Urn I contains 6 whites and 4 blacks balls. Urn II contains 2 white and 2 black balls.
From urn I 2 balls are transferred to Urn II. A sample of size 2 is then drawn without replacement from urn II.
What is the probability that the sample will contain exactly 1 white ball?
I don't even know where to start.
Best Answer
Here's a hint.
After the transfer, you'll have either $4W$ and $2B$, $3W$ and $3B$, or $2W$ and $4B$.
So, first calculate the probability of each of these states for Urn II. (For example, to get $4W, 2B$ you'll need to draw $2W$ from Urn I. What is the probability of that happening? The probability of drawing $2W$ from Urn I is $\frac{6}{10} \frac{5}{9} = \frac{1}{3}.$)
Then, calculate the probability of drawing $1W, 1B$ from Urn II, given each of the configurations (and their associated probabilities). So, given $2W$ were drawn from Urn I, the probability of getting $1W,1B$ is
$$P((1W,1B) | 2W) = \frac{1}{3}\left[\frac{4}{6}\frac{2}{5} + \frac{2}{6}\frac{4}{5}\right] = \frac{8}{45}.$$
Then figure out the others.