"Of three urns, the first contains 2 white and 4 black cards, the second contains 8 white and four black cards, and the third contains 1 white and 3 black card. If one card is selected from each urn, find the probability that exactly 2 white cards are drawn"

2. Use Poisson's rule.
Let $A_1$: the card drawn from the first urn is white.
$A_2$: the card drawn from the second urn is white.
$A_3$: the card drawn from the third urn is white.
We have:
$p_1=p(A_1)=\frac{1}{2}, \ q_1=\frac{1}{2}$
$p_2=p(A_2)=\frac{2}{3}, \ q_2=\frac{1}{3}$
$p_3=p(A_3)=\frac{1}{4}, \ q_3=\frac{3}{4}$

So, the probability that exactly two cards are white is equal to the coefficient of $x^2$ in the expression
$(p_1x+q_1)(p_2x+q_2)(p_3x+q_3)$
That means $p_1p_2q_3+p_1q_2p_3+q_1p_2p_3$.
Now, plug in $p_1,p_2,p_3,q_1,q_2,q_3$.