# Collection all cards in a cardgame

Imagine we have a cardset of 140. Every card has 1/140 chance of being drawn every time. When a card is drawn, it is shuffled into the deck of cards again. This would be like when you buy a set in a store, you can get the same card multiple times. How many cards do I have to draw, before I have drawn every card in the deck?

This is the coupon collector's problem. The expected number of draws is $140 (1/1 + 1/2 + \ldots + 1/140)$ or about 773. Basically, on the first draw you're guaranteed to get a card you haven't seen before. Then to get a second unique card will take on average $140/139$ draws (since there is a $139/140$ probability that you'll draw a card you haven't seen before on each draw), to get a third unique card after you've gotten the second will take $140/138$ draws, and so on, until you have 139 of the different 140 cards and it'll take on average $140$ draws to get the last card.