Question

1.2 In a group of 60 students, 20 study history, 24 study French and 8 study both history and French.
Are the events ‘a student studies History’ and ‘a student studies French ‘independent?

Answer 1

Let 𝐴 be the event ‘a student studies History’ and let 𝐵 be ‘a student studies French’. These two events are independent, by definition, if probability of their intersection equals to product of separate probabilities, 

$$P(A\cap B)=P(A)P(B)$$

We have that

$$P(A)={20 \over 60}={1 \over 3}$$

$$P(B)={24 \over 60}={2 \over 5}$$

$$P(A\cap B)={8 \over 60}={2 \over 15}$$

$$P(A)P(B)={1 \over 3}\cdot {2 \over 5}={2 \over 15}=P(A\cap B)$$

Therefore, the events ‘a student studies History’ and ‘a student studies French ‘ are independent.