Questions tagged [poker]

Poker is a card game with several variants (Draw, Texas Hold'em, ...), the common goal being to collect face value patterns such as "four of a kind", "full house" or "straight" or all of a suit ("flush") with five cards. Questions about combinatorics, probabilities, strategies (bluffing and betting).

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What's the number of possible five-card poker hands drawn without replacement?

So we know a standard deck has 52 cards and it's asking the number of possible 5 card hands we could get from it without replacement which means without putting them back in the deck. I thought about ...
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Conditional Probability in poker, Bayes Theorem

In a four person pokergame you are dealt a "king of heart" and "8 of spades". Three cards lay on the table none being a king or 8, meaning there is 41 cards left. Q: What is the ...
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Can this Sum be reduced? It is calculating the expected value of a poker hand when you "Run it multiple times" for the last card.

There is a scenario in poker (Texas Hold'em) where if two players go "all in", they have the option to deal the remaining cards multiple times and it results in multiple pots that they could ...
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To show that there are two people in the tournament with the same starting hand and position in a Texas Hold'em poker tournament.

A Texas Hold'em poker tournament has $1557$ players, in $9$ distinct positions at $173$ tables. Each table has a standard $52$ card deck, and each player is dealt $2$ cards. Hands are considered "...
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1 vote
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Probability of Full House in 5 Cards Given a Pair of Aces

What is the probability of a full house in a 5 card hand, given the first 2 cards are aces? My thought process is that there are 4 other events that could occur: A: Ace p: any card of given face App —&...
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-4 votes
1 answer
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What are the odds of drawing a pocket pair in poker? [duplicate]

In poker, you make a pocket pair when the ranks of your first two cards match. This does not include making a pair on the flop or in a full 7-card hand. There are six ways to draw a pocket pair of a ...
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0 votes
1 answer
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probability of getting two pairs in poker [duplicate]

If it is assumed that all (52C5) poker hands are equally likely, what is the probability of being dealt two pairs? (This occurs when the cards have denominations a, a, b, b, c, where a, b, and c are ...
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Standard deviation of a sample of poker hands from Mathematics of Poker book

This is a question related to the book "The Mathematics of Poker" by Bill Chen and Jerrod Ankerman. The author is explaining how to calculate the standard deviation per 100 hands based on a ...
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Calculate the probability that a poker hand of 13 cards does not contain at least one suit

I know that this problem can be correctly calculated using inclusion-exclusion principle. However, I thought solving for the complement (i.e. the probability that it contains all suits) would be the ...
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3 votes
1 answer
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Probability of having n number of cards of the same kind from drawn N cards

Introduction Before starting, I would like to specify a few things: The deck has 52 cards. From 1 (Ace) to 10, J, Q, K, all in four different suits: spades, diamonds, club and hearts. I don't add ...
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Which card should we change (and to what card) to maximize the probability of getting two pairs?

In a standard $52$-card deck we change 8♧ with 10♤ and 9♧ with J♤ from another deck of cards. If we change another card, which card should we change (and to what card) to maximize the probability of ...
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How is calculated the frequency of a pair, a flush of 2 cards, a straight of 2 cards and a straight flush of 2 cards (52 cards, hand of 2, 3, 4, etc)?

Introduction Before starting, I would like to specify a few things: The deck has 52 cards. From 1 (Ace) to 10, J, Q, K, all in four different suits: spades, diamonds, club and hearts. I don't add ...
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2 votes
1 answer
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Finding the probability of a full house in poker with two wild cards

Problem: Suppose that we have a standard poker deck and we add $2$ cards to it. These two cards added are jokers and are considered wild cards. What is the probability of getting a full house? Note: ...
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Probability of getting two pairs from a modified deck

In a deck of cards with 52 cards we change clubs 8 with spades 10 and clubs 9 with spades jack from another deck of cards. What is the probability of getting two pairs if you draw 5 cards? In a ...
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1 vote
2 answers
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What is the probability of four of a kind in poker with wild cards?

Below is a problem I made up and did. Problem: Suppose that we have a standard poker deck and we add $2$ cards to it. These two cards added are jokers and are considered wild cards. What is the ...
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poker hand conditional probability

Consider a deck of cards, where the backs are marked in such a way that you can distinguish a pip card (Ace, 2,...,10) from a face card (King, Queen, or Jack). You deal a poker hand to a friend. Given ...
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How to calculate the probability of getting a pair on the flop in Texas Hold 'Em

I'm wondering how exactly I would go about calculating the odds of getting exactly a pair on the flop (so two pairs and three of a kind do not count). I'm assuming that the player's hand is not a ...
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What is the probability of flopping 2 pair with paired hole cards in Holdem?

I was looking into it and the only thing that I'm not sure is whether in the equation for this probability has to be multiplied by 3 or 2. 48/50 x 3/49 x 44/48 x2 =10.78% Can someone please clarify?
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I draw 7 cards from a standard card deck, what is the probability 5 of them are the same suit?

Apologies for the formatting, I am not familiar with this platform. I believe I have solved it but I looking for assurance. Let X be a random variable indicating the number of cards of the same suit (...
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2 votes
1 answer
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What is the probability of getting exactly two pairs in a poker hand of $7$ cards?

Problem: What is the probability of getting exactly two pairs in a poker hand of $7$ cards? Note: Only $5$ cards count, and the cards that count will be determined by the player holding the cards. ...
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1 vote
1 answer
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What is the probability of hitting 2 pair on the flop with upaired cards in Holdem poker?

I have mainly seen that the probability is 2.02%, but it always somewhere said it does not include if boards pairs. So I was thinking is the probability of getting 2 pairs if board pairs is equal to: ...
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In Texas Hold’em does flipping the river before the turn change the game? [duplicate]

This question has a caveat that the game is played with the community cards dealt prior to the start of the hand, which isn’t standard poker play. The community cards are dealt: burn 1 card, 3 cards ...
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3 votes
2 answers
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What is the probability that a five-card poker hand contains all 4 suits?

I know that my answer is wrong but I don't know why it is wrong. What is the probability that a five-card poker hand contains all 4 suits? What I did was I this: $$\frac{C(4,1)\,C(13,1)\,C(3,1)\,C(...
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1 vote
2 answers
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Probability of getting three of a kind in a Poker hand

I have seen some answers to the question as $$\frac{\displaystyle\binom{13}{1}\binom{4}{3}\binom{12}{2}\binom{4}{1}^2}{\displaystyle\binom{52}{5}} $$ Why don't we use $$\frac{\displaystyle\binom{13}{1}...
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6 votes
1 answer
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Why is the probability of getting a pair in a five-card poker hand so complicated?

The working-out of this question has really confused me. I know the basics of probability but I don't get the calculations here. I think 13C2 * 4C2 determines the number of possible pairs, and the ...
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Texas Hold 'Em: What is the probability of both players being dealt the same pair, with this pair being a deciding factor in an earlier hand

A game of Texas Hold Em is down to the last 2 players. What are the odds that both players are dealt the same pair (specifically 10-10)? What are the odds that this situation results in a Two Pair ...
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4 votes
2 answers
153 views

Probability of a flush given that the first two cards are of the same suit.

there. I'm looking to write probability of a flush (suppose it contains royal flush, straight flush, etc.) given that the first two cards are of the same suit. Here is the solution, let $F$ be the ...
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1 vote
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Probability of a five of a kind with a certain strategy

Let's imagine you throw five dices. After the throw, you can choose the amount of dices that you want to rethrow (can be a number between 0 and 5). What is the probability to do a five of a kind with ...
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Conceptual Doubt: Probability of 4 of a kind hand

I have seen people's solutions to calculating probability of getting 4 of a kind in a poker hand, and I know the answer is $\frac {(13\cdot48)} {^{52}C_5}$. Now I understand that out of $13$ sets of ...
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0 votes
2 answers
172 views

Probability that a 5-card poker hand is a straight

I'm trying to find the probability that a 5-card poker hand contains 5 numbers in a numerical sequence. For the first card, there are 52 options. For the second, there are 4 on either side of the ...
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1 vote
1 answer
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Given that I have an ace of spades in my 5 card poker hand, what is the probability of having at least one more ace in my hand? [closed]

My thought is that it would be $1-\frac{\binom {48}{4}}{\binom {51}{4}}$ because it is as if we are getting 1 minus the probability of having no aces in a 4 card hand for a 51 card deck with only 3 ...
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4 votes
4 answers
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Full House in Poker -- Why do we treat all possible hands to be equally likely?

The problem asks us to find the probability of a full house in a well-shuffled deck of $52$ cards. The solution in the textbook states in the first line "All of the $52\choose 5$ possible hands ...
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1 vote
0 answers
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How to calculate distinct 5-card hands in 7-card (Texas Hold'em) Poker using equivalence classes and combinatorics

There are quite a few questions and answers here on specific holdings in poker and their frequency. I am more generally interested in the number of distinct 5-card hands that are possible in 7-card ...
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0 votes
1 answer
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Why are there no combinations for certain hands when the dealer doesn't qualify for Ultimate Texas Holdem? [closed]

I am trying to understand why the probabilities are 0 for the "Four of a kind" and "Full house" hands when the dealer doesn't qualify for the game described at https://wizardofodds....
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Combinatorics involving a poker hand

How many 5 card poker hands can be dealt from a standard deck of 52 cards with 3 hearts and 2 diamonds? I have tried this but I am not sure if it is correct: ${13\choose3}{13\choose2} = 22308$
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Why do these two probability solutions lead to different answers?

DISCLAIMER: this is a question already on the site, but I am seeking to gain clarity on a different approach to this problem. Question is the probability of a full house (i.e., $3$ of a kind + $2$ of ...
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1 vote
2 answers
234 views

Expected number of aces in a poker hand

So I'm having difficulty understanding a certain idea. Say I have a standard $52$ card deck (fairly shuffled, etc.) and I pull $5$ cards from it. We ask, what is the expected value of the number of ...
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0 votes
1 answer
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Stud Poker Probability Knowing 2 Cards

Good afternoon, I find many topics on how to calculate the probability of a full house, a straight, and so on. But I did not find a source where the counting proceeds with the knowledge of some cards. ...
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1 vote
1 answer
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Extremal poker: the most favored that a currently weakest hand can be

We are playing heads-up Texas hold 'em poker. I want to find a situation where my hand is currently weaker than my opponent's (in the sense of hand rank if no more cards were to be drawn), but has the ...
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How many full house sets are there if we take suits into consideration

Suppose we had a standard $52$ card deck. How many different full house sets are there if we take suits into consideration? For example: $\rm(8\clubsuit, 8\heartsuit, Q\heartsuit, Q\spadesuit, Q\...
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1 vote
1 answer
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Find the probability of flopping a straight, given you have 2 hole cards that can flop a straight

Find P(flop straight | dealt 2 hole cards that can flop a straight) At first I simply did: $\frac{12}{50} \times \frac{8}{49} \times \frac{4}{48} = 0.003$ but that seems awfully low. Then I tried ...
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3 votes
4 answers
210 views

How are all possible poker hands of equal probability?

Please pardon the elementary question, for some reason I'm not grocking why all possible poker hand combinations are equally probable, as all textbooks and websites say. Just intuitively I would think ...
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Poker problem: Find the probability you are dealt 2 hole cards that can FLOP a straight.

My brute force approach: We notice that some ranks have more "straight feasibility" than other ranks. An ace has a range of $8$ potential 2nd cards $\{2-5, 10-K\}$, whereas a $2$ has a range ...
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Tree diagram for Full House in Poker

So I wanted to follow up on question I asked yesterday (that was well answered), but I wanted to look at it in a more visual way. I was asking how many ways there are to make a full house in Poker ...
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4 votes
1 answer
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Number of ways to get a full house in Poker

I'm trying to understand how to represent the number of ways to get a full house in poker without using the combinations method. I read that this is true: $\frac{52\times3\times2\times48\times3}{3!2!}=...
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0 votes
1 answer
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Probability of getting a full house in five-card poker with 4 cards added? [closed]

a) If I draw 5 cards randomly what are the chances of getting a full house ? b) Which other card can you remove to maximize your chances of getting full house? So you have in total 56 cards, the 4 ...
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1 vote
1 answer
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Poker cards combinatorics question

Question Suppose you draw 7 cards from a standard deck of 52 cards. What is the probability that you draw three pairs of cards if the three pairs must have distinct denominations? the three pairs don'...
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1 answer
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What is the probability that the hand contains more than one ace, given that it contains an ace of spades?

A poker hand of 5 cards is dealt from a normal deck of 52 cards. What is the probability that the hand contains more than one ace, given that it contains an ace of spades? I'm assuming that I need to ...
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0 votes
1 answer
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Can someone explain why the probability of getting a royal flush is only based on the choosing the suit?

So $|S| = 52C5, |E| = 4C1$ therefore $P = \frac{1}{649740}$. If you only select the suit, how do you guarantee that the cards OF the suit will be 10, J, Q, K and Ace? It seems so random that only $4C1$...
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4 votes
1 answer
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What is the Group of symmetries of Texas Holdem Poker hands?

In Texas Holdem poker, each player is dealt two cards from a standard deck of 52 playing cards. (2,3,4...Queen,King,Ace). What is the group of all the symmetries of all (two-card) poker hands? (Given ...
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