There're 9 books in a library. 5 are used, 4 are new. (all the books are the same)
3 students in January rent one book each and return it the same month. (means if they rented a new, when they return it, it becomes used).
3 new students in June rent one book each. What's the probability that at least one student will rent a used book in June?
I know how to solve it for a single student, but the fact that there are 3 students is confusing me.
In January, there are several options that are mutually exclusive:
The students all rent new books, so after that 8 are used, 1 is new. In that case there will always be a student in June that gets a used book. The probability of all three students getting new books: $\frac{4}{9}\frac{3}{8}\frac{2}{7}$.
Two students rent new books, one gets a used book. After that 7 are used, 2 are still new, and we also have that there will always be a student in June that gets a used book. The probability of the former is $\frac{\binom{5}{1}\binom{4}{2}}{\binom{9}{3}}$
One student rents a new book, two get a used one. After this 6 are used, 3 are new. The probability for this is $\frac{\binom{4}{1}\binom{5}{2}}{\binom{9}{3}}$. This has to be multiplied by $1$ minus the probability that all three students in June get a new book (which is now possible), so times $1 - \frac{3}{9}\frac{2}{8}\frac{1}{7}$.
They all get used books, which happens with chance $\frac{\binom{5}{3}}{\binom{9}{3}}$. Then there are still 4 new books in June, and the chance that we only rent new books then is $\frac{\binom{4}{3}}{\binom{9}{3}}$, so we have to multiply with $1 - \frac{\binom{4}{3}}{\binom{9}{3}}$ to get the chance at least one has a used book.
Now add up these probabilities.