A bin has $n$ balls. $b$ balls are black and $w$ balls are white. I pick randomly $k$ balls, where $k$ can be $1$, $2$ or $3$ with equal probability. What is the probability that all selected balls are black?
My approach was as follows. Let $S$ define an event that all selected balls are black. $$Pr(S) = \frac{1}{3}\left(\Pr( \text{all_black} | \text{picked 1 ball}) + \Pr( \text{all_black} | \text{picked 2 balls}) + \Pr( \text{all_black} | \text{picked 3 balls}) \right) = \frac{1}{3}\left(\frac{b \choose 1}{n \choose 1} + \frac{b \choose 2}{n \choose 2} + \frac{b \choose 3}{n \choose 3}\right)$$
Is this correct?