# Probability of Two Pairs in Five-Card Poker Hand

• MHB
• alexmahone
In summary: There are still {13\choose 3} left and we need to pick the last card. So, the probability of having two pairs is $\dfrac{{13\choose 3}\cdot {4\choose 2}\cdot {4\choose 1}}{{52\choose 5}}=\dfrac{1}{5}$.
alexmahone
What is the probability that a five-card poker hand contains two pairs (that is, two of each of two different ranks and a fifth card of a third rank)?

My attempt:

Let us first pick the 3 different ranks. There are $$\displaystyle {13\choose 3}$$ ways of doing this.
Out of each rank consisting of 4 suits, we must pick 2 cards, 2 cards and 1 card respectively.
So, no. of ways $$\displaystyle ={13\choose 3}\cdot {4\choose 2}\cdot {4\choose 2}\cdot {4\choose 1}$$

Total no. of ways of selecting a five-card poker hand $$\displaystyle ={52\choose 5}$$

$$\displaystyle p=\dfrac{{13\choose 3}\cdot {4\choose 2}\cdot {4\choose 2}\cdot {4\choose 1}}{{52\choose 5}}$$

This doesn't match the answer given in the textbook. Where have I gone wrong?

Alexmahone said:
What is the probability that a five-card poker hand contains two pairs (that is, two of each of two different ranks and a fifth card of a third rank)?

My attempt:

Let us first pick the 3 different ranks. There are $$\displaystyle {13\choose 3}$$ ways of doing this.
Out of each rank consisting of 4 suits, we must pick 2 cards, 2 cards and 1 card respectively.
So, no. of ways $$\displaystyle ={13\choose 3}\cdot {4\choose 2}\cdot {4\choose 2}\cdot {4\choose 1}$$

Total no. of ways of selecting a five-card poker hand $$\displaystyle ={52\choose 5}$$

$$\displaystyle p=\dfrac{{13\choose 3}\cdot {4\choose 2}\cdot {4\choose 2}\cdot {4\choose 1}}{{52\choose 5}}$$

This doesn't match the answer given in the textbook. Where have I gone wrong?

If there are two pairs, then that already accounts for 4/5 cards in the hand. Let's break this up into the 4 cards that contain the pairs and the one card that doesn't.

There are 13 possible ranks, as you said, and we are picking 2 of them to match. We need 2,2 or 3,3, etc. Those kind of matches. Then for each of those 2 ranks they could be 1 of 4 possible suits. How would you express that?

Once you pick the two pairs, we need to pick the last card. How many ranks are left available?

(The $\binom{4}{2}$ terms and the $\binom{4}{1}$ term are all correct. You need to modify your first term and add one more term.)

## 1. What is the probability of getting two pairs in a five-card poker hand?

The probability of getting two pairs in a five-card poker hand is approximately 23.5%. This means that out of all possible combinations of five cards, about 23.5% of them will result in a hand with two pairs.

## 2. How is the probability of getting two pairs calculated in poker?

The probability of getting two pairs in poker is calculated by dividing the number of combinations that result in a hand with two pairs by the total number of possible combinations of five cards. This can be represented by the formula: P(two pairs) = (C(13,2) * C(4,2) * C(4,2) * C(11,1) * C(4,1))/C(52,5), where C(n,r) represents the combination of n items taken r at a time.

## 3. What is the difference between getting two pairs and getting a full house in poker?

The main difference between getting two pairs and getting a full house in poker is the number of pairs and three-of-a-kind involved. In a full house, there are three cards of the same rank and two cards of another rank, while in two pairs, there are two sets of two cards with the same rank. The probability of getting a full house is lower than that of getting two pairs.

## 4. Can the probability of getting two pairs change throughout a game of poker?

The probability of getting two pairs can change throughout a game of poker, as the cards on the table and in each player's hand are revealed. This can affect the overall probability of getting two pairs, as well as the likelihood of getting specific combinations of two pairs (e.g. two pairs of aces and kings).

## 5. How can understanding the probability of two pairs in poker help improve my game?

Understanding the probability of two pairs in poker can help improve your game by allowing you to make more informed decisions. For example, if you have two pairs in your hand and the probability of getting a full house is low, you may be more confident in your hand and bet more aggressively. It can also help you determine the strength of your hand compared to other players and make strategic decisions based on the likelihood of certain combinations appearing on the table.

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