An urn contains five white and four black balls. Four balls are randomly selected and
transferred to a second urn. A ball is then drawn from the second urn, and it happens
to be black. Find the probability of drawing a white ball from the remaining three in
the second urn
I am attempting this using conditional probability. To calculate $P(W | B)$,
\begin{align*}
P(W | B) = \frac{P(B|W)P(W)}{P(B)}.
\end{align*}
$P(W)$ is the probability of there being a white ball in the second urn, which is 5/9. $P(B)$ is the probability of a black ball being in the second urn, which is 4/9. $P(B | W)$ is the probability of there being a black ball in the urn assuming there's already a white one, which I think is 1/2 because out of the remaining 8 balls in urn 1 exactly 1/2 are black.
Putting this together we get $P(W|B) = (1/2)(5/9)/(4/9) = 5/8$.
How does this look?
Best Answer
You can directly calculate $P(W|B)$. To understand it better assume that you skip the second urn part and you directly draw 4 balls from the first urn one by one. As after the first draw we have a black ball, we're left with $5$ white and $3$ balls. The probability of drawing at least one white ball in the three attempts is "one minus" the probability of all of the three draws being black. Hence:
$$P(W|B) = 1 - \frac{3}{8} \times \frac{2}{7} \times \frac{1}{6} = \frac{55}{56}$$