The Venn diagram shows the number of students studying al Mathematics (M) and the number of students studying Physics (P) in a college. 35 students do not study either subject. 8 The total number of students is 121. It can be shown that x=27. One of the 121 students is selected at random. Find the probability that this student studies Mathematics, (2 marks)

## Step1: In such a problem where two events are independent, we need to use the formula of Conditional Probability. The Conditional Probability is, the likelihood of an event or circumstance happening, based on whether another related event or circumstance has yet occurred or hasn't.## Step2: Let's denote event "student studies Mathematics" as and event "student studies Physics" as . Given our represents the group studying both Physics and Mathematics, we shall express them as events . ### The formula for conditional probability is \(\textbf {P(M|P) = \frac {P( M \cap P)}{P(P)}}\)## Step3: To get \(P(M|P)\) – the probability that this student studies Mathematics, given they study Physics, we need to determine both \(P(M \cap P) \) and \(P(P)\). - \(P( M \cap P)= \frac{27}{121}\) is the probability of picking a student who studies both Math and Physics. This is obtained by dividing (those studying both Math and Physics) by total number of students which is 121.- \(P(P)\) corresponds to the probability of picking a student who studies Physics, which is obtained by the summing up all students who study Physics including those who also study Maths and then divided by the total number of students i.e \(\frac{(x + (121 - (x + (35 + x))))}{121} = \frac { 79 }{121 }\)## Step4: We substitute these values into the conditional probability formula to get the final answer.