In a class of 30 students, 18 study mathematics, 12 study physics, and 8 study both mathematics and physics. If a student is selected at random from those who study mathematics, what is the probability that the student also studies physics?

Step 1: Recognize this as a conditional probability problem.

We are told that the selected student already studies mathematics.

Therefore, the relevant sample space is the 18 students who study mathematics, not all 30 students.

Step 2: Identify the students who satisfy both conditions.

There are 8 students who study both mathematics and physics.

Step 3: Use the conditional probability formula.

P(physics | mathematics) = P(physics and mathematics) / P(mathematics)

Using the student counts directly:
P(physics | mathematics) = 8/18

Step 4: Simplify the fraction.

8/18 = 4/9

Step 5: Convert to a decimal if desired.

4/9 ≈ 0.4444

Therefore, given that the student studies mathematics, the probability that the student also studies physics is 4/9, or approximately 0.4444.