1. The problem statement, all variables and given/known data The problem I'm having involves looking for a specific formula for the probability of matching a deck of playing cards via the result of a shuffle. I want to know how many times I must shuffle for the probability of the shuffle matching one of the past resulting decks to be at least 50%. 2. Relevant equations At first, I considered the binomial equation, but because each shuffle introduces a new deck to compare subsequent shuffles to I found myself at a loss. So the equation I have examined is P=1-(1-p)^n 3. The attempt at a solution I thought about the probabilities step-wise looking for a pattern. For example, say I do the same with a 32 sided dice. At first, I choose any side to be the one I am looking for a match. So in one roll, the probability of rolling a match is 1/32. But in two rolls, it would be (1/32)+(31/32)(2/32), because it assumes 31 out of 32 times I need to reroll, but now I have two sides I will consider a success. For 3 rolls, it would be (1/32)+(31/32)(2/32)+(31/32)(31/32)(3/32)? Correct? I just don't know whether my take on probability is correct and would love any comment.