Given three random variables , $X \in \mathbb R$ (a real number) , $Y\in \{0, 1\}$ , and $Z\in\{0, 1\}$ ,
let $(X\mid Y=0, Z=0)\sim\mathcal N(\mu_0, 1)$ and $(X\mid Y=0, Z=1) \sim\mathcal N(\mu_1, 1)$ ,
What can we say about the distribution of $X\mid Y=0$?
We can not say much about $P(X\mid Y=0)$ without knowing the Probability of $Z$.
We have to use this formula , where we currently have 2 "unknowns" multiplying the 2 "knowns" :
$$P(X\mid Y=0)=P(X\mid Y=0, Z=0)P(Z=0)+\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P(X\mid Y=0, Z=1)P(Z=1)$$
When $P(Z=0)$ is almost $0$ , then that term will "vanish" & the other term will "Dominate".
When $P(Z=0)$ is almost $1$ , then that term will "Dominate" & the other term will "vanish".
Other-wise , Both terms will Contribute.
Even when we want to get Probability of $X$ , we will have 4 "unknowns" :
$$P(X)=P(X\mid Y=0, Z=0)P(Y=0)P(Z=0)+\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P(X\mid Y=0, Z=1)P(Y=0)P(Z=1)+\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P(X\mid Y=1, Z=0)P(Y=1)P(Z=0)+\\ \ \ \ \ \ \ \ \ \ \ P(X\mid Y=1, Z=1)P(Y=1)P(Z=1)$$
When we know these "Multipliers" , of course , it is easy to get what we want , though that is a Different Question.