Question 8·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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The table shows the distribution of 160 interns and their departments at a company. Each intern is categorized in exactly one department.

Department	Frequency
Biology	40
Chemistry	32
Physics	24
Engineering or Computer Science	64

If one of the 160 interns is selected at random, the probability of selecting an intern who is categorized as Engineering, given that the intern is not categorized as Biology and not categorized as Physics, is $\frac{7}{24}$ . How many of these interns are categorized as Computer Science?

The phrase given that signals conditional probability: limit who could be picked before choosing the denominator. Use the given probability to find the Engineering count, then subtract it from the row that combines Engineering and Computer Science. A Desmos regression can solve the one-unknown equation. The tempting slip is to use all the interns as the denominator.

Hints

Hint 1
Given that narrows the sample space, meaning the interns who could still be picked. Remove both excluded departments. Which rows remain, and how many interns do they contain altogether?
Hint 2
A conditional probability compares the Engineering count with the eligible group, not with everyone at the company. If $x$ is the Engineering count, what fraction has $x$ on top?
Hint 3
The $64$ -intern row combines Engineering and Computer Science. After finding the Engineering count, subtract it from $64$ to get the Computer Science count.

Step-by-step

Restrict the group, then split the combined row
Step 1
Count the interns who meet the condition
The given-that condition rules out Biology and Physics. Each intern has exactly one department, so the eligible interns are the $32$ in Chemistry and the $64$ in Engineering or Computer Science. Type $32+64$ in Desmos; it shows $96$ . This is the group to pick from, not all $160$ interns.
Step 2
Write the conditional probability
Let $x$ be the number of Engineering interns. A conditional probability puts the target count over the count that meets the condition. So use all $96$ eligible interns as the denominator, but only Engineering interns as the numerator:
$\frac{x}{96}=\frac{7}{24}.$
Step 3
Find the Engineering count
Type $\frac{x_1}{96}\sim\frac{7}{24}$ in Desmos. The $\sim$ tells Desmos to find the value that makes the sides equal; $x_1$ is its solve-for version of $x$ . Under PARAMETERS, Desmos shows $x_1=28$ . So $28$ counts Engineering interns, not Computer Science interns.
Step 4
Find the Computer Science count
The table combines Engineering and Computer Science into $64$ interns. Subtract the Engineering count: type $64-x_1$ beneath the regression. Desmos shows $36$ , so $36$ interns are categorized as Computer Science. Choice A.

Question 8·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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