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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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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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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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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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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Game theoretically optimal play can it depend on randomness?

I am studying GTO of poker but my question should apply to general game situations. When I solve some situation using any of the avaialble GTO programs out there, I get a result which say for instance,...
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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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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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25 views

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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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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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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35 views

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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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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Number Of Two Pair (Poker) Combinations Allowing For Pairs Or The “Junk” Card To Be Of The Same Rank

The standard two pair formula is ${13\choose2}{11\choose1}{4\choose2}{4\choose2}{4\choose1}$ but this assumes that in the poker hand AABBC the rank of AA, BB and C cannot be the same (to avoid for ...
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Probability of hitting a quad when the flop has a three of a kind in Texas Hold'em

Let's assume that I am playing Texas Hold'em with 5 of my friends (total 6 players). I have pockets Kings (KK). Now the flop has 3 A (AAA). I want to figure out if anyone on the board has beaten me ...
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Expected Value : Poker

You are dealt $4$ cards face down on a table. You know that $2$ are red and $2$ are black. Let’s say you are given $100 to bet on this game and you are able to bet after each card is dealt and before ...
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Whats the intuitive difference between ${n \choose 2}$ and taking ${n \choose 1} {{n-1} \choose 1}$?

Example: I am trying to find the number of combinations of full house in poker. In that case I take ${13 \choose 1}{4 \choose 3} {12 \choose 1} {4 \choose 2}$. In a different scenario suppose I want ...
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134 views

Probability that at least one Ace is dealt in poker with $6$ players

If two cards are dealt to each of the 6 players from a 52 deck of cards what is the probability that there is at least one ace dealt? I've tried to solve this using the following reasoning: In total ...
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30 views

Multiplayer poker at least one gets flush

I am trying to calculate probability if $3$ card Indian poker is being played between $8$ players drawing $3$ cards each what are chances that at least one player has either flush or straight or royal ...
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How many five-card hands contain exactly one pair of the same rank and three other cards of different ranks?

How many five-card hands contain exactly one pair of the same rank and three other cards of different ranks? This is my thought process: I first multiply it by $13 \binom{4}{2}$, since there are $13$ ...
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141 views

Find the probability of getting a heart on the first draw and a club on the second draw while drawing 2 cards successively and without replacement

I am currently having some difficulty with a card probability question (I guess combinatorics will haunt me for the whole life...). The question is to find the probability of getting a heart on the ...
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Probability that a 5-card poker hand contains exactly four cards of the same suit?

I'm not that confident in my ability, so I've come here. This is what I have: $\frac{C(4,1) * C(13,4) * C(3,1) * C(13,1)}{C(52,5)}$ My thinking is that $C(4,1)$ selects one suit from the 4 suits and $...
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35 views

Poker Counting Problem

In Poker, I receive $2$ cards and they are a pair of $8$s. What are the chances of me getting, just a pair, 2 pair, and three of a kind(but nothing higher) as the other three cards are placed down? I ...
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194 views

Probability of at least 3-of-a-kind in a 7-card poker hand [duplicate]

This problem asks to explain why the method used is overcounting or undercounting cards in a 7-card poker hand that contains at least two 3-of-a-kind (3-of-a-kind means three cards of the same rank) ...
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2answers
105 views

Probability of a five-card poker hand containing 2 pairs

I am trying to solve this question in my own way. I tried counting like this- First card can be chosen in 52 ways, second in 3 ways because it has to be of same kind as first, third in 48 ways, fourth ...
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25 views

Poker Combination problem

A little while ago I made a post on here and this is basically a follow up to it as I ran into some problems. Basically two people , player A and B, are playing a version of poker where there's 3 ...
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1answer
114 views

Probabilities associated with each possible value of random variable ; 5-card poker hand

A five-card poker hand may contain from zero to four aces. If $X$ is the random variable denoting the number of aces, enumerate the range space of $X$. What are the probabilities associated with each ...
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169 views

Probability of getting 3 of a kind in a 5 card poker hand using combinations

I have a question about finding a probability of 3 of a kind using combinations. This is a solution I originally came up with: $$\frac{\binom{13}{1}*\binom{4}{3}*\binom{12}{1}*\binom{4}{1}*\binom{11}{...
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Poker - Combination Questions

I'm struggling a bit with this question so any help will be very much appreciated! The premise is that two people, call them A and B, are playing a variant of poker where there's 3 shared cards, and ...
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81 views

Probability that a hand of 10 cards dealt from a normal pack of 52 contains…

Exactly one ace given that it contains exactly two queens? I am struggling how to solve this maths problem. I know that a hand of $10$ cards dealt from a normal pack of $52$ is $52\choose 10$ $=$ $...
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141 views

Two pairs from a standard deck of 52 cards [duplicate]

You are dealt $5$ cards from a standard deck of 52 cards. A standard deck of 52 cards has 13 ranks in each of four suits.) How many ways can you be dealt the $5$ cards so that they contain two cards ...
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61 views

Drawing 3 of a kind in a 7 card hand

I have a probability question if anyone is willing to help. In a $7$ card hand without replacement what is the probability of getting at least a $3$ of a kind (standard deck A-K, $4$ suits). I was ...
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How to Calculate 5 to 11-Card Poker Straights When Dealt 11 Cards from 8-Deck Shoe

I'm looking for how to calculate the odds of a 5-card straight poker hand formed from 11 cards dealt from an 8-deck shoe, and then the odds for a 6-card straight out of those 11 cards, and then the ...
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60 views

What are the odds that two decks of poker cards have the first $10$ cards in common? [closed]

I know that the number of possible deck combinations is $8.06581752 \times 10^{67}.$ What I would like to know is how to calculate the odds of two $52-\text{card}$ poker decks being shuffled, then ...
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In a friendly game of Texas Hold ’em, four people are dealt two cards each…

In a friendly game of Texas Hold ’em, four people are dealt two cards each. Considering there are 12 face cards, how many ways are there for everyone to get two face cards? I need help on this problem....
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How to find probability of one pair poker?

I have the following problem: So my main question is how am I supposed to choose the rank of the cards when trying to calculate the probability of one pair? I don't have the usual 52 deck of cards so ...
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Hand of Poker Probability

I'm not sure about how to solve the following problem Randomly draw 5 cards from a deck of cards. What is the probability of getting two pairs and one of the 5 cards is Heart-7 (♥7)? I solved it this ...
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What are the odds of a straight and three three-of-a-kinds in poker?

How do you calculate the odds of this happening in poker? There are 4 players. Each of them had a pair each of the cards K,Q,A,10 in pre flop itself. The cards on the table are 5, 10, A, Q J So the ...
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bounded Cantor pairing?

I know this is not the coding stack exchange but the first three paragraphs just pertain to the background (I hope it is not too jargon heavy). My question itself is more mathematical. Also I don't ...
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Probability of drawing an ace or 2 after an ace [closed]

From a standard 52 card deck, we draw one card at a time without replacement. After having drawn the first ace, the game continues until either We draw another ace, or We draw a 2, and the game then ...
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30 views

Probability of a group of players getting at least one straight in a 5-card poker game.

With inclusion of straight flush and royal flush, there are $10$ ways of getting a straight in a single suit and each of the $5$ cards that form a straight has $4$ different suits. Hence, there are $...
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1answer
71 views

Number of ways to pick at least three of a kind in 5-card poker - what's wrong with C(49,2) for the last two cards?

In finding the number of ways to get a 5-card poker hand that contains at least three of a kind, what's wrong with the following $$n=\binom{13}{1}\binom{4}{3} \binom{49}{2}$$ So, we have 13 numbers to ...
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1answer
68 views

Probability of Two Pairs ( Cards game )

Question: Calculate the probability of getting a two pair hand ( e.g., two 8’s, two Queens, and a Knight ) My answer: The probability of getting a two pair hand is : $$ \frac{13\cdot\binom{4}{2}\...
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Counting pair in a poker hand - counting confusion

I want to be good enough to do math competitions, and one of the main topics in them is always combinatorics. So I'm going through a combinatorics book, "A Walk through Combinatorics." The confusion ...
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Numbering system for poker hands

I figure that this is surely either a solved problem or it is provably impossible. But I'm not able to track down an answer. There are ${52 \choose 5}$ = 2,598,960 distinct ways to deal five cards ...
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127 views

Calculating the number of poker hands that have exactly 4 different denominations

I'm been working through some examples of questions mentioned in the title and came across this one that I wasn't entirely if I was approaching this correctly. My initial approach to solving this ...
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1answer
48 views

Texas hold'em pecularity with different number of players

While simulating Texas hold'em poker hands, to find the probability that certain two cards would give the best hand after river, I found a pattern that I can't explain. I made $100,000$ simulations ...
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1answer
35 views

How to calculate the number of unsuited hands in Texas Hold'em opening hand

I was just going back and refreshing my knowledge on binomial coefficients/probability and I was trying to calculate some poker odds since I would be able to look them up and verify my attempts. I ...

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