# 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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### poker expectation clarification

I ran across a passage in "The Mathematics of Poker" explaining bluffing strategy. Here is a paraphrased version. Two players are playing heads-up ($1$ vs $1$) poker. They're on the final ...
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I know that when calculating the probability of getting one pair in 5 card poker, it is done like: $$P(\text{one pair}) = \frac{\binom{13}{1} \cdot \binom{4}{2} \cdot \binom{12}{3} \cdot \binom{4}{1}^... 0 votes 0 answers 58 views ### Probability of two pair for a poker dice I understand that the well known way to solve this is Choose the rank for the two pairs from 6 options - 6C2 Choose the dice for the first pair - 5C2 Choose the dice for the second pair - 3C2 ... 3 votes 1 answer 141 views ### How do I calculate the probability of getting a 5 card straight in 7 card poker? So here's my progress on this problem. The first straight to examine is \begin{pmatrix} A & 2 & 3 & 4 & 5 & n & m\end{pmatrix}. If we start with all one suit, there are \... 0 votes 1 answer 50 views ### How to prevent Nash-EQ solver getting stuck in an infinite loop? I would like to write a Nash equilibrium solver for a simplified Poker game. I was reading that the way some Nash equilibrium solvers for Poker work is that they iterate until convergence: For a given ... 0 votes 1 answer 30 views ### What are the odds that 2 players heads-up in poker will both not make a pair or better by the river, assuming they only check? Here is my current thinking, please let me know if it is correct. The result doesn't seem intuitive and I feel like I'm missing something, hence I am reaching out here. 52 - 4 = 48 cards remaining. (4-... 0 votes 2 answers 35 views ### Exponentials of stochastic band matrices Edit: I should probably say the motivation: I was interested in modeling the probability of winning a heads-up poker match if the players will play until one player goes bust. Because the game is zero ... 1 vote 1 answer 38 views ### Probablity of 5 card holdem hand given 7 cards The probabilities of getting a straight flush, 4 of a kind, full house, ... , and so on, when dealt 5 cards from a randomly shuffled deck are easily calculated as straight flush .000015, four of a ... 5 votes 1 answer 143 views ### Probability of poker pair - what's wrong with this? Thanks to others here, I know the correct way to calculate the probability of one pair in 5-card poker is P(one pair) = C(13, 1) * C(4, 2) * C(12, 3) * [C(4, 1)]^3 / C(52, 5) = 0.4225690276... Why ... 1 vote 1 answer 27 views ### What are the odds of getting 2 sets in texas holdem? I'm looking to create a variant of holdem allowing 6 and 7 card hands in special situations. Namely, 3 pair, 2 set, and quads over set (a full mansion). My probability skills are rusty, but I'm still ... 3 votes 1 answer 73 views ### What are the odds of getting 3 pair in texas holdem? I'm looking to create a variant of holdem allowing 6 and 7 card hands in special situations. Namely, 3 pair, 2 set, and quads over set (a full mansion). My probability skills are rusty, but I'm still ... 2 votes 0 answers 62 views ### Probability of flush given that you observe 4 diamonds out of 5 cards I came across this probability problem. In a Texas Hold’em game, you are dealt 2 cards in your hands and the 3 flop cards are shown. You observe 4 diamonds out of the 5 cards. What are your chances ... 0 votes 0 answers 23 views ### Modified Deck. Unique Rules. Poker Probability. 64 cards. 4 standard suits. The ranks go from 1-12. However there are 16 cards that have a rank of "0"... The 0 rank cards have some unique attributes that modify the rules: They are double ... 1 vote 1 answer 51 views ### Counting problem for finding the probability of two pairs in a five card hand. I am having trouble understanding what I have wrong in this stats problem for my hw. I am trying to find the probability of getting two pairs in a 5 card hand from a deck of cards. So first we know ... 2 votes 1 answer 160 views ### How to rank a new type of Hold 'Em Poker hand (hard) I am trying to figure out how to rank a new type of poker hand, henceforth called a 'Block'. A Block consists of 6 cards, and so is not possible in regular Poker - only Hold'em. Each Block consists of ... 1 vote 0 answers 23 views ### Probability of a straight ending on specific number in 7-Card Poker I am trying to calculate the probability of a straight that ends on a specific number, e.g. 6:{2, 3, 4, 5, 6} or Q:{8, 9, 10, J, Q}. Calculate Total number of straights: Assume x in [5,13(K)]. ... 2 votes 1 answer 76 views ### Combinations Straight Flush in Texas Hold'em Poker On the Wikipedia page about Poker probabilities for 7-card poker hands (Link) it says the following for the Royal Flush: \begin{equation*} \binom{4}{1} \binom{47}{2} \end{equation*} First we choose 1 ... 0 votes 1 answer 139 views ### Probability of a poker hand of winning Just by curiosity I was looking for the probabilities of poker hands as a simple array of 5 cards, and what I found in many websites are the frequencies asociated to each combination of shown in a ... 3 votes 1 answer 145 views ### Has the game of texas hold em poker been beaten? I am no mathematician, I guess this is obvious. I hope this question is appropriate. So, I know that in poker the opponents psychology matter, but imagine that you are playing against a perfectly ... 4 votes 2 answers 134 views ### Why does calculating probability of 2 pairs seem so different from for a full house? For 5 card poker without discards/draws, the probability of being dealt a 2 pair hand is about 0.047539, and for a full house is about 0.001441. 1 I first learned about calculating for a 2 pair hand.... 3 votes 2 answers 922 views ### What is the probability of getting a full house in a 7 card poker hand I'm trying to calculate the probability of getting a full house in a 7 card poker hand. This is obviously the number of 7 card full house hands divided by \binom{52}7. To try and calculate the ... 4 votes 3 answers 104 views ### Three-of-a-kind Poker Hand Problem Problem: Three-of-a-kind poker hand: Three cards have one rank and the remaining two cards have two other ranks. e.g. {2♥, 2♠, 2♣, 5♣, K♦} Calculate the probability of drawing this kind of poker ... 1 vote 1 answer 123 views ### Can Nash Equilibrium strategy be exploited, if another player deviates? Consider a 3 player game (e.g. poker for 3 people). Nash equilibrium is a set of strategies s_1, s_2, s_3, such that the expected profit value for each player s_i E_i(s_1, ..., s_i', ..., s_N) ... 0 votes 1 answer 32 views ### Deriving the Semi-Bluffing Formula in Poker I am recently taking this MIT Open course, and trying to derive the Semi-Bluffing formula on my own. The formula can be found on Page 52 of the slide. I am a bit confused about why there is a 2 in ... 1 vote 0 answers 32 views ### What n-tuple am I applying the multiplication principle to when counting Texas Hold'em pocket cards with a pair? Textbook problem (quoted): In Texas Hold'em poker a player receives 2 cards first. These are called the player's pocket cards. Find the number of pocket card configurations (that is, the number of ... 0 votes 1 answer 57 views ### 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 ... 0 votes 1 answer 115 views ### 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 ... 0 votes 2 answers 66 views ### 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 ... 2 votes 1 answer 114 views ### 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 "... 1 vote 1 answer 210 views ### 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 —&... -4 votes 1 answer 223 views ### 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 ... 0 votes 1 answer 114 views ### 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 ... 1 vote 1 answer 110 views ### 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 ... 0 votes 1 answer 46 views ### 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 ... 3 votes 1 answer 172 views ### 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 ... 0 votes 0 answers 49 views ### 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 ... 5 votes 1 answer 158 views ### 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 ... 2 votes 1 answer 92 views ### 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: ... 0 votes 1 answer 67 views ### 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 ... 1 vote 2 answers 237 views ### 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 ... 0 votes 2 answers 96 views ### 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 ... 0 votes 1 answer 252 views ### 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 ... 0 votes 0 answers 14 views ### 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? 0 votes 1 answer 631 views ### 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 (... 1 vote 1 answer 156 views ### 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. ... 0 votes 1 answer 167 views ### 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: ... 0 votes 0 answers 42 views ### 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 ... 3 votes 2 answers 918 views ### 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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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}...