Answer to Question #213195 in Statistics and Probability for livy

Question

Question 1.2 [4] An online survey of a large number of soccer fans determined that 65% follow European competitions and 55% follow local (South African) competitions while all of the fans follow at least one of the two activities. What proportion of soccer fans follow European competitions exclusively?

Answer 1

A=follow European competitions

P(A)=0.65

B=follow local (American) competitions

P(B)=0.55

It is given that all of the fans follow at least one of the two activities. It means

$P(A \cup B) = 1$

By addition theorem

$P(A \cup B) = P(A) + P(B) -P(A \cap B) \\ P(A \cap B) = 0.65 + 0.55 -1 \\ P(A \cap B) = 0.2$

The proportion of soccer fans follow European competitions exclusively is computed as follows:

$= P(A) -P(A \cap B) \\ = 0.65-0.20 \\ = 0.45$

45% of soccer fans follow European competitions exclusively.