# Conditional probability of two conditional independent events in Bayesian Networks

I have a common Bayesian Networks example

I need to estimate probablity of JohnCalls given MaryCalls. Other words I need to find

$P(JohnCalls = True|MaryCalls=True)$

From product rule, I know that

$P(JohnCalls = True|MaryCalls=True) = \dfrac{P(JohnCalls, True|MaryCalls=True)}{P(MaryCalls=True)}$

But how can I calculate $P(JohnCalls, True|MaryCalls=True)$ I understand, that it should depend on $Alarm$ variable, but I don't know how.

The equation you have immediately after "From the product rule, I know that..." is incorrect. The definition of conditional probability states that for two random variables $X$ and $Y$,
$$P(X = x | Y= y) = \frac{P(X=x, Y=y)}{P(Y=y)}.$$
$$P(JohnCalls = True|MaryCalls=True) = \dfrac{P(JohnCalls = True, MaryCalls=True)}{P(MaryCalls=True)}.$$
Note that the expression $P(JohnCalls, True|MaryCalls=True)$ doesn't even really make sense.