Probability, balls and 2 kinds of boxes There are 5 boxes. In two of them there are 5 white and 3 black balls, in three of them there are 4 white and 6 black balls. We pick out randomly one box and choose 3 balls. What is the probability that exactly 2 out of these 3 balls are white?
At the beginning in these 2 boxes there is 5/8 positive result and in other 3 boxes there is 4/10 positive result. Further I don't really understand what to do.
 A: The event "exactly $2$ white out of $3$" can happen in two disjoint ways: (i) We chose a $5$ white $3$ black box and got $2$ white out of $3$; or (ii) We chose a $4$ white $6$ black box and got $2$ white out of $3$.
To find the answer to our problem, we calculate the probability of (i), calculate the probability of (ii), and add up.
We calculate the probability of (i) and leave (ii) to you.
The probability we chose a $5$ white $3$ black box is $\dfrac{2}{5}$. We calculate the probability of getting $2$ white $1$ black, given that we are sampling from a $5$ white $3$ black box. 
There are $\dbinom{8}{3}$ equally likely ways to choose $3$ balls from an $8$-ball box. Now count the number of choices that have $2$ white $1$ black. The $2$ white can be chosen in $\dbinom{5}{2}$ ways. For each of these ways, the black ball can be chosen in $\dbinom{3}{1}$ ways. So there are $\dbinom{5}{2}\dbinom{3}{1}$ ways to choose $2$ white and $1$ black.
It follows that the probability of (i) is
$$\frac{2}{5}\frac{\binom{5}{2}\binom{3}{1}}{\binom{8}{3}}.$$ 
