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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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1answer
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How many ways are there to win Settlers of Catan?

The object of the board game "The Settlers of Catan" is to obtain 10 "victory points". There are five ways to obtain victory points: Settlements, worth 1 victory point. Each player starts the game ...
1
vote
1answer
36 views

Version of Fitch Cheney's card trick

Fitch Cheney's card trick is well known. Alice picks five cards from a deck. Bob takes them, gives one back to Alice and arranges the other four in some order. Chuck then enters the room, looks at ...
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0answers
25 views

Probability equivalence of sum of card drawn from a deck

Two people draw 5 cards each from a well-shuffled pack of cards. What is the probability that the sum of numeric values of the two people is the same? Ace is 1, King is 13, Queen is 12 and Jack is 11. ...
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1answer
36 views

Probability of having drawn ace of spades after drawing a card and putting it back in the deck

I have been wondering if the probability of drawing a card from a 52 card deck, obviously 1/52 probability, is different from the following. If you get to draw a card from a 52 card deck and right ...
1
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1answer
32 views

Probability of a certain 5 card hand from a standard deck

A poker hand consists of 5 cards randomly dealt from a standard deck of cards without replacement. What is the probability that you're dealt a hand that contains exactly one pair of red Queens with ...
1
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1answer
3k views

How many different ways are there to get a 21 in Black Jack?

Using one deck of cards. Taking into account different suites. However, the way they are arranged would not be accounted for. Ace of Spades and Queen of Diamonds would be the same as Queen of ...
1
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1answer
44 views

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, ...
0
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0answers
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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0answers
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. ...
4
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2answers
683 views

How do you calculate the odds that the odds will be right?

So in this instance I have a standard deck of 52 cards and am playing a high/low game with it (ie turn over the top card, guess if the next card is higher or lower) and maintain a record of all the ...
1
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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
56 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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2answers
275 views

Is it possible to get the largest conceivable score in Hearts?

I should explain the necessary rules: in the four-player game of Hearts, the object is to get as few points as possible. The points you receive are determined by the cards you pick up through a hand: ...
1
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1answer
45 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 ...
0
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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 ...
2
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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 ...
2
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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 ...
1
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1answer
75 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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3answers
784 views

Probability of getting at least 1 ace in 5-card hand.

I know the correct answer is $1 - P(\text{no aces}) = 1 - \frac{\binom{48}{5}}{\binom{52}{5}}$ But I cannot think of why $\frac{{\binom 4 1}{\binom {51} 4}}{\binom {52} 5}$ is wrong. Does this ...
3
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1answer
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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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 ...
2
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1answer
51 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 ...
7
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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 ...
1
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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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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 ...
0
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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. ...
0
votes
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 ...
0
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2answers
790 views

What is the minimal number of different symbols in the game “Dobble”?

There is a game called "Dobble" (or "Spot It!" in some countries) which implies an interesting problem I couldn't solve. The game consists of some amount of cards $c$, which have $s$ distinguishable ...
3
votes
4answers
128 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 ...
0
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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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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 ...
2
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4answers
71 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}{...
0
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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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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 ...
0
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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 ...
5
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2answers
399 views

Is it possible to be “too good” at Spider Solitaire?

There was a similar question here: Losing at Spider Solitaire However, what I'm asking is different. The game has a rule that it would not deal the next ten cards, unless there is already a card in ...
0
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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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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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0answers
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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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
47 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
33 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?...
3
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3answers
61 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,...
0
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1answer
48 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{\...
4
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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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9answers
13k views

What is the algorithm to generate the cards in the game “Dobble” ( known as “Spot it” in the USA )?h

In the game Dobble ( known in the USA as "Spot it" ) , there is a pack of 55 playing cards, each with 8 different symbols on them. What is remarkable ( mathematically ) is that any two cards chosen at ...
3
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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 ...
2
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0answers
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 ...