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Questions tagged [card-games]

Questions regarding probabilities and combinatorics of card games. The number of cards in the deck, number of players, and other parameters can vary.

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Discrete mathematics: chances of getting specific card on card-game

The game is played with a deck of $40$ cards. Three of them are dealt to each of two players, so my hand, the cards that were given to me, is formed by three cards of the deck. At the same time, ...
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21 views

Two packs of cards each containing cards numbered 1 to n are shuffled and placed on the table

Let $ n ∈ N $. Two packs of cards each containing cards numbered 1 to $ n $ are shuffled and placed on the table. One by one, two cards are simultaneously turned over from the top of the packs. What ...
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42 views

What is the probability that other players have Hearts or Q of Spades on hearts game?

I'm developing a 4 player Hearts game that has three AIs. So, I need some probabilities to do so. Rules: Initially, 52 cards are distributed among all 4 players. Each of them has 13 random cards. ...
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1answer
38 views

Probability that one player has all the card and one player has no card of spades?

I am developing a 4 players AI based hearts game. Every player has random 13 cards of the 52 cards. I need some probabilities to make the AI. In the first move (Everyone has 13 cards), I have 4 cards ...
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1answer
55 views

Discrete Math problem with a deck of cards

I have a standard deck of 52 cards. How do I find the number of hands of 13 cards that contain 4 cards of the same rank? (A,2-10, J,Q,K) My initial thought was to choose 10 cards from 49, because 9 ...
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1answer
25 views

What is the probability of getting the following hand?

What is the probability of getting the following hand? (Red 10, Black J, Red Q, Black K and Red A) or (Black 10, Red J, Black Q, Red K, Black A), we do not consider ordering in this case. This is ...
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1answer
44 views

Which is better probability-wise two pair or pocket pair two pairs

In Texas holdem on the flop is it better for someone to have two pairs where the pair is a pocket pair combined with a pair on the flop, or is it better to have a two pair where the two hole cards ...
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1answer
55 views

Probability of a poker hand existing that is at least as good as a three of a kind when choosing 10 cards?

I am working on creating a magic trick and it got me wondering about this related probability question. Ten cards are chosen at random from a standard $52$ card deck. You may use only these ten ...
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1answer
41 views

Compute number of combinations

I got really stuck with this task: Nine of ten cards, among which there is an ace of hearts, are distributed to three players so that the first one receives 3, the second - 4, and the third - 2 ...
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2answers
32 views

Combinatorics for a 15 card deck. Not choosing two sets with 6 drawn?

I have a combinatorics problem for a game I am playing that I can't seem to wrap my head around. If anyone could take a stab on it that would be greatly appreciated! I have a deck of 15 cards. The ...
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1answer
74 views

How to get an ace of hearts?

A card is drawn at random from a pack of $52$. According to the value of this card ($A =1, J=Q=K=10$) as many further cards are drawn. What is the probability that the ace of hearts is among those ...
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56 views

$52$ cards are dealt among 4 players, determine the probability that a player gets all the spades

$52$ cards are dealt among 4 players, determine the probability that a player gets all the spades Number of ways the cards can be dealt among $4$ players = $52 \choose {13,13,13,13} $$= \frac{52!}...
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1answer
18 views

Probability in a deck of Spanish cards

I have a deck of Spanish cards. (It would work the same with poker decks). If I ask a person to give me a number $X$ and to another to give me a card type $T$ In a given permutation of the decks, I ...
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2answers
48 views

Optimal way to stack deck against adversarial opponent?

A two-player card game is played with a deck of cards numbered 1-52, which is shuffled and placed face down. Each player draws a card from the top of the deck, then both players reveal their cards and ...
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1answer
40 views

Probability and cards

In Nepal, we play a card game mixing $3$ deck of cards. We deal $21$ cards to each player. What is the probability of getting $3$ of a kind (both number and suit)? I got $2.8\%$ but it's dubious. ...
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1answer
76 views

Optimal way to stack deck against uniformly random opponent?

A card game is played by splitting a face-down shuffled deck evenly between two players. The deck consists of cards numbered 1-52. Each player reveals the top card of their deck, and the player whose ...
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1answer
21 views

Maximizing the product of two numbers in a set given their differences

This is an interview problem from Glassdoor that I've been unable to solve. You have all the clubs from a deck, 13 cards, and you can choose 2 from the deck and get paid their product, where all ...
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4answers
126 views

Selecting $2$ cards from a full deck of cards.

I thought of a problem earlier and I am quite clueless on how to solve it, or begin solving it, because I cannot find a way to easily compute the amount of combinations of $2$ cards that sum to a ...
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3answers
40 views

Odds of drawing any 3 identical cards from 27-card deck (9 unique x 3 copies each) in 9 draws

I have a 27-card deck with 3 copies of each of 9 distinct cards. If I'm drawing 9 cards, what are the odds that I draw all 3 copies of any one of the 9 possible options? [Also known as "What are the ...
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1answer
38 views

Odds of having at least one pair in a 4-card hand

I'm trying the calculate the odds of having at least one pair(or "better" such as three or four of same cards) in a 4-card hand. I made the calculations in 2 different ways but the math doesn't add up....
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4answers
69 views

How adding a joker makes events dependent?

I have a question about probability. Imagine we have a deck of 40 cards with 4 different suits of ten cards. Define two events: $A \equiv $ We take a card and it's an ace $\Rightarrow P(A)=\frac{1}{...
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1answer
29 views

General formula for probability of drawing certain hands of cards

I have a shuffled deck of $d$ cards containing $g$ “good” cards. I'm going to draw a hand of $h$ cards (without replacement) and I want it to contain exactly $w$ good cards. (Change the variables if ...
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1answer
44 views

Recompute expected value of the sum of value of five face down cards when the lowest value card is replaced by a new face down card

Recently I heard of this puzzle/riddle type question that was asked in an application interview and I am unable to solve it. The problem is as follows Stage 1: The interviewer holds a regular deck ...
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0answers
60 views

Probability of a straight in a card game I'm designing

I am developing a card game with a friend and I want to calculate the probability of different win conditions being achieved (I am putting together an excel sheet). I have trouble calculating the ...
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0answers
107 views

Could the Collatz Conjecture be related to Solitaire?

When observing different variations of the Collatz Conjecture, I found that stumbling across loops instead of watching the trajectory decay to one happens a lot. When playing a game of Klondike ...
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1answer
25 views

Probabilities of hand draws in Keyforge

Keyforge is a new game by Richard Garfield (who created Magic: The Gathering, King of Tokyo, etc.) that utilizes 36 card decks composed of three houses, 12 cards of each. I'm trying to figure out ...
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1answer
29 views

Combinatorics deck/dice

A player has 52 cards and dice. Every time he picks a card (without seeing it) throws the dice. How many cards at least does he have to draw to make sure that discovering them there are at least 4 ...
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2answers
37 views

Playing Cards question

Say I have a hand of X playing cards drawn from a standard 52-card deck, where X can be 1 through 10. Then say I also have a second deck of the 52 standard playing cards, and draw a card from there. (...
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1answer
46 views

Calculating combinations of a straight in Texas Hold 'Em

Similarly to a previous question that I've asked, I have now attempted to calculate the amount of all possible seven card combinations that would result in a straight (and nothing better). The ...
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1answer
31 views

Calculating combinations of three of a kind in Texas Hold 'Em

The question sounds quite simple - in a game of Texas Hold'Em (or simply out of ${52 \choose 7}$), how many combinations of cards are there, that would result in three of a kind being the highest hand?...
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3answers
60 views

Your friend starts first. What is the probability that you win?

Imagine you’re playing a game of cards with a friend. You decide to keep it a simple game: taking turns, you both take a card from a standard $52$-card deck (first your friend takes a card, then you,...
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1answer
47 views

Dual 52 card deck, pair of exact card match (position and rank) probability

Suppose we have 2 well shuffled decks and alternately draw 1 card from each deck without replacement, placing the cards in a 2 deep 52 wide pattern, and looking for exact card matches between the 2 ...
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2answers
29 views

Six-card hand with $2$ cards of each suit - did I calculate this correctly?

Suppose you have a deck of $36$ cards - $3$ different suits, $12$ cards per suit. If you draw a $6$-card hand, what is the chance of a hand with $2$ cards of each suit ($2-2-2$)? I would do $$\frac{\...
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2answers
89 views

A simple model for the card game Dominion

This problem is inspired by the card game Dominion. We play a card game. At the beginning, we get one card "Copper". At each round, we draw a card uniformly at random form cards that we already have....
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36 views

Card probability (15 cards, 9 unique suits, odds of specific possibilities acros three 5 card hands)

A randomly shuffled deck of cards consists of 15 cards total of 9 suits. We'll designate the suits as A1, A2, A3, B1, B2, B3, C1, C2, & C3. The deck contains one card each from suits A1, A2, B1,...
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43 views

Card Expected Value with Option to Skip Cards

In a normal deck of cards, you can either reveal the top card or guess whether that card is black. If you reveal the top card, you get to see what the card is and the game continues with one less card ...
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1answer
225 views

3 cards are drawn at random from a pack of well shuffled 52 cards.find the probability that all the 3 cards are of the same suit.

Three cards are drawn at random from a pack of well shuffled $52$ cards. Find the probability that all the $3$ cards are of the same suit. My two attempts: $\displaystyle\frac{ {13 \choose 3} }{ 52 }...
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1answer
50 views

Drawing self-improving cards: get good or remove bad first?

Suppose there is a single player card game in which there is a deck of $n$ cards. A card is either red, green or blank. Initially, the deck consists of $r_0$ red, $g_0$ green and $b_0$ blank cards ...
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1answer
39 views

combinatoric problem: probability of different wins (outcomes) in the some way similar to the the BINGO game

In game, we randomly generate four grids (cards) $3\times 5$ (row × col), with each column containing 3 numbers randomly selected without replacement from 18 possibilities: first columns from ...
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1answer
94 views

Estimate number of cards needed to be drawn from deck before full house

Imagine we have a well-shuffled deck of cards and we keep drawing cards until there is at least one full house in the drawn cards. How many cards will we draw on average? I would be interested in both ...
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1answer
60 views

How many 5-card hands from a standard 52-card deck contain exactly 1 king and exactly 1 heart?

Is my solution correct? There are two cases: Case 1: (the chosen heart is not a king) So there are $3 \choose 1 $ ways to choose the king, and $12 \choose 1 $ ways to choose the heart, and then $...
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1answer
30 views

What is the probability that only one of the cards will have the matching suit?

A group of 4 friends are playing a card game using a standard deck of 52 cards. Each friend receives one card from the deck, and the next card is flipped up. Only one of the friend has a card that ...
2
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1answer
28 views

Indicator Random Variables for Card Game (2 Red in a row)

Working through some practice problems in Introduction to Probability, Blitzstein. (I found similar problems to this one but want to make sure I understand this concept.) You have a well-shuffed 52-...
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2answers
40 views

Probability of Infection in Who Goes There / The Thing (the game)

In the game Who Goes There, you are fighting for survival with other players (up to 6 can play) and fending off The Thing, an alien who infects and attacks your camp. To be infected, you take an ...
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Calculating probability distributions for card game

I am creating a card game for a project and I need to calculate the probability and distribution when playing the game. I do not know how to go about it, and would appreciate some insight on starting. ...
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0answers
29 views

Probability of picking a card randomly vs always picking the first card?

I shuffle a deck of cards. Each card appears there only once. Now, I need to pick a card randomly. I can do this in two ways: literally, pick a card somewhere in the beginning/middle/end of the deck ...
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1answer
94 views

Expected value of a card game with 6 cards

A deck contains 3 cards with +1 value and 3 cards with -1 value. The dealer shuffles the cards and deals them face up one by one. After each card is dealt, you have the option of stopping the game. ...
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144 views

How to calculate probability for card game

I am creating a card game for a project and I need to calculate the probability and distribution when playing the game. I do not know how to go about it, and would appreciate some insight on starting. ...
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1answer
122 views

Drawing Cards and Expected Value

In many strategic board games, there is a stack of cards with events. I'm interested in the number of cards that need to be drawn until you can expect a certain card or a combination of certain cards ...
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2answers
67 views

Probability of winning/losing a lonely card game when you shout an integer progressively each time you draw

Take a standard face-down 52-deck of cards, including 13 ranks of four suits (for simplicity, assign value 1 to 13 from ace to King). Draw the cards one by one, shouting the integers from 1 to 13 (and ...