# 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♠

-

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.

-

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

-

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?

-

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.

-