Is $p(X \in A|\frac{Y+Z}{2}) = p(X \in A|Y,Z)?$

Question

Let $X,Y, and \space Z$ be random variables. Let $A$ be a subset of $U$ such that $p(X \in U)=1$

Is $p(X \in A|\frac{Y+Z}{2}) = p(X \in A|Y,Z)?$

Do these two expressions represent the same thing? If so why is that true? Thank you!

Edit: Fixed notation according to Nicholas R. Peterson's tips

Answer 1

Hint: Clearly, if you know $Y$ and $Z$, then you know their average; is the reverse true?

Try thinking about it in this light, and see if you can come up with an answer. Let me know if you get stuck.

With that said, let me interject here that your notation is no good; are you intending $p$ to be a probability measure, or a density function?

If it is a probability measure, then this is meaningless: you can't compute the probability of a random variable... you need to compute the probability of an event. So, it could be something like $P(X\in A\mid\frac{Y+Z}{2})$, etc.