Conditional Probability Proof Question

Question

I have not used this website in a while. I just had a question about probability.

Let B be an event of a sample space Ω with P(B) > 0. For a subset A of Ω, define Q(A) = P(A|B). For E and F, events of Ω (with P(F ∩ B) > 0), show that Q(E|F) = P(E|F ∩ B).

I am not sure how I can use the definition of conditional probability to solve it. Can someone give me a hint and tips?

Thank you

Answer 1

From one side, $$ \mathbb Q(E\mid F)= \frac{\mathbb Q(E\cap F)}{\mathbb Q(F)} = \frac{\mathbb P(E\cap F\mid B)}{\mathbb P(F\mid B)} =\text{can you continue from here?} \tag{1}\label{1} $$ From the other side, $$ \mathbb P(E\mid F \cap B) = \frac{\mathbb P(E\cap F\cap B)}{\mathbb P(F \cap B)} \tag{2}\label{2} $$ Rewrite \eqref{1} by definition of conditional probability and compare with \eqref{2}.