# Questions tagged [probability]

Questions about probabilistic game mechanics and how to calculate and estimate random events, as well as how probability affects games and gameplay.

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### How do you calculate the likelihood of drawing certain cards in your opening hand?

In Magic, at the start of the game, you draw 7 cards. How would you calculate the likelihood of drawing a specific card in your opening hand? For example, let's say I have a 60 card deck, and I'm ...
24k views

### How can I estimate my chances to win a Risk battle?

I'm looking for an way to estimate the percent chance of winning a specific Risk battle. Assume the maximum number of dice will be used. It doesn't need to be a perfect calculation, just an ...
2k views

### How many spins does it take to complete a game of Hi Ho! Cherry-O?

I recently picked up this game for my 2-year-old daughter, who is learning how to count. I find this game fairly infuriating to play by the actual rules. It contains a spinner with the 7 following ...
510 views

### Are there statistic/probabilty applications specifically made for card game designers?

I am developing a card game that involves building out sequences of matching symbol pairs and also has a lose state for a (different set) of symbols that is hidden to the player. Trying to create an ...
635 views

### What are the odds in Scrabble of not being able to make a legal move on opening turn

I've been playing the game for 50 years and this happened to me recently: I had the following rack PBQZJDH And so I could not play a legal move. I've been trying to work out the odds of this (no ...
3k views

### In backgammon, can you resign, and how can you tell when you should?

In backgammon, there comes a time when all pieces have passed each other; there is no more interaction, you are simply rolling it out until you can bear off all of your pieces. If one player is well ...
2k views

### How do the Monopoly odds change when the board is played in reverse?

One semi-popular house rule is to play Monopoly as usual, but move the pieces counter-clockwise instead of the usual clockwise - that is, hit the higher-end properties first and move down toward the ...
1k views

### Fairness of Cowry Shells as Dice

I just learned about the use of cowry shells as binary dice and find the idea enchanting. However, I'm concerned about the apparent "fairness" of the dice. Obviously, because there are multiple ...
7k views

### What are the chances of going out on the first hand in Rummy?

What are the chances of any particular player being able to go out in the first round of rummy? For two players the hand size is 7. For all other numbers the hand size is 5. Going out requires a ...
20k views

### What are the odds of getting the following hands in euchre?

I really like euchre, but I stink at statistics. Last time we were playing euchre, people would get some crazy hands, and we kept saying, "What are the odds?" So now I ask you all, what are the ...
9k views

### How to calculate the probability to get all cards of Exodia, the Forbidden One by drawing them from my Deck during the Draw phase?

In Yu-Gi-Oh!, there are some cards that allow to the duelist to win a duel automatically. In this case, I'm focusing my question on Exodia, The Forbidden One. To win a duel using Exodia, you must ...
619 views

### Is there a method that gets beneficial diminishing returns when adding more dice, yet stays random?

Short Story: I want to find a dice system where multiple dice are rolled and compared at once, but rolling more dice is only slightly more effective than rolling fewer dice, while still being fairly ...
523 views

### If I put a copper and add 4 more random cards into my hand from my starting deck, does it change the odds of getting the 5:2 split?

I was thinking about a situation where someone accidentally reveals a copper from their starting hand while shuffling their deck or discovers that they only have 9 cards in their starting deck (and ...
449 views

### How many submarines do I need to take down an aircraft carrier?

In Axis and Allies, how many submarines do I need to have a greater than 60% chance to take down a lone aircraft carrier?
857 views

### King of Tokyo: Complete Destruction

What is the expected number of turns required to get the required roll for Complete Destruction, given no evolutions or other cards?
21k views

### Which Faction Power will result in the greatest bonus troops and/or fewest troop losses?

Without taking into account any other packets that might be opened over the course of a game (no spoilers please, only use the starting rules, starting scars, starting stickers). 1) What is the best ...
886 views

### French Tarot (card game): Feasibility of playing a King on the first hand

I had a little debate with some friends about the feasibility of a certain type of play and wanted to get the communities opinion. The question is whether playing a King(Regent) on the first hand is ...
2k views

### Magic The Gathering: probability of casting a 3 drop on turn 3

So, I have a probability related question. In MTG, which is the probability of landing a 3-drop on turn 3, but using the correct combination. For example, I want to cast a Geist of Saint Traft on ...
2k views

### If I really need a card on my start hand, how many mulligans make sense? [duplicate]

There is a funny modern deck where you play Treasure Hunt, but you really need to have it on the starting hand. How many mulligans should I take, if my deck contains 60 cards and contains 4 copies of ...
5k views

### What are the odds of drawing a given card after discarding my initial hand?

Let's say that I have 4 of a card in my 60 card deck. What are my odds of drawing at least 1 of that card in my initial 7 card hand? If I discard those 7, what are the odds of drawing 1 of the 4 in ...
63 views

### How do I determine a character's probability to escape from combat in DungeonQuest (third edition)?

I have 4 cards with the following numbers on them: 1, 2, 3, 4. My opponent has 4 cards with the following numbers on them: 2, 3, 4, 6. We both randomly draw a card from our decks, if my number is ...
5k views

### Risk Legacy: Is it better to roll 1 or 2 defense dice when defending a bunkered location?

In Risk Legacy, there is a Bunker that always adds (+1 to the highest defense die, except when the high die is a 6). Is it better to roll 1 or 2 defense dice when defending a bunkered location.
125 views

### In Spades, what is the probability a suit is distributed in a way, given I hold n cards from that suit?

In Spades, (or any other trick taking game), when I hold n cards of a certain suit, what is the distribution of the other 13-n cards in that suit? In particular, what is the probability that ♦'s are ...
1k views

### What is the average number of turns in a game of Monopoly?

I can't find any research or public information on this topic. Edit: It was requested that I clarify the specifics of my request (e.g. the number of players and rule specifics). Rather than pigeon-...
373 views

### How do you calculate hand probabilities for 10 cards per hand (8 suits); e.g.: Cripple Mr Onion?

Cripple Mr Onion is a card game that involves 8 suits (104 cards in toto), with 2-7 players being dealt 10 cards each (without replacement). I've tried several different websites for hypergeometric ...
158 views

### Does Avis or Iago have a better chance of winning a match?

In my quest to find the answer to this question, I have decided to start smaller. The rules of Button Men are here, and the characters are listed below. Avis: d4, d4, d10, d12, dX (where X is d4, d6, ...
827 views

### Probability of being dealt a 7-card flush in Rummy

Playing 7-card rummy with 4 players, I was dealt a 7 card Spade flush (not straight) and on the first card picked from the deck pulled an 8th card spade and immediately won (it was 23456 8910). What ...
901 views

### How do cards like Brainstorm affect the probability of drawing an out?

I posted this on a Magic forum and got zero responses. I'm playing a High Tide combo deck. Let's say I am Brainstorming with 50 cards in my deck, 23 of which are outs that make me win. I think the ...
1k views

### Calculate Mind Funeral efficiency

I was wondering about the formula on how to calculate the average number of cards milled by Mind Funeral. Target opponent reveals cards from the top of his or her library until four land cards ...
2k views

### Risk (revised edition 2008): Is it better to roll 1 or 2 Defense dice with the Bonus Defense die Major Reward?

The Risk revised edition rules have a Major Reward that grants the defender a bonus defense die. This die will replace the lowest defender's die rolled (unless the bonus die is the lowest). What are ...
1k views

### Calculating the probability of NOT drawing a traitor card in Dead of Winter

If you're playing a 3 player game and hand out 1 card to each player, out of a deck of 3 ally and 1 traitor cards, how likely are the players all allies? I think the math is you have a (3/4)^3 or ...
3k views

### What's the probability of having a combo on the first turn in MTG?

I'm creating a combo deck, and I'm interested in the probability of having a certain combo on the first turn. For this combo, I need 4 different cards. There are 4 copies of each of these cards in ...
116 views

### Token Drawing [All At Once (per player) vs Round Robin] -- Mathematician / Statistician needed for a question

Hello Fellow StackGamers, In Paris Connection (PC), you go player by player drawing a starting number of trains, with each player drawing their entire complement all at once. One of our players ...
166 views

### What are my odds of drawing the rest of my deck

Do to the deterministic nature of the game, if I have a decent memory I can easily know what cards are left in my deck, if not their order. I have two cards in my hand that let me draw a card (1 each) ...
439 views

### Why is 'equity' much more commonly used than 'winning probability'? What is the mathematical formula that links 'equity" and 'winning probability'?

In backgammon, people always speak about 'equity', but never about 'winning probability'. To me, 'winning probability' seems much more simple and much more intuitive than 'equity'. So why is 'equity' ...
727 views

### In Pinochle, what is the probability of a thousand aces and a double run in one hand?

What are the odds of getting a hand containing: A♣ A♦ A♥ A♠ A♣ A♦ A♥ A♠ Tx Kx Qx Jx Tx Kx Qx Jx for some suit x, and another four unspecified cards. Rules Background Double-deck Pinochle is a four ...
879 views

### What are the probabilities of getting any or all five cards of Exodia using a specified monster card effect

In Yu-Gi-Oh!, there exists a monster effect card called Morphing Jar, which has the following effect: FLIP: Both players discard their entire hands, then draw 5 cards. There are more (magic/trap/...
2k views

### Calculating the probability of a combo hand [duplicate]

Before I ask the question, I should mention I already saw this and just don't get it. I have a 40 card deck with cards I'll label: A (3 copies) B (3 copies) C (3 copies) D (2 copies) E (1 copy) F (2 ...
717 views

### Probabilities: Backgammon - final move

So I was playing a lot of backgammon lately, and I noticed something. It seemed like I was "going out" on doubles more often than not. And my initial reaction was "Shouldn't that be, at best, a 1 in 6 ...
276 views

### What's the expected damage from playing Terror of the Peaks, when I have Mirror March out?

If I play Terror of the Peaks while I control Mirror March, what is the expected damage? That is the average of winning 1, 2, 3, ... 'Infinity' flips? I'm sure it's a converging sum, but I'm not sure ...
132 views

### Mahjong board formation and drawing rules

In Mahjong basic rule (this one for example), the board is setup in a spiral formation and the tiles are drawn in order from the 4 draw piles. With all the tiles facing down and shuffled, the ...
65 views

### What is the optimal strategy for deciding dice to reroll when trying to reach a target value?

Let's say that I have N dice (standard d6) and I'm trying to reach a value of x or higher. I roll all of the dice once and I don't get my target value. I can do one reroll and can freely choose ...
283 views

### Probability of drawing duplicates of a card in magic [duplicate]

Was trying to figure this out, and realized I don't know the exact math: In the card game magic you have a deck of 60 cards. You can have 4 copies of any individual card, and you are dealt a starting ...
76 views

### What is the value by percentage of Lindel's reroll special ability?

Here is an image of Lindel's card. After rolling dice ofr an attribute test, you may reroll one die; you must accept the second resul As you can see his stats are 6, 6, 6, 5. He needs to roll ...
2k views

### Use exactly 36 combinations instead of two dice

I like playing Settlers of Catan with one extension: "Catan: Event Cards" The deck consists of 36 dice cards to represent the 36 different combinations that can be rolled with 2 dice. This means that ...
51 views

### Is the Hand pattern probabilities equal to suit pattern probabilities?

In Spades, I wish to know the probability a certain suit is distributed in a particular way. For example I wish to know what is the probability that the diamonds are distributed either 4-4-4-1 or 5-4-...
2k views

### What are the odds of being dealt particular melds in Double Deck Pinochle?

This is a followup to another question: What are the odds of getting a double run in four-handed pinochle? I'd love to know the odds of being dealt particular melds in Double Deck Pinochle. I found a ...
49 views

### Mahjong Probability Graph Question [closed]

Picture from pierebean's post in In Mahjong , what is the chance to get a hand that allows a player to finish on their first turn? I was wondering what the graph on the right meant? How does the ...