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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32 views

Probability of getting a specific set of rolls atleast once after n dice rolls.

I am trying to calculate average number of turns it will take to win in Catan given a set of hexes. I am stuck at calculating probability of an event given n rolls. Each roll uses 2 6-sided dice. You ...
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1answer
45 views

Multiset definition of "all" [closed]

The multiset $\{n,n,n\}$ is a multiset containing elements that are all $n$. It does not contain non-$n$ elements. The empty multiset $\{\}$ also does not contain non-$n$ elements. Is it also a ...
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How can you measure how "shuffled" a deck of cards is?

A few days ago I asked for some methods of measuring how shuffled a deck of cards was. Predictably there were a lot of suggested methods, which got me thinking, which is the best one? I think it'd be ...
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119 views

How do I measure how "random" a deck of cards is? [closed]

I'm doing a project on shuffling cards and I've come across an interesting problem. At some point I need to look at a deck of cards and be able to "measure" how random the deck is. I'm using ...
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probability of drawing at least 3 cards of the same number among 4 [closed]

The question is the following: You draw 4 cards from a deck of cards, what is the probability that you get at least 3 of the same number. (I'd appreciate calculations as I need to generalise this for ...
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47 views

How to calculate chances of beating two opponents with one card?

SETUP: Standard 52 card deck. Game is O-hell. Trump game where any card from one arbitrary suit beats any other card from another suit. 3 players, A, B & C. A has the lead. Each player has one ...
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0answers
111 views

Puzzle: Good card of 2021*2021 cards of 2021 colors

Assume there are 2021 kinds of colors, and for each color, we dye 2021 cards with this color. Now we put these 2021*2021 cards in a straight line. We define a card as "good" card, if the ...
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1answer
29 views

Probabilities of repeated card draws

I draw from a deck of 40 cards 9 cards (edit: without replacing them). 3 of the cards in the decks are a Joker. What is the chance I drew at least 1 Joker? My initial thought was: "Easy!, the ...
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0answers
64 views

Probability of everyone having 4333 distribution in bridge hand with given information

In a game of bridge, there are $4$ players, each is dealt a hand of $13$ cards coming from $4$ suits each of which has $13$ cards in that suit. We say that a player has a 4333 distribution if they ...
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0answers
72 views

Is this weird solitaire card game a real game or just a multi-generational prank?

I have a question about a weird, unknown (and possibly impossible) game a friend of mine has taught me. It is a solitaire game that's played with a regular deck of 52 cards but it is suspiciously ...
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1answer
30 views

Arrangements of a standard deck of $52$ playing cards if the denominations of the cards are ignored, so that only the suits are distinguished?

I am struggling to see why my solution is wrong for this question. My solution: Number of arrangements of length $k$ is $4^k$ and, since the maximum length is $52$ we have that the total possible ways ...
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0answers
43 views

Proability with card games of sequence of length 2

Consider the following two-player card game using a standard deck of cards. • Players A, B each choose a length-two sequence of colours (e.g. RB, BR). • Cards are then revealed from a standard deck of ...
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146 views

Provably Unwinnable Spider Games

I would like to know what fraction of Spider Solitaire games (played with 104 cards and four suits) are provably unwinnable. It seems various people have various ideas about what constitutes an ...
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2answers
96 views

Probability with card deck flips

Here is my problem : we flip cards from a 52-card standard well-shuffled deck until the first club appears. I am looking to calculate the probability that the next card at the $k+1$th flip is also a ...
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42 views

Draw n cards k times. Probability that we got exactly all red cards at least once?

We have a standard deck of 52 cards, that has 26 red and 26 black cards. We draw n cards from the deck at the same time, then return them and reshuffle. We repeat that k times. The question: What is ...
4
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1answer
45 views

Martingale and card withdrawal

I have deck of 52 shuffled cards from which I draw cards one by one and an indicator $I_k$ where $ I_k \in \{0, 1\}$ that shows the k-th card is a King or not. My filtration is $\mathcal{F}_n = \sigma(...
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1answer
293 views

Sum of the total number of "moves" needed to sort the n cards for each permutation.

We have n cards numbered from 1 to n. We get pre-shuffled cards and we have to sort them in ascending order according to some rules: we look at the number of the first card in the sequence (let's ...
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0answers
47 views

What is the Chance of Poker Flush $5/52$?

What is the Chance of Poker Flush $5/52$ ? I think the answer should be $$\frac{C_4^1C_{13}^5}{C_{52}^5}.$$ But the writer of my textbook said the answer is $$\frac{C_4^1C_{13}^5-C_4^1-C_4^1C_9^1}{C^...
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46 views

What is the probability of drawing three cards, twice in a row, and having both results be any same suit set in a 53 card deck (52+joker)

I found that the probability of drawing three cards and have them be of the same suit i.e. HHH, DDD, CCC,SSS from a 53 card deck (one joker included) is $(52/53)*(12/52)*(11/51)=0.048835$ I want to ...
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1answer
29 views

Probability of drawing three different suits with a joker included in a single deck.

I found here how to calculate the probability of drawing three cards from a standard 52 card deck and have all three be different suits. It's $39/51⋅26/50=169/425$ which is 0.397647... If a joker was ...
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2answers
64 views

Probability of drawing an ace from a modified deck

Given the following question, I have a deck of $52$ cards. I shuffle the deck, and drop $20$ cards randomly into a shredder. You then draw two cards from what remains. What is the probability that ...
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33 views

Probability of picking card

You have 2 standard decks of cards. You randomly pick 1 card from the first deck and 1 from the second. What is the probability that card from the second deck has a rank strictly higher than the rank ...
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1answer
561 views

Why is the probability of getting a pair in a five-card poker hand so complicated?

The working-out of this question has really confused me. I know the basics of probability but I don't get the calculations here. I think 13C2 * 4C2 determines the number of possible pairs, and the ...
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3answers
28 views

PMF of the turn T at which the 4th ace is drawn

Cards are drawn one by one from a shuffled 52-card deck. Find the PMF of the turn T at which the 4th ace is drawn. Solution given: The event T=t occurs if no ace is drawn in turns t+1 to 52 and an ace ...
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1answer
24 views

Understanding the thinking process behind drawing cards

So we draw cards from a deck and we want to know the probability of taking a king of hearts. The first thinking process: We draw 2 cards together and we do not return them: $$P(A) = \frac{1\times \...
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1answer
24 views

Intuition behind negative covariance of two card draws?

As seen in this link, the covariance of two draws from a standard set of 52 (with the mapping of A, J, Q, K to 1, 11, 12 and 13 respectively) is negative. Is there an intuitive reason as to why it is ...
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0answers
12 views

What is the possibility that 6 of my cards are double cards of continuity?

I was playing a card game where I thought of the problem and could not solve. The rule is that you have to put down the bigger cards than the previous player did. The order is: 3,4,5,6,7,8,9,10,J,Q,K,...
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0answers
34 views

Probability drawn card does not match turn number

We play a game with a (shuffled) standard 52 poker cards deck where each card value is in $\{A,2,3,4,5,6,7,8,9,10,J,Q,K\}$. For simplicity, relabel the 13 values as $\{1,2,3,4,5,6,7,8,9,10,11,12,0\}$....
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60 views

How to Confirm the Mathematical soundness of the bidding table of the Card game Vira

Vira is a Swedish card game of the trick-taking type E.g. similar to Contract Bridge or Spades. It is a 3 player game with each player dealt 13 cards and the remaining undealt stock of 13 cards to be ...
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31 views

Guessing Card Game

Consider a deck with 26 black and 26 red cards. You draw one card at a time and you can choose either guess on whether it is red beforehand or simply observe the result. If the card is red you get $1\$...
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1answer
35 views

Cards laid out in rows

A deck of 52 cards is laid out in 4 rows of 13 cards each. I need to find the probability that All hearts will be in two rows All hearts will be in every row I have a few guesses about the answers, ...
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1answer
66 views

Two piles of 26 cards

There is a 52-card deck. The deck is split into two piles of 26 cards each. I need to find the following probabilities: $1)$ Each pile contains two aces $2)$ The first pile contains more hearts than ...
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1answer
91 views

a curiosity about cards, why this is true?

We split the standard 52-card deck between 13 people, each of them gets 4 cards. In any case it is possible to take away one card from each person to fulfil the following condition: the 13 cards taken ...
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2answers
43 views

Counting the number of ways of getting 4 aces

Suppose I am playing bridge with 3 others, and I want to find the number of ways I could get $4$ aces. The 13 cards are dealt to me first. I had two solutions I thought of, but the numbers are ...
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1answer
39 views

Numbers possible sequences of playing cards with multiple restrictions.

I was trying to work out how many sequences of 5 standard playing cards I could make with the following characteristics. The card in position number 5 is a 5 (of any suit) However the card in position ...
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1answer
83 views

Modelling the variance of dollar values of the rare/mythic slot in a Magic: the Gathering booster pack

I'm trying to model the variance of a certain kind of card that is pulled from a Magic: the Gathering trading card pack. My simplified model is this: I open a booster pack containing exactly one card....
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24 views

About the tie in card game 'Le Her'

I am analyzing a game called 'Le Her', some of you will be known to be a classic card game in the calculus of probabilities at the times of Bernoulli and Montmort. But I have a question about the game,...
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1answer
36 views

N cards, choose 2, what is the probability card 1 comes before card 2.

Initial example: Ace, Jack, and King. What is the probability the Ace will be drawn before the Jack? Answer: 50/50 Reason: 6 possibilities (3!), 3 where Ace is drawn before, 3 where Ace is drawn after ...
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4answers
69 views

Probability of getting a two pair with an ace as last card

What's the probability of getting a two pair(2 pairs of similar cards) in a 5 hand with the last card being Ace. An example would be 2 6's, 2 3's, and an ace. I tried, $$ \frac{13 \cdot \binom{4}{2} \...
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1answer
96 views

Optimal Strategy and Expected Value of card game

The game We have $10$ cards faced down each numbered $1-10.$ For $1\$$ You may select any $2$ cards and I will tell you which is the the minimum card and its value. At any point you may stop the game ...
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2answers
117 views

Expectation value of sum of 3 poker cards (variation)

Assume you have a deck of 52 cards with the values K = 13, Q = 12, J = 11, 10 = 10,..., 2 = 2 and A = 1. Suppose you draw 3 cards $X_1, X_2$ and $X_3$ at random from the deck of 52 cards. Then you ...
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1answer
79 views

Probability of different card hands

I'm looking at doing a game for an upcoming expo and I'm having a hard time with the math for it. I'm wanting to lay out 25 cards (all suited the same so let's say all Hearts) and give out prizes ...
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1answer
72 views

Card Trick of the 'Prime Obsession' book

I saw this problem at the first chapter of the book called Prime Obsession by John Derbyshire: Take an ordinary deck of 52 cards, lying on a table, all four sides of the deck squared away. Now, with ...
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0answers
41 views

What is the distribution of the number of inversions in a riffle shuffle with fixed cut?

Fix $a,b \ge 1$. Starting with a deck of $a+b$ distinct cards in order, consider cutting the deck into the first $a$ cards (collectively $A$) and the last $b$ cards (collectively $B$), and then ...
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1answer
73 views

Expected number of cards

Given a standard 52-card deck, what is the expected number of cards drawn (without replacement) before you get 4 of a kind (4 aces, 4 kings, etc.) I tried to think of the problem from the perspective ...
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2answers
176 views

Expected number of cards turned over before seeing first Ace?

A standard 52-car deck is shuffled, and cards are turned over one-at-a-time starting with the top card. What is the expected number of cards that will be turned over before we see the first Ace? (...
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0answers
60 views

Pigeon hole problem for custom deck of cards

Suppose you have a custom deck of cards. In a traditional deck of cards, each card has 1 of 4 suits and 1 of 13 numbers (ignoring jokers). "Suits" and "numbers" are what I will ...
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2answers
205 views

Which Deck wins the most?

I have two decks of cards, both shuffled randomly and placed face-down. Deck A has 40 cards. 10 are red, while the remaining 30 are black. Deck B has 80 cards. 20 are red, while the remaining 60 are ...
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2answers
50 views

Combinatorics involving a poker hand

How many 5 card poker hands can be dealt from a standard deck of 52 cards with 3 hearts and 2 diamonds? I have tried this but I am not sure if it is correct: ${13\choose3}{13\choose2} = 22308$
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54 views

If you took 18 cards at random from a 54 card deck (jokers included), what is the probability you get 4 of the same card?

Question is pretty self explanatory, I'm just not sure if I'm right. My brain isn't functioning and I want to sleep. Heres the full question in a tiny bit more detail; Suppose you took 18 cards at ...

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