If X and Y are two independent random variables then P[X ≥ 2|Y > 1] < P[X ≥ 2].true or false ? justify

Answer 1

False. When X and Y are independent, the conditional probability becomes a marginal probability. In this case, P(X >= 2 | Y > 1) = P(X >= 2). So the inequality doesn't hold.

In general, if the events A and B are independent, then P(A|B) = P(A).

Answer 2

This is false. The reason is that, if they are independent, then the outcome of event Y will not affect the outcome of event X. That being the case, then any probability regarding the outcome of X will be the same, regardless of what happens with Y. So P[X ≥ 2|Y > 1] = P[X ≥ 2].