# probability - 2 cards with same rank

From a deck of 52 cards,What's the probability that he gets a combination of 2 cards with same rank. Eg: 3♥ 3♠

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The "first" card doesn't matter, as only the second card has to have the same rank. After removing one card, there are 51 cards left in the deck. 3 of them have the same rank as the card that was removed. Hence, the probability of getting dealt a pair is 3/51 = 1/17.

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Hint: First you pick a card. Then you have to pick a second one with the same rank to make a pair.

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There are $13$ ranks. In each rank there are $4$ cards, and the number of ways of choosing $2$ cards from a set of $4$ is $\binom42$. Thus, there are $13\binom42$ pairs of cards of the same rank. How many pairs of cards are there altogether? And what do you do with these two numbers to get the desired probability?

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There are the same number of possibilities of getting a pair as there are of selecting $2$ elements from $13$ piles. To see this, you can partition the deck of cards into $13$ piles of $4$ cards each, such that picking any $2$ piles gives you exactly one pair, and such that each pair can be obtained in this way.

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The probability of getting a pair is expressed as such... First you have to pick any card. The chance you pick a card is 52/52. Then you need to pick the same card again. There are 3 cards of the same rank left so the probability you pick one is 3/51, because after the first card is picked, it isn't replaced. You then multiply 52/52 times 3/51 to get your answer.

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