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 and learn that the person mainly uses a Mac. Find the probability that the person is a graduate student.

Short answer

The probability that the randomly selected person is a graduate student is approx$0.3833.$

Step-by-step solution

Given information

$60\%$of the students mainly use MACs.

$67\%$are undergraduates.

$23\%$are graduates who mainly use a MAC.

Explanation

G: for graduates

U: for undergraduates

MAC: for MAC users

The probabilities are as follows:

Probability for MAC users,

$P\left(\mathrm{MAC}\right)=0.60$

Undergraduates' probability,

$P\left(\mathrm{U}\right)=0.67$

Graduates and MAC users have a higher chance of succeeding.

$P\left(\text{Gand}MAC\right)=0.23$

Because the person who was chosen at random primarily uses a Mac.

Use the conditional probability formula:

$P(\mathrm{G}\mid \mathrm{MAC})=\frac{P\left(\text{Gand MAC}\right)}{P\left(\text{MAC}\right)}=\frac{0.23}{0.60}=\frac{23}{60}\approx 0.3833$

Therefore, the probability that the randomly selected person is a graduate student is approx$0.3833.$