A committee is formed by randomly selecting from eight nurses and two doctors. What is the minimum committee size that ensures at least a 90 % probability that it will not be comprised of nurses only?
Do we need to analyze this with cases?
Thank you!
Best Answer
There are $\binom{10}{n}$ committees of size $n$. There are $\binom{8}{n}$ nurses only committees of size $n$. So the probability a committee is nurses only is $$\frac{\binom{8}{n}}{\binom{10}{n}}.\tag{1}$$
We want the ratio (1) to be $\le 0.1$. The numbers are small enough that probably a reasonable strategy is to try $n=2,3\dots$ until we get to or under $0.1$.
It would be more efficient to work backwards from $n=8$. The probability (1) is under $0.1$ at $n=8$ and $n=7$, but not at $n=6$.