Question

Suppose 60% of all college professors like tennis, 65% like bridge, and 50% like chess; 45% like any given pair of recreations. (a) should you be suspicious if told 20% like all three recreations?

Answer 1

Let T represent tennis, B represent bridge and C represent chess.

Suppose it is true that 20% of all college professors like all three recreations, then


`P(T\cup B\cup C)=1 \\ \\ P(T\cup B\cup C)=P(T)+P(B)+P(C)-P(T\cap B)-P(T\cap C)- \\ P(B\cap C)+P(A\cap B\cap C) \\ \\ =0.6+0.65+0.5-3(0.45)+0.2=1.95-1.35=0.6`

Because the union of all probability is not 1, this is not a valid distribution and hence, we would be suspicious.