# 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.

513 questions
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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 ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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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### 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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### 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 ...
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### 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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### 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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### 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: ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...