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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Median poker hand (Texas Hold’em)

I try to find the median poker hand (Texas Hold’em). The following is given: 1) There are 52 cards 2) Assuming seven of them are chosen randomly 3) Create the best possibility with 5 of these 7 ...
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2answers
479 views

Why is this not a valid algorithm for counting poker hands?

In my probability class, we had an example asking what is the probability of getting three of a kind when randomly drawing $5$ cards from a typical deck ($13$ denomination and $4$ suits, total of $52$ ...
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110 views

Why is a Pair of Tens better than a Pair of Aces in Texas Hold 'Em?

A couple years ago I developed a program to calculate the optimum betting amount for a round of Texas Hold 'em by using the Kelly criterion. In the process of computing the probability of winning for ...
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22 views

Poker Pot Odds EV Problem

My issue is with how EV is calculated. I think it's wrong in the approach how it treats the Pot. What's put in the Pot belongs to the Pot. Sure! But that's kinda stupid because the Pot will partially ...
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1answer
19 views

Given that you have exactly one pair, what is the probability you have two aces?

You are drawing 5 cards from a 52 card deck in a game of Poker. Here's what I've got so far, but I'm a bit stuck on how to proceed. Let $S = \{h \in 2^D : |h| = 5\}$ and $P(h) = \frac{1}{52 \choose ...
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38 views

What is the probability of getting STRAIGHT FLUSH in a $13$-card poker game?

What is the probability of getting STRAIGHT FLUSH in a $13$-card poker game? Here is my attempt: A straight flush is five cards in sequence and of the same suit, but not ace king queen jack ten. ...
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1answer
42 views

What is the probability of getting FOUR OF A KIND in a $13$-card poker game?

What is the probability of getting FOUR OF A KIND in a $13$-card poker game? Here is my attempt: The setup for the required poker hand would either be: $$AAAABCDEFGHIJ,$$ $$AAAABBBBCDEFG,$$ or $$...
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3answers
48 views

What is the probability of getting NO PAIRS in a $13$-card poker game?

What is the probability of getting NO PAIRS in a $13$-card poker game? Here is my attempt: The setup for the required poker hand would be: $$ABCDEFGHIJKLM$$ where $A, B, \ldots, M$ are distinct ...
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1answer
81 views

Another card game probability calculation

The game: dealer and multiple players, lets say 6 players, 1 dealer. Each participant received 2 cards. Hands: each hand consists of 4 cards, the player's two cards and the dealer's two cards. ...
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1answer
22 views

Best Poker Hand probability when dealt 3OAK with 5 opponents

So I've gone through and found all possible number of hands and probabilities for each hand (royal flush, straight flush, straight, flush, full house, four of a kind, three of a kind, 2 pair, 1 pair). ...
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29 views

Multiple Deck vs Single Deck Poker Game Calculation

I was reviewing some calculations for a $4$-card poker type table game and got confused about single vs multiple deck approach to calculating probability. If someone could enlighten me, I would ...
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1answer
40 views

Need help understanding poker hand probability/combinatorics.

I have some trouble understanding a difference when calculating the probability for a full house versus two pairs. For two pairs the number of possible hands is: $$ \binom{13}{2} \binom{4}{2} \binom{...
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2answers
54 views

Calculate One Pair probability discarding 1, 2, 3, 4 or 5 cards

We start with a standard poker deck (52 cards, 4 suits, 13 ranks). In our first hand (5 cards) all the 5 cards are different (different rank). Now our first hand have 5 cards with 5 differents ranks. ...
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1answer
37 views

Finding number of possible hands in 5 card stud when order is matter

I'm trying to determine the game 5 card poker when order is matter all my trails ended with fail except the one pair. The way I did one pair assuming it is $\{k,k,x,y,z\}$ $^{13}P_1 \times ^4P_2 \...
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2answers
44 views

5 cards in sequence, don't understand answer

Find the probability that a 5 card poker hand will be 5 cards in sequence regardless of suit, consider ace high, not low. given answer: \begin{align} \frac{{9 \choose 1} {4 \choose 1} ^ 5}{52 \...
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1answer
26 views

Interpretation of Pot Odds (mainly in poker)

In the middle is 10 dollars, someone bets 5. I have to call 5 dollars in order to win 10 dollars. My Pot Odds - 5:10 = 1:2 I can interpret it as that i can lose 2 games and win 1 and I ll be break-...
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2answers
48 views

8 cards are drawn without replacement. Find the probability of obtaining exactly $3$ aces, or exactly $3$ kings, or both.

$8$ cards are drawn without replacement from an ordinary deck. Find the probability of obtaining exactly $3$ aces, or exactly $3$ kings, or both. The answer is $\frac{2~\binom 4 3 \binom {48} 5 - \...
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1answer
68 views

6 Card Poker hand with given probabilities

Each player draws a hand of six cards from the standard pack. two triples: three cards of one value, and three of another, for instance, 3♠ 3♥ 3♣ J♠ J♥ J♦; three pairs: two cards of each of three ...
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1answer
38 views

Counting the frequency of a flush hand in $7$-card poker

I'm trying to count the frequency of a flush hand in $7$-card poker. Since a flush hand could be thought of as having $5$ cards with the same suit while the other $2$ doesn't matter, I wrote down as ...
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2answers
35 views

How to efficiently calculate probabilities of not getting two pair?

Suppose two guys are playing 2 card draw poker. I am trying to find the probability of neither players drawing a pair. Let $C_1,C_2,C_3,C_4$ denote the cards drawn. We know that any card will do ...
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1answer
108 views

An optimal sequence of length 13

I'm looking for an optimal (or much better than I have now) increasing sequence $t_1, t_2, .., t_{13}: t_i \in N, 0 \le t_i<t_{i+1}$ where $3t_{12} + 4t_{13}$ is minimized, subject to the ...
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0answers
38 views

ways to change a function locally for my poker texas holdem bot

I apologize for the lack of math format and possible ambiguity. If I had more maths knowledge, this could be a two-sentence question, but I don't have that. I want to find a local optimum for my ...
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1answer
39 views

Where am I going wrong on this poker math problem?

Someone gives you two dice. They offer to pay you \$37 if you roll two 6s. However, it will cost you \$1 a roll. What is the EV of this wager? My work: 1/36 = 35:1 = 0.0277 for \$37 = 1.027 ...
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44 views

Expected payout of an optimal hand in Poker

Expected payout of a hand I created a video poker hand evaluator and I am not getting the optimal hand and I know my mistake is because of the way I am calculating the expected payout. Above is a ...
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1answer
28 views

Calculating probability that you are dealt a full house

What is the probability that you are dealt a "full house"? (Three cards of one rank and two cards of another rank.) why can't I calculate the probability in the following way? The number of ways I ...
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1answer
38 views

5 card poker hand double counting cards

This equation calculates the number of ways to get a pair in a 5 card poke hand. How come this equation doesn't need to be divided by two? My rational for dividing by 2 is as follows: The first two ...
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2answers
36 views

Ways to get a pair in poker

When drawing a pair the number of ways to get it is: $${13 \choose 1}{4 \choose 2}{12 \choose 3}(4)^3$$ But I don't understand why the following doesn't work: $${13 \choose 1}{4 \choose 2}{12 \...
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1answer
15 views

Probability of dealer having a same-value card as me in Black Jack

In a game of Black Jack, before any additional cards are given out (so everyone has exactly two cards), what are the chances that the dealer has, say, a King, given that one of my cards is a King ? ...
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0answers
20 views

Chance of being in a range with observed population tendencies

Actually its a poker related question, but hopefully turning it into a real life example I can get an answer. Let assume we have 2 bag of different colored candies, we have 1326 candies in the two bag ...
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1answer
39 views

One jack and one club

How many ways a player can be dealt exactly one jack and exactly one club ? Of course, A player can be dealt 5 cards in a normal poker game So we have 4 different jacks and 13 different club cards. ...
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1answer
104 views

Poker hands combination

In a poker game where we have a standard deck ($52$ cards) and each player is dealt $5$ cards, how many hands of each of the following types are possible ? (Type 1, "Low face") one face card and $...
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114 views

5 cards are drawn one at a time without replacement from a deck of 52.

Calculate the probability of landing a straight flush. A royal flush is considered a straight flush. I can't be sure, but I think the answer changes depending on the first card you choose. Like if ...
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0answers
75 views

Probability of an Indisputable winner at Texas Holdem

What are the fraction of hands that can be classified as "indisputable winners" after the river is revealed in Texas Holdem? By "hand" I mean the 2 hole cards you have that no one else can see plus ...
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2answers
148 views

5-Card Poker Two-Pair Probability Calculation

Question: What is the probability that 5 cards dealt from a deck of 52 (without replacement) contain exactly two distinct pairs (meaning no full house)? Solution: $$\frac{\binom{13}{2}\binom{4}{2}\...
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54 views

How many poker hands until statistically significant winner

How many poker hands do I have to play to determine a statistically significant winner? What is the best approach to get a 95% confidence interval? To give some more context: I have been building a ...
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98 views

Video Poker - RTP

This site shows that using a 'simple' strategy or an 'optimal' strategy in Jacks Or Better 'fullpay' pay table(see below) video poker results in $99\%$ return-to-player (RTP). RTP is defined as the ...
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44 views

Poker Combinations Probabilities

I've been asked to write a C# program that computes specific poker probabilities. And I am struggling with how to do it. The game is a Texas hold em. It's a 32 cards deck. You have two cards. You ...
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60 views

Poker game probability [closed]

From a deck of 52 cards, pick 3 cards randomly, what is the probability that there is a pair among the three cards? (A pair means that two of the cards are the same number) I guess it allows to have ...
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1answer
67 views

How many $3$-of-a-kinds are possible if one ace is missing?

So I know how to do the simpler 'How many $3$-of-a-kinds are possible' question - it has been asked on here multiple times too. But what if say an Ace is missing. How would you go about starting the ...
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2answers
42 views

Explanation for solution of a combinatorial problem

The given problem is: An ordinary deck of cards is dealt to four people: Joe, Bob, Jim, and Larry. If Larry has exactly one ace, what is the probability that Jim has all the remaining aces? My ...
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139 views

Probability of Poker Hands with Joker

Need help with a homework question: If a five card hand from a standard deck of 52 with an added joker (wildcard) is drawn: What is the probability that a hand contains at least one pair? ...
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1answer
39 views

Simulated probability doesn't match closed-form answer

I am trying to calculate the probability of a hand of 13 cards containing at least five pairs. Three of a kind counts as one pair; four of a kind counts as two pair. I simulated millions of hands and ...
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2answers
642 views

What is the expected number of suits in a hand of 4 cards?

To find the expected number of suits the formula is $E(Num Suits) = 1*P(1 Suit) + 2*P(2 Suit) + 3*P(3 Suit) + 4*P(4 Suit)$ For the probability of getting 4 suits I got ${13 \choose 1}^4 {4 \choose 4}/...
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1answer
449 views

How many poker hands have exactly two pairs?

I found an interesting solution to the combinatorial question of "How many poker hands have exactly two pairs?" and I cannot figure out (or find) the reasoning of the solution. The answer I found in ...
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1answer
75 views

What's more likely with 7 dice - Straight, Full House, or 3-of-a-Kind

Here's a problem I'm having. I have 7 dice. I want to rank the difficulty of achieving a particular 5 dice combination after all 7 dice have been rolled. A straight is when at least 5 of the seven ...
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1answer
124 views

Expected value of the call option in Caribbean Stud Poker: examples

For the game Caribbean Stud Poker, compute the expected value of the call option for the following cases. I try to solve part (a). There are $\mathsf{A}\mathsf{K}\mathsf{J}$ there. If $\mathsf{Q}$ ...
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62 views

On Casino Holdem game

For the game Casino Holdem, I am told to compute the expected value of the call option for the following hole cards and flop. (8♠,9♣) and (10♣,J♣,2♠) Can anyone help me on this problem. The rule ...
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34 views

Find the probability that at least one of poker opponents hits a card

In Texas hold'em, when you see the flop there is about 33% chance that one of your 2 cards hits something on the flop, maybe not exactly but let's say it's 33% that your opponent did hit something on ...
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3answers
218 views

Probability of get 3 cards out of 4

How to calculate a probability of choose 3 off-suited and off-ranked cards out of 4 given cards from a standard 52-cards deck (the situation when all the cards off-suited/off-ranked should be avoid). ...
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45 views

How can I find the probability of drawing cards in these cases?

Given 2 players A and B. We will draw 5 cards for A and then 5 cards for B. Find the probability of this event: A receives Two Pair (contains two cards of the same rank, plus two cards of another ...