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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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7answers
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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 ...
5
votes
1answer
368 views

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 ...
5
votes
4answers
5k views

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 ...
11
votes
3answers
754 views

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, ...
1
vote
1answer
8k views

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 ?
4
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3answers
1k views

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 ...
14
votes
4answers
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 ...
13
votes
2answers
280 views

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) ...
7
votes
1answer
26k views

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 ...
11
votes
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 ...
8
votes
1answer
906 views

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 ...
5
votes
3answers
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 ...
4
votes
2answers
513 views

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, ...
14
votes
3answers
2k views

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, ...
6
votes
2answers
97 views

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, ...
4
votes
1answer
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. ...
4
votes
0answers
119 views

Largest subset of $S^n$ without a loop for each element for a particular 3-element commutative magma $S$.

Let $S$ be the $3$-element commutative magma with the operation that gives the same element when the operands are the same, and the excluded element when the operations are different (i.e. $S=\{\{a,b,...
3
votes
2answers
95 views

Simulating a fair die with a 5-card hand

This question is inspired by an earlier post: Card game and dice rolling: Finding random variables with similar probability mass function I want to simulate a fair 6-sided die using 5-card hands from ...
1
vote
2answers
872 views

non-routine application of inclusion-exclusion

What are the circumstances when inclusion-exclusion can't be routinely applied, and some adjustments have to be made ? This question arises from a problem of finding the probability of getting a ...
1
vote
2answers
13k views

What are the probabilities of getting a “Straight flush” in a poker game?

I'm not a pretty much fun of Poker, but I'd like to study that game. What are the probabilities of getting a Straight flush in a Poker game considering this factors? Number of playersHow are cards ...
0
votes
4answers
116 views

Probability of 5 card hand

You have a $5$ card hand from randomly shuffled standard deck of $52$ cards: $X$ - Event that hand exactly contains $1$ spade $Y$ - Event that hand exactly contains $1$ ace How do I calculate $P(X \...
0
votes
2answers
661 views

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 ...
0
votes
2answers
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 ...
33
votes
10answers
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 ...
6
votes
4answers
595 views

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 ...
4
votes
1answer
235 views

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 ...
7
votes
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 ...
4
votes
1answer
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 ...
2
votes
2answers
444 views

Straight Flush probability with a huge hand.

It's easy to calculate the probability of a straight flush when you're dealt $5$ cards. I'd like to ask for the probability of the same when you're dealt half the deck. I seek $P(\text{straight flush}...
1
vote
0answers
52 views

How many cards do you need to win this Set variant

I enjoy the card game Set, and have come up with a few variants based on the concept of assigning card "values" to stacks of cards (that is, each stack of cards is considered equivalent to a ...
12
votes
1answer
4k views

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 ...
4
votes
1answer
1k views

Minimum number of out-shuffles required to get back to the start in a pack of $2n$ cards?

So I'm stuck on this problem. If you perform a faro out-shuffle (i.e. a perfect "riffle shuffle" where the top and bottom cards stays in place) on a pack of 52 cards ($n=26$), you can get back the ...
3
votes
3answers
437 views

Probability: Is the next card the ace of spades or heart two?

A card deck with 52 playing cards is well mixed and the cards are indicated successively, until the first ace appears. Is it more likely that the next card is the ace of spades or heart two? My idea ...
3
votes
3answers
3k views

Probability of having exactly 1 pair from drawing 5 cards

I have an exercise as follows: There is a collection of cards consisting of 52 cards (13 types and 4 colours each type). We draw 5 cards from the collection. Then what is the probability of having ...
2
votes
3answers
315 views

Pick a card from a set with the help of a dice

Some days ago i was playing "Settlers of Catan", an homemade version. In this game there is an event where a players pick, randomly, a card from another player. So here is the problem: can we use the ...
2
votes
5answers
827 views

Probability of drawing exactly 13 black & 13 red cards from deck of 52

We have a normal deck of $52$ cards and we draw $26$. What's the probability of drawing exactly $13$ black and $13$ red cards? Here's what I have so far. Consider a simplified deck of $8$ (with $4$ $...
2
votes
1answer
134 views

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 ...
2
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1answer
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 ...
2
votes
1answer
100 views

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 ...
1
vote
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?...
1
vote
2answers
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 ...
1
vote
2answers
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 ...
1
vote
2answers
204 views

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 ...
0
votes
2answers
139 views

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 ...
0
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1answer
760 views

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 ...
10
votes
1answer
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....
10
votes
3answers
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 ...
4
votes
1answer
703 views

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 ...
2
votes
2answers
258 views

Bulgarian Solitaire: Number of root loops and which numbers have multiple root loops

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......
2
votes
2answers
193 views

How many turns, on average, does it take for a perfect player to win Concentration (adjusted)?

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 ...