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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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up vote 5 down vote accepted

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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