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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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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. ...
11
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0answers
498 views

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 ...
9
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0answers
112 views

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 ...
6
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46 views

Deck of 5 cards Shuffling Problem but only allowed to choose two adjacent cards

Say I have a deck of 5 cards that are labeled 1, 2, 3, 4, 5. 1 being at the top and 5 being at the bottom. The rules of this game are as follows. You can take only take two adjacent cards (1 2, 2 3, 3 ...
4
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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,...
4
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0answers
175 views

An improbable Euchre game — A 9$\spadesuit$ kind of evening

Setup During a game of cutthroat euchre (3 players instead of 4), my girlfriend won the game by playing as her last card, the 9$\spadesuit$. Given that euchre is played with a deck consisting of ...
4
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0answers
534 views

Four of a kind -dilemma

CONDITIONS Imagine, that we have five standard 52-card decks. We will now mix all these five decks into a one (afterwards "big deck", 260 cards in total). Now our task is to turn all the cards open, ...
4
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105 views

How many possible shuffles can be won perfectly?

It is known that the possible shuffles of a deck of cards is $52!$, or ~$80658175170943878571660636856403766975289505440883277824000000000000$ different combinations. I have become aware of a game ...
4
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0answers
102 views

Seven-Card Stud with Random Hand Selection

I was recently confronted with a number—$2727707$, actually—that started a short train of thought while I was placed on hold. (This seems to happen quite often: both the observation of unusual ...
3
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0answers
51 views

Probability of a pulling a card (king) followed by any other card(s) that are not kings

Im trying to learn how to calculate the probability of events and Im unsure how to evaluate one of my practice questions. "What is the probability of pulling a king out of a pack of 52 cards followed ...
3
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0answers
46 views

Find most varied match assignments for a 4-player card game

I'm a programmer and confronted with a particularly hard (at least for me) problem I couldn't find an answer for. This is not a school task or anything. It is something I need personally. I've ...
3
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0answers
67 views

How to punish a riffle shuffle?

It is common knowledge that for a deck to be considered randomly sorted, at least seven riffle shuffles should be used. However, in my experience, very few people will take the time to complete seven ...
3
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0answers
93 views

Shuffle Deck $7$ times or $8.55$ times?

Persi Diaconis showed in $1992$ that in order to shuffle a deck of $52$ cards, you need at least $7$ riffle shuffles. However, in the paper he published, he showed that we needed $\frac{3}{2}\log_2n$...
2
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0answers
43 views

Card Expected Value with Option to Skip Cards

In a normal deck of cards, you can either reveal the top card or guess whether that card is black. If you reveal the top card, you get to see what the card is and the game continues with one less card ...
2
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0answers
83 views

Is a perfect game of Set always possible?

For anyone not familiar with the game of Set, I'll refer you to the description on this question. My question is this: The game ends when there are no more cards remaining in the deck and there are ...
2
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0answers
157 views

Post-test probabilities in the game The Resistance

Some of you might be familiar with the social deduction game, The Resistance. It's a fun game of asymmetric knowledge where traitors try to hide within a group and covertly fail a mission. It's also a ...
2
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0answers
51 views

Probability of making a fullhouse with a $39$ card deck

If i have a stripped deck consisting of only $3$ suits instead of all $4$ suits, what is the probability I am able to make a full house? I am not sure where to begin but I have the following $${_{13}...
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0answers
120 views

The most probable score in Microsoft Spider Solitaire

My dad plays a lot of Spider Solitaire on his computer. He told me he noticed he solved it very often in $299$ moves. I'm not a mathematician, but I think someone must have thought of the ...
2
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0answers
131 views

Sufficient number of riffle shuffles for randomizing a deck of n cards.

It has been proven previously that for a deck of $52$ cards, $7$ randomized riffle shuffles are enough to randomize the deck to the point that any further riffle shuffles will not meaningfully get the ...
2
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0answers
252 views

The hardest game of mahjongg

I was playing Mahjongg solitaire the other day. It got me thinking... The board has $2n$ pieces at the beginning and assuming that the game is winnable. The game would be trivial if there would be ...
2
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0answers
101 views

Is there a better way for calculating the odds for the horserace card game?

horserace card game in progress Game: The 4 aces ("horses") are laid face-up at one end of the table. 6 cards face-up are laid ("race track") in a straight line perpendicular to the aces. The race ...
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0answers
57 views

Ideal Card Game

I have invented a very interesting card game. All the cards from 2 to 10 (in four colours) are divided evenly between the two players (the deck is shuffled before dealing the cards, of course). Now ...
2
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0answers
79 views

How many shuffles are really needed for bridge?

According to the Gilbert-Shannon-Reeds model (which apparently models reality well), one should riffle shuffle seven times to achieve a suitably randomized $52$ card deck. However, it occurs to me ...
2
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0answers
358 views

Winning strategy for solitaire?

I'm talking about the Klondike solitaire, turning three cards at once to the waste and placing no limit on passes through the deck. I know there isn't always a winning strategy, a counterexample can ...
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107 views

Could the Collatz Conjecture be related to Solitaire?

When observing different variations of the Collatz Conjecture, I found that stumbling across loops instead of watching the trajectory decay to one happens a lot. When playing a game of Klondike ...
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0answers
65 views

card war game probability

The "card war" game is played with 2 players and 8 decks (52 cards each, 416 total), the purpose of any player is to score a higher value card, where the values are ordered: ace < two < ... &...
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0answers
58 views

Card probability for more complex hands

My friends and I were playing a game -- as part of this game you can draw cards to gain a benefit. Now, over the course of this we got into a debate over the probability of various things occurring. ...
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0answers
391 views

Blackjack starting Combinations

I gather from several sources that the number of starting combinations of dealer up card and the 2 player hole cards is 550. I was thinking along the lines that there are 10 distinct values. Of which ...
1
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0answers
31 views

New expected value given condition (card game)

I have 15 distinct cards, each taking on a value of 1 to 15 (i.e. Card 1 has a value of 1, Card 2 has a value of 2, etc). I understand that expected value of each card is 8. Now say there are 5 ...
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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 ...
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0answers
143 views

Combinatorics applied to a playing card game

I want to analyze the features of a deck of Magic the Gathering cards game, which many of you know for sure. My goal now is to create an Excel sheet, where you can enter the composition of the deck (...
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0answers
108 views

Big Two's classification according to game theory

Is the game Big Two, as described in https://www.pagat.com/climbing/bigtwo.html, classified as: a game of perfect or imperfect information? deterministic or stochastic? EDIT: I am fairly certain it ...
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77 views

Combinatorics while drawing cards

He have a deck with $52$ cards and $4$ suits, $H$,$D$,$C$,$S$, from which you take $26$ cards. There are some combinations that, when taken with the same suit, give a prize. Let's those combinations ...
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0answers
70 views

When does it become impossible to lose a game of FreeCell?

After spending more time than I should playing FreeCell, I've been wondering about the following: At what point in the game does it become impossible for someone to lose? "losing" probably isn't ...
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0answers
152 views

Poker Hand Probabilities in Doomtown Reloaded

The short version of my question is as follows: What is the best way to compute the probability of different hand ranks after redraw in poker? Can this easily generalise to a non-standard deck? ...
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0answers
83 views

Finding the 'best' way in a card-arrangement-game

Let $n\ge 2\in\mathbb N$. Suppose that we have a card on which $1$ is written, a card on which $2$ is written, $\cdots$ , and a card on which $n$ is written. Now these $n$ cards are arranged from left ...
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239 views

Card combinatorics problem

Suppose we have a standart deck of cards. It is fixed and can have maximally 52 cards, but may be less. Also we have several subsets of this deck, each subset consist either of all cards of the same ...
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0answers
26 views

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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0answers
21 views

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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0answers
42 views

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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0answers
60 views

Probability of a straight in a card game I'm designing

I am developing a card game with a friend and I want to calculate the probability of different win conditions being achieved (I am putting together an excel sheet). I have trouble calculating the ...
0
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0answers
36 views

Card probability (15 cards, 9 unique suits, odds of specific possibilities acros three 5 card hands)

A randomly shuffled deck of cards consists of 15 cards total of 9 suits. We'll designate the suits as A1, A2, A3, B1, B2, B3, C1, C2, & C3. The deck contains one card each from suits A1, A2, B1,...
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0answers
22 views

Calculating probability distributions for card game

I am creating a card game for a project and I need to calculate the probability and distribution when playing the game. I do not know how to go about it, and would appreciate some insight on starting. ...
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0answers
29 views

Probability of picking a card randomly vs always picking the first card?

I shuffle a deck of cards. Each card appears there only once. Now, I need to pick a card randomly. I can do this in two ways: literally, pick a card somewhere in the beginning/middle/end of the deck ...
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0answers
146 views

How to calculate probability for card game

I am creating a card game for a project and I need to calculate the probability and distribution when playing the game. I do not know how to go about it, and would appreciate some insight on starting. ...
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0answers
13 views

Probability of doubles on 2 shuffled deck of cards and gaining doubles

I am not asking for questions on homework or anything related to school or work, I am asking for help with a card game I was playing with my friends. I was playing a card game called Tractor(its a ...
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0answers
28 views

How to shuffle more than 1 deck of cards

I know that after "ruffle shuffling" 1 deck (54 cards) 7 times, every configuration is nearly equally likely. How should I shuffle (for example) 3 decks (162 cards) assuming I can shuffle together ...
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0answers
22 views

Probability of reaching 20 by drawing X cards from a single deck

A Blackjack inspired question i thought of after reading several existing questions, which lead me to this formula http://probability.infarom.ro/blackjack.html which is fairly simple for calculating ...
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0answers
31 views

Calculating probabilities for drawing specific cards from multiple decks?

I've been trying to get an achievement in a game called 'Sentinels of the Multiverse' (for context, its a superhero-themed card game). The achievement requires you to get insanely lucky with your ...
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0answers
35 views

Is bridge a fair game: references/heuristics?

Though not knowledgeable on card games or game theory, I've just learnt the rules of bridge, and I'm starting to wonder whether it can be proven that bridge is a fair game. Of course, to give an ...