a.) There are $4$ distinct items in the set. What is the probability of picking all $4$ items after picking $n\ge4$.
b.) How many items do you need to pick to collect all four with a probability of at least $.9$.
My answer for part a. (which I think is wrong):
The sample space is the all the possible ways of picking $n$ items from the set of $4$ items. That number is equal to the number of non negative integer-valued solutions to this problem:
$a_1+a_2+a_3+a_4=n$
that number is $\binom{n+4-1}{4-1}=\binom{n+3}{3}$
All possible ways of picking $n$ items and having at least 1 of each of the $4$ items is the number of positive integer-valued solutions to this problem:
$a_1+a_2+a_3+a_4=n$
That number is $\binom{n-1}{4-1}=\binom{n-1}{3}$
So the probability of picking all $4$ items after picking $n$ items is:
$\frac{\binom{n-1}{3}}{\binom{n+3}{3}}$
Using this answer, I find that the probability of picking all $4$ items after picking $4$ random items is:
~$.02857\ne\frac{4}{4}\frac{3}{4}\frac{2}{4}\frac{1}{4}$
Please explain where I went wrong.