# 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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### What is the math behind the game Spot It?

I just purchased the game Spot It. As per this site, the structure of the game is as follows: Game has 55 round playing cards. Each card has eight randomly placed symbols. There are a total of 50 ...
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### How to mentally flip a coin?

If you've ever played rock-paper-scissors, and you are reading this on math.stackexchange, you probably know that always playing $1$ of the $3$ choices at random (more precisely: uniformly at random ...
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### Mathematics behind this card trick

Suppose I have $21$ playing cards. I distribute them in $3$ columns and tell you to choose mentally a card. Then just indicate in which column the card is. I pick up one of the columns which doesn'...
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### When can we quit a game of War?

Consider the game of War. (The rules are below.) It would be nice to be able to end the game early. Suppose, for example, one player has 50 of the 52 cards. It is very likely that he's going to win. ...
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### Probability Discrepancy in drawing 2 cards from a Deck of 52

A colleague and I are having a hard time figuring out this probability question and was wondering if anyone could provide an explanation / insight into it. The question is simple: what is the ...
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### What is the probability that a solitaire game be winnable?

By "solitaire", let us mean Klondike solitaire of the form "Draw 3 cards, Re-Deal infinite". What is the probability that a solitaire game be winnable? Or equivalently, what is the number of ...
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### In the card game Set, what's the probability of a Set existing in n cards?

Given $n$ randomly drawn Set cards on a table from a standard 81-card deck, how can I determine the probability of one or more Sets existing on the table? First, for those who may not be familiar ...
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### Combinatorics: Number of possible 10-card hands from superdeck (10 times 52 cards)

I have the following problem from book "Introduction to Probability", p.32 A certain casino uses 10 standard decks of cards mixed together into one big deck, which we will call a superdeck. Thus, ...
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### Card guessing game

There is a pile of $52$ cards with $13$ cards in each suit (diamonds, clubs, hearts, spades). The cards are turned over one at a time. At any time, the player must try to guess its suit before it is ...
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### Is the probability of picking an Ace as the second card of a deck the same if the deck is shuffled after the first card?

Here's the situation: You have a deck of 52 cards that is already shuffled. You pick the first card, and the first card is not an Ace. Is the probability of drawing an Ace as second card the same if ...
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### Probability of getting A to K on single scan of shuffled deck

Let us say we have a regular 52-card well-shuffled deck. We scan through the deck (first to last) till we find an Ace. Then we continue (from that Ace) till we find a 2. Then we scan (from the 2) ...
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### A card game with no decisions

A friend showed me a mindless card game he plays, in which the initial state of the deck completely determines whether he wins or loses. The game is played as follows: Shuffle a standard $52$ card ...
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### Losing at Spider Solitaire

Spider Solitaire has the property that sometimes none of the cards in the final deal can "go" and so you lose, regardless of how much progress you have made beforehand. You would have known that you ...
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### A question about Poker (and probability in general)

Okay, so I've been thinking about this question for a long time, and I'm starting to think that there isn't an answer. So please read the question, and if there is an answer, tell how you came to it, ...
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### Can you explain this card trick?

Can you explain the card trick that is explained here? Edit: Here's a summary of the trick as explained in the video: Start by asking a spectator to pick any three cards they like out of a ...
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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 ...
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### A magic trick - find out the fifth card if four is given

Here is a magic trick I saw. My question is how the magician and his partner did it. Given the simple French deck of cards, with $52$ cards. A person from the audience chooses randomly five cards ...
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### Algorithm for a deck manipulation

Let's say you have a randomised deck of $N$ different cards. An $M$-action ($M\le N$) is defined as follows: you look at the top $M$ cards of the deck, put as many of them as you choose on top of the ...
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### Counting how many hands of cards use all four suits

From a standard $52$-card deck, how many ways are there to pick a hand of $k$ cards that includes one card from all four suits? I know that for any specific $k$, it's possible to break it up into ...
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### How many turns, on average, does it take for a perfect player to win Concentration?

Let's say a person with perfect memory is playing the card-matching game Concentration. He/she randomly lays out 2n cards (ie. n card pairs) sorted randomly and repeatedly takes turns consisting of ...
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### Cards game and probabilities

From a standard 52-cards deck we randomly pull out 2 cards. One is black and the other is a 4. We choose one of the two cards. What is the probability: The card we pick is a 4 The card ...
843 views

### On the probability of obtaining 27 sets in the card game Set

For people not familiar with the card game Set, see its entry on Wikipedia and/or one of the related questions here on Math SE. It might be faster to just play the game a couple of times though, see e....
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### At what rate does the entropy of shuffled cards converge?

Consider a somewhat primitive method of shuffling a stack of $n$ cards: In every step, take the top card and insert it at a uniformly randomly selected one of the $n$ possible positions above, between ...
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### Probability of a Full House for five-card hand.

So I know this can be solved easily by counting the total ways to make a full house and dividing that by the total possible hands, but I want to know why another way I thought of to solve it is wrong. ...
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### At time n, randomly choose a natural number ≤n. How long is it until a single number is chosen three times?

To clarify, the number ≤n is chosen uniformly at random at each step, and n chooses from the natural numbers beginning with 1. I wish to determine the expected value of $n$ at which a natural number ...
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### Combinatorial Card Flip Game

Recently, I've been given an interesting type of problem that I am having a difficult time proving. The problem revolves around a game, with the rules as follows: "You have a deck containing n cards ...
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### Asymptotic value of card drawing game

A deck consisting of $r_0$ red cards and $b_0$ black cards is randomly shuffled. The host turns up the cards one at a time; if it is red, you get $\$1$; otherwise you pay the host$\$1$ (and you're ...
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### Modify the rules of Gomoku (Five-in-a-row) or Connect Four type games to enforce the fairness among players

One colleague and me were discussing this problem during lunch today, and I did a little bit digging for several hours after returning to my office. Fact: For an $(m,n,k)$-game, there does not exist ...
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### Bulgarian Solitaire: Size of root loops

I first learned of Bulgarian solitaire from one of Martin Gardner's books a while ago and have since investigated it somewhat. Google-searching has revealed that surprisingly little work has been done ...
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### Probability with card game

I need help calculating the chances of winning this strange game that I'm going to explain right now: ...
283 views

### Which outcome is more likely, getting four kings within 2x cards or a specific pair of triples (from a group) within x cards?

Suppose 2 people (call them A and B) challenge each other to a card game. A fair, well shuffled, standard 52 card deck is used for each hand. Community (shared) cards are drawn one at a time without ...
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### Probability that a random 13-card hand contains at least 3 cards of every suit?

A random 13-card hand is dealt from a standard deck of cards. What is the probability that the hand contains at least 3 cards of every suit? (Introduction to Probability, p.36) My solution: There ...
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### The probabilty of a new arrangement of 52 cards deck?

I just read this article, it claims that if you just shuffle a 52 card deck, you will mostly be creating an arrangement that no human have ever seen before. But this doesn't seem right since every ...
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### I don't understand why this card trick works

So a friend of mine showed me a very interesting card trick that I don't quite understand. You have 52 normal cards from any regular deck you can buy anywhere. The Ace is worth 1 while 2 through 10 ...
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### Probability question involving sets of cards

I have an infinite deck built out of sets of 10 cards (in other words 10*n cards). The sets are identical so one '2' is identical to another '2'. A player draws 6 cards. If he draws: any '1' AND a '...
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### With a standard deck of 52 cards, how many different 25 card spreads are available?

With a 52 card standard deck, if you need to deal 25 cards, how many different spreads are possible? For example, with a 2 card spread: ...
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### A combinatorial card problem

Suppose you have a standard bridge deck of 52 cards. We will say you "have a pair" if you have two consecutive cards in the deck with the same rank (2,3,4,5,6,7,8,9,10,J,Q,K,A). If a deck is ...
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### Mental card game

Each of n persons draws a card from a shuffled deck of $k$ cards numbered from $1$ to $k$. There are at least as many cards as persons. The winner is the person who is holding the largest card. If ...
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### How to calculate a fullhouse after dealing 5 cards?

In case you are not familiar with a fullhouse, it is when you have XXXYY (three of a kind + a pair). I am not that good calculating odds, but after time I manage ...
217 views

### Proof of solutions to “24” operations game

Consider a deck of cards with values $1$ through $13$, each with multiplicity $4$, so that $$S = \left( \bigcup_{i=1}^{13} \{ i \} \right) \times \{1,2,3,4\}$$ Supposedly, there exists a game in ...
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### Proving a certain type of poker game can always be won

Ok, so I just finished a Discrete Math class where we learned about all types of proofs and counting and graphs, etc. So I know the basics, but never thought I'd need to use it in the real world (not ...
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### Number of 5-card Charlie hands in blackjack

A five-card Charlie in blackjack is when you have a total of 5 cards and you do not exceed a point total of 21. How many such hands are there? Of course, the natural next question concerns six-card ...
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### Card Problem from Gardner

The following problem is from aha! Insight by Martin Gardner. On a draw of $n$ cards from a deck without replacement, what is the probability that seven of the $n$ cards all have the same suit? ...