There are $10$ balls each of $3$ colors ($30$ balls total). We pick $20$ balls. What is the probability that $20$ balls represent $3$ unique colors?
My answer: $\dfrac{27\choose{17}}{30\choose{20}} = 0.2808$
This is obviously wrong, because this number should be greater than $0.5$ (intuitively), as the probability of picking balls with $2$ unique colors will obviously be less than $0.5$, and the probability of picking balls with $1$ unique color is obviously $0$.
Explanation: I pick $3$ balls, $1$ each of $3$ unique colors and then pick the remaining $20 – 3 = 17$ balls out of $30 – 3 = 27$ balls.
Why is the answer wrong? What will be the correct answer?
Best Answer
For this type of questions, I often find myself looking at the opposite event: what is the probability that the twenty balls contain less than three colors? Obviously, since there are ten balls of each color, at least two colors will be represented. There are only three ways to select twenty balls containing two colors only (namely, by choosing one of the three colors not to appear in the selection of twenty balls). We thus find that the probability of selecting two colors equals:
$$\frac{{3 \choose 1}{20 \choose 20}}{30 \choose 20} = \frac{3}{30045015}$$
As such, the probability of all three colors being represented equals:
$$1 - \frac{3}{30045015} \approx 0.99999990015$$