Let $Y$ and $Z$ be discrete random variables and $W$ and $X$ be continuous.
Both $Y$ and $Z$ take values in {$0, 1$}.
Let $$p(w|X = x, Y = y, Z = z) = \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{(w-z)^2}{2}\right)}$$
$$p(x|Y = y, Z = z) = \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^2}{2}\right)$$
$p(Y = 0|Z = 0) = 0.1$
$p(Y = 1|Z = 0) = 0.9$
$p(Y = 0|Z = 1) = 0.3$
$p(Y = 1|Z = 1) = 0.7$
$p(Z = 0) = 0.2$
$p(Z = 1) = 0.8$
Find:$$\space p(Y = 1)$$
How would I start to solve this?
I recognize that the joint distributions are normal distributions, but not sure what to do with this information.
Hints: $$P(Y=1) = P(Y=1 \mathrm{\ and\ } Z=1) + P(Y=1 \mathrm{\ and\ } Z=0)$$ $$P(Y=1 \mathrm{\ and\ } Z=0)= P(Y=1 | Z=0)\cdot P(Z=0)$$