)$=1-P\left(U\right)=1-0.67=0.33$, P (M)$=0.60$and $P(G\cap M)=0.23$

Put all the values in the rule.

We get $P\left(G\cup M\right)=0.33+0.60-0.23=0.70$

Hence, the probability of those who are graduates or using Macs is$0.70$

Part (b) Step 1: Given Information

We are given the values of the students who are undergraduate and who use Macs and also those who are both graduate and use Macs, and we have to find out the probability of those who are graduate or use Macs.

Part (b) Step 2: Explanation

Applying the union rule of probability,

which is$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)+P(A\cap B)$

Here, the probability of students who are undergraduates and use primarily PCS is denoted by $P\left(U\cap P\right)$. Now, to find its value,

P(U)$=0.67$, P(P)$=1-P\left(M\right)=1-0.60=0.40$

and P$(U\cup P)=1-0.23=0.77$

Put all the values in the union rules.

we get$P\left(U\cap P\right)=0.67+0.40-0.77=0.30$.

Hence, the probability is$0.30$.