What is the expected number of times you would draw to have seen every card in a deck assuming a draw consists of a random sample of exactly 26 cards?

Question

You would obviously need at least at least two draws but probably wouldnt expect seeing every card even after three draws. I took combinatorics but forget how to solve this.

Answer 1

It seems you’re referring to a $52$-card deck.

Your comment under the question contradicts the title of the question. Since the expected number of draws required to see all cards (asked for in the title) is easier to calculate than the number of draws required to have a $50\%$ chance to see all cards (asked for in the comment), I’ll compute the former. The formula for this is derived in the question Mike Earnest linked to, Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen. In our case, $n=52$ and $m=26$, so the expected number of required draws is (Wolfram|Alpha computation)

$$ \sum_{j=1}^{52}(-1)^{j-1}\binom{52}j\frac1{1-\frac{\binom{52-j}{26}}{\binom{52}{26}}}\approx7.1\;. $$