At a certain university, 20% of all students are freshmen, 18% are sophomores, 21% are juniors, and 41% are seniors. As part of a promotion, the university bookstore is running a raffle for which all students are eligible. Ten students will be randomly selected to receive prizes (in the form of textbooks for the term).
- What is the probability the winners consists of 2 freshmen, 2 sophomores, 2 juniors, and 4 seniors? I know this is 0.0305
- What is the probability the winners are split equally among underclassmen (freshmen and sophomores) and upperclassmen (juniors and seniors)?
- The raffle resulted in no freshmen being selected. This would be $1-.20^2$ right?
I'm going to assume your part (a) is correct.
For part (b): Note that the probability of an upperclassman winning is $21\%+41\%=62\%,$ which also makes the probability of an underclassman winning $38\%.$ As a result, the probability of 5 upperclassmen and 5 underclassmen winning (an even split) is $\binom{10}{5}(0.62)^5(0.38)^5\approx\boxed{0.1829}.$
For (c): The easiest way to compute this is to actually determine the probability that one of the winners isn't a freshman $(100\%-20\%=80\%),$ and then raise this probability to the tenth power for each of the $10$ winners - this gives a probability of $(0.80)^{10}\approx\boxed{0.1074}.$