What does this notation mean : $P(X > 4|X > 2)$.

Question

Considering a random variable defined for a probability mass function on $x = {1,2,...}$ , in a question in my textbook I encounter the notation $P(X > 4|X > 2)$. What does this mean? I understand what say $P(X > 4)$ would mean.

Answer 1

It's a "conditional probability". You should read "P(X|Y)" as "probability of X given that Y did occur". In your example: "what is the probability that $X>4$ if you already know that $X>2$?"

Also, the most important formula for such is $P(X|Y) = \frac{P(X \text{ and } Y)}{P(Y)}$

Answer 2

The notation $P(X > 4|X > 2)$ stands for: "the probability that $X>4$, given that $X>2$". It is a conditional probability and it is therefore given by the formula: $$ P(X > 4|X > 2)=\frac{P(X > 4)}{P(X > 2)} $$