Question:
A bag contains $5$ white and $4$ black balls and another bag contains $7$ white and $9$ black balls . A ball is drawn from the first bag and two balls are drawn from the second bag. What is the probability of drawing one white and two black balls?
My Approach:
Case $1$: 1 white ball from bag $1$ and $2$ balls from bag $2$ . So probability is $$\frac{\binom{5}{1}\times \binom{9}{2}}{\binom{9}{1}\times\binom{16}{2}}$$
Case $2$: I can't understand how to solve this part.
Let's draw from the first bag at first.
Then:
$$P(wbb)+P(bbw)+P(bwb)=\frac59\frac9{16}\frac8{15}+\frac49\frac9{16}\frac7{15}+\frac49\frac7{16}\frac9{15}$$
where e.g. $bwb$ stands for the event that at first a black ball is drawn (from the first bag) secondly a white ball (from the second bag) and thirdly a black ball (from the second bag).