Q:

Bayes theorem confusion...

A denotes event that present was hidden by mom.
B denotes event that present was hidden by dad.
E denotes event that present was hidden upstairs.
F denotes event that present was hidden downstairs.
P(A)=.6
P(B)=.4
P(E|A)=.7
P(F|A).3
P(E|B)=.5
P(F|B)=.5
Find P(E)
Can you explain why the answer is:
P(E) = P(A)P(E|A)+P(B)P(E|B)
     = (.6)(.7)+(.4)(.5)
     = .42 +.2
     = .62

A:

The present could be upstairs in two different (and disjoint) ways: (i) Mom hid the hiding and hid the present upstairs or (ii) Dad did the hiding and hid it upstairs.
What is the probability of (i)? In symbols, it is $\Pr(A\cap E)$. But we have the general formula
$$\Pr(X|Y)\Pr(Y)=\Pr(X\cap Y)=\Pr(Y\cap X).$$
Putting $X=E$ and $Y=A$ gives us $\Pr(A\cap E)$. 
Do a similar calculation for the probability of (ii), that is, $\Pr(B\cap E)$.
Then add up.

