Given that A and B are dependent events which of the followi

Given that A and B are dependent events, which of the following is true:

a) P(A and B) is always less than or equal to both P(A) and P(A|B)

b) P(A and B) and P(A|B) are always less than or equal to P(A)

c) P(A and B) and P(A) are always less than or equal to P(A|B)

d) Only Two of the above options are correct

where

*P(A and B) = probability that both A and B occur

*P(A|B) = probability that A occurs given B has occurred

Solution

a) This is TRUE, because

for dependent events p(A and B) = p(A)*P(B|A) = p(B)*P(A|B), and since both p(A), p(B), p(A|B) and p(B|A) being probabilities lie between 0 and 1, hence P(A and B) is less than both p(A) and p(A|B).

b) This is FALSE, because

p(A|B) can be greater than p(A), because if B has already occured then total no of possibilities reduces - ie denominator of the probability reduces hence the value of p(A|B) can be greater than p(A).

c) This is TRUE as per a) and b) given above.

d) This is TRUE as a) and c) are TRUE.