# card matching

• posted

I) Given an infinite deck of cards, each distinct, all ranked. Lay them out in a row, face up, left to right, in order.

Shuffle a second, identical deck. Deal it one card at a time, face up, directly below the first.

On average, how many pairs match?

II) Given an infinite deck, composed of replicates of a standard 52 card deck; that is, an infinite number of copies. Arrange it into separate decks, each ordered by suit and rank. Lay them out as above, face up, in a single row.

Shuffle another identical deck. As before, deal it out below the first, one card at a time.

On average, what fraction of the pairs match?

• posted

For each card the chance of matching is 1 / N where N is the number of cards in the deck. For any card to match multiply by N, but that's not quite right since it doesn't consider the chances of multiple matches.

In this case the chances of a match are always 1/52 since consuming a card still leaves an infinite number of cards. So the total number of matches over the infinite deck is infinity / 52.

• posted

Too easy, hey? So the average is N/N = 1 match. Even with an infinite deck!

Good catch, I forgot about multiple matches. Don't know the answer exactly, but it must involve an infinite series.

Right, but again, the correct answer must account for multiples.

• posted

Actually, the multiple match issue is a red herring since we are counting the average number of matches and not the chance of a match. I was not thinking clearly when I wrote that about the first case. The only issue is if the Nth card in deck A matches the Nth card in deck B counting the total number of expected matches over the infinite deck.

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