The Venn diagram shows the number of students studying Mathematics (M) and the number of students studying Physics (P) in a college - OCR - GCSE Maths - Question 14 - 2018 - Paper 1

To find the conditional probability that a student selected at random studies Mathematics (M) given that they study Physics (P), we use the formula:

$P(M|P) = \frac{P(M \cap P)}{P(P)}$

Where:

• $P(M \cap P)$ is the number of students studying both Mathematics and Physics, which is x (found to be 27).
• $P(P)$ is the total number of students studying Physics, which is the sum of students studying only Physics and those studying both:

$P(P) = ext{Only P} + ext{Both} = 18 + 27 = 45$

Now, substituting the values into the probability formula:

$P(M|P) = \frac{27}{45}$

This fraction can be simplified:

$P(M|P) = \frac{3}{5}$

Thus, the probability that a randomly selected student studies Mathematics given that they study Physics is ( \frac{3}{5} ).