I'm working on the following proof:
I have three random variables, where $X\in\{0,1\}, Z\in\{0,1\}$, and $Y$ is a continuous random variable. I have two assumptions: $$P(X=1\mid Y=\bar{y})>P(X=1\mid Y=y') \text{ for some } \bar{y}>y'$$ $$\text{and}$$ $$P(Z=1\mid X=1)>P(Z=1\mid X=0)$$
My question is, are the assumptions above enough to prove that $P(Z=1\mid Y=\bar{y}) > P(Z=1\mid Y=y')?$ I've been trying to prove using Bayes' formula, but I'm getting nowhere.
Thanks!