# 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.

59 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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### Derangements with repetitive numbers

Hi to all i have a very simple problem Lets assume that i have a well shuffled deck of 52 cards. I start drawing the top card always and when the card matches it's rank i lose. J=11 Q=12 K=13 If ...
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### Probability of winning the game 1-2-3-4-5-6-7-8-9-10-J-Q-K [duplicate]

A similar question to mine was answered here on stackexchange: Probability of winning the game "1-2-3" However, I am unable to follow the formulas so perhaps someone could show the ...
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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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### A bridge hand void in one suit

A bridge hand consists of 13 cards from a standard deck of 52 cards. What is the probability of getting a hand that is void in exactly one suit, ie consisting of exactly 3 suits ?
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### Probability of winning the game “1-2-3”

Ok, game is as follow, with spanish cards (you can do it with poker cards using the As as a 1) You shuffle, put the deck face bottom, and start turning the cards one by one, saying a number each time ...
7k views

### 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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### 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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### 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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### 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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### 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 ...
1k views

### Probability - Expected number of draws to get all 52 cards at least once drawing in groups of size n

Imagine you have a deck of cards and want to be fairly sure that you draw each card once (with a perfectly fair, complete, and random deck on each draw, of course). You are drawing cards in groups of ...
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### In a Zener card test using a 25 -card pack and no replacement, what is the expected score?

The Zener cards were invented by Karl Zener and used by J B Rhine in his experiments on extrasensory perception (ESP) at Duke University in the 1930s. They comprise cards of each of five types, ...
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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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### In a Zener card test using a 25 -card pack and no replacement, what is the expected score if we try to minimise correct guesses?

The Zener cards were invented by Karl Zener and used by J B Rhine in his experiments on extrasensory perception (ESP) at Duke University in the 1930s. They comprise cards of each of five types, ...
995 views

### Game Theory Matching a Deck of Cards

Moderator Note: This question is from a contest which ended 1 Dec 2012. Suppose we have a deck of cards labeled from $1$ to $52$. Let them be shuffled in a random configuration, then made visible. ...
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### How many 5-cards poker hands are there containing at least 3 of the 4 suits?

How many 5-cards poker hands are there containing at least 3 of the 4 suits? I don't how know how to properly handle this question. There 13 ranks of each card, and 4 suits My Attempt: Begin with ...
156 views

### In the card game “Projective Set”: Compute the probability that $n$ cards contain a set

In the game of Projective Set, it turns out that any seven cards contain a projective set. For fewer than 7 cards, how can we determine the probability that one or more sets exist (in terms of the ...
1k views

### 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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### 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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### In the card came “Projective Set”, show that 7 cards do always contain a set. [duplicate]

In the game of Projective Set, it turns out that any seven cards contain a projective set. How can one prove this? And for fewer than 7 cards, how can we determine the probability that one or more ...
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 ...
196 views

### Optimal stopping in red vs black card game deck of 52 cards

I have a optimal stopping problem that is solved by recursion. I was stumped by this question in an interview once. I am hoping someone can walk me through the reasoning so I can reproduce it on ...
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### Probability of picked cards to be smaller than the largest picked card

I have an assignment for my algorithms module that requires us, amongst other things, to find the equations for the following question. Edit - Question Updated You have n cards with pairwise ...
313 views

### BlackJack Card Counting Probabilities

Note: This is related to this question but I'm going to frame the problem in a completely different way (hopefully a more productive way). Lets say we have a 2-deck blackjack game, half of the cards ...
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### Do the last three remaining cards in a game of Set always form a set?

Question (brief introduction to the game Set is given after the question) When a game of Set gets to a point that there are only three cards left on the table, and all other cards were already ...
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### 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?...
996 views

### Deck of Cards Probability Question - Probability of Getting At Least 2 Queens

There was actually another question like this but the final answer a person mentioned was incorrect and I was confused as to how he got it. Can any answers explain how they got there? I'd like to ...
515 views

### How much would you be willing to pay to play this card game?

Consider a normal $52$-card deck. Cards are dealt one-by-one. You get to say when to stop. After you say "stop" you win $\$1$if the next card is red, lose$\$1$ if the next card is black. Assuming ...
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### guess the color of the next card, what is the payoff of this game?

There are 26 red cards and 26 black cards on the table which are randomly shuffled and are facing down onto the table. The host turns up the cards one at a time. You can stop the game any time (even ...
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### Prove that this card game between 2 people always ends in a finite amount of moves.

Update: Bram28 and Barry Cipra have given very good answers! I've also fixed a lot of my examples because of the better solutions. Also, as per Barry Cipra suggestion, I've created a follow-up ...
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### Conditional probability within a poker hand

Given a 5 card poker hand from a standard deck, I'm looking to calculate the probability of getting: all 1 suit, 2 different suits, 3 different suits or 4 different suits, with the additional ...
842 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....
742 views

### 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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### BlackJack Card Probability when Counting Cards

In a single deck blackjack game - if you're not counting cards - the probability that the next card will be a 10/J/Q/K is 16/52. I'm trying to figure out how to adjust the probabilities when you are ...
For an explanation of what Bulgarian Solitaire is, look here. I have worked out the full graphs for $1 \leq \text{number of cards} \leq 13$ in the past, and all of them had just one root loop......