Dealt 3 cards. Odds of being dealt any pair?

Question

This is not to aid a gambling habit. I am simply curious how to do this math.
You get dealt 3 cards. What are the odds of having any pair? (We can exclude 3 of a kind)

Total number of hands = $\begin{pmatrix}52 \\ 3\end{pmatrix}$ = 22100

What do I do next? (Added from response below)

How many ways can I get a pair of 2's, for example? $\begin{pmatrix}4 \\ 2\end{pmatrix}$ = 6 And there are 13 types of pairs I can get. So, 13x6 = 72. So, there is only a 72/22100 chance of being dealt a pair?

Supplemental: If there are 5 players, what are the odds at least 1 person is holding a pair?

Answer 1

Let's count the hands where you don't get a pair: For the first card, there are $52$ possibilities, for the second there are $48$ (since the cards of the first type are forbidden now) and for the third hand there are $44$ possibilities. Now for the hand, the order doesn't matter, so there are $$\frac{52\cdot 48\cdot 44}{6} = 18304$$ hands without a pair. Now the number of hands with a pair is $$\binom{52}{3} - 18304 = 3796.$$ Thus, the probability of getting a hand with a pair is $$\frac{3796}{22100} \approx 17.2\%.$$

EDIT: The above solution counts three of a kind as a hand with a pair. In the case that $3$ of a kind are forbidden, we have to subtract the $$\frac{52\cdot 3\cdot 2}{6} = 52$$ hands with $3$ of a kind. Now there are $$3796 - 52 = 3744$$ "good" hands, so the probability of getting a pair, but not three of a kind, is $$\frac{3744}{22100} \approx 16.9\%.$$

Answer 2

• Number of total combinations of 3 cards = $52\choose3$
• Combinations of 3 cards with only a pair (and not a three of a kind) = $13\times {4\choose2}\times 48$ (There are 13 possible ranks of cards (from Ace to King) that could form the pair, ${4\choose2}$ possible ways in which that pair could be formed by different suits, and 48 possible cards for the other card - all except the card that forms the pair)

Thus, the probability of getting only a pair (and not a three-of-a-kind) is $p_{pair}=\frac{13\times {4\choose2}\times 48}{52\choose3} = \frac{72}{425}$

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For the extension question, @Azimut pointed out that my approach below is incorrect. I've left it here for posterity.

For $n$ people in general, the probability that at least one person has a pair is equal 1 subtracted by the probability that no person has a pair. The probability that each person does not have a pair is $(1-p_{pair})$, and for $n$ people not to have a pair it would be $(1-p_{pair})^n$

For your given example, the required probability would be

$$1-(1-p_{pair})^5\approx0.6047$$

Answer 3

Assume you are dealt 3 cards in a row (so no other cards go missing from the deck between the cards you receive). The chances of getting a pair are simply 1 - (the chances of getting no pair). The first card is some card with probability 1. There are now 3 cards in the deck, which if dealt to you, would give you a pair, so the chances of not getting a pair on the second card are 48/51. Now there are 6 cards in the deck which will give you a pair if you receive them, so the chances of not getting a pair on the third card is 44/50. Overall chances of getting no pair is 1*(48/51)*(44/50), so the chances of getting a pair is 1 minus that product.