Question

Mac or PC? A recent census at a major university revealed that$40\%$ of its students primarily used Macintosh computers (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 who said that they used PCs as their

main computers. Suppose we select a student at random from among those who were part of the census.

(a) Assuming that there were $10,000$ students in the census, make a two-way table for this chance process.

(b) Construct a Venn diagram to represent this setting.

(c) Consider the event that the randomly selected student is a graduate student who uses a Mac. Write this event in symbolic form using the two events of interest that you chose in (b).

(d) Find the probability of the event described in (c). Explain your method.

Short answer

Part (a) The table is

Part (b) Venn diagram is

Part (c) Event is $P(MAC\cup Graduate)$

Part (d)$P(MAC\cup Graduate)=0.1$

Step-by-step solution

Part (a) Step 1. Given Information

According to a study, Macintosh computers were used by$40\%$ of the students (Macs). The rest mostly used computers. $67$ percent of the school's students were undergraduates at the time of the census. The rest of the group was made up of graduate students. Furthermore, $23\%$ of respondents were graduate students who stated that PCs were their primary machines.

Part (a) Step 2. Concept Used

A Venn diagram is a visual representation of relationships between things or limited groups of objects using circles. Circles that overlap have similar qualities to circles that do not overlap. Venn diagrams are graphic representations of the similarities and differences between two concepts.

Part (a)  Step 3. Explanation

MAC's share of the market is $42\%$

67 percent of students are undergraduates.

Graduates as a percentage of the total population and PC=$23\%$

In the census, there are $10,000$ students. As a result, the two-way table becomes,

Part (b) Step 1. Explanation

Venn diagram becomes

Part (c) Step 1. Explanation

The probability of a college student using MACs and graduating is represented by the probability. This can be expressed symbolically as,

$P(MAC\cup Graduate)$

Part (d) Step 1. Calculation

$Probability=\frac{Numberoffavorableoutcomes}{Totalpossibleoutcomes}$

There are a total of $20,000,000$ possible outcomes.

Therefore, $P(MAC\cup Graduate)=\frac{1000}{10000}=0.1$