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

(NOTE : Final revision of my original answer) The odds of the first tileset not having a single valid word is exactly 91,595,416 / 16,007,560,800 or .5722%, with it occuring once every 174.76378 games....
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What are the odds of beating this obscure solitaire game?

Depending on the deck's starting arrangement, your game either ends in a win or loss, or it never ends. To estimate the probabilities, I simulated 10^8 games (using the Python program below), under ...
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Token Drawing [All At Once (per player) vs Round Robin] -- Mathematician / Statistician needed for a question

No player has a greater chance of drawing an even or uneven distribution than any other. One way of looking at it is to consider permutations of the tokens, where they are laid out in some sequence ...
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How do the Monopoly odds change when the board is played in reverse?

The key change in reversing the direction of movement is the fact that players now leave Jail in the opposite direction. The oranges and reds are particularly profitable because they are within 1-2 ...
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Is there a method that gets beneficial diminishing returns when adding more dice, yet stays random?

You're actually asking two questions: How can I design a function that computes a result on a dice roll that gives asymptotically decreasing benefit to adding more dice? How can I have a system ...
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How likely am I to have a key card in my starting hand in Brawl?

The first mulligan is free in Brawl[CR 903.11g], so it's 3 mulligans leaves you with 5 cards. I shall reply accordingly. After removing the Commander, 59 cards remain in the deck. 58 of those aren't ...
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Fairness of Cowry Shells as Dice

This has been a problem since the days when dice, hand-carved out of sheep's knucklebones, could never be considered mathematically fair. Historically, there have been two principal ways of improving ...
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What are my odds of drawing the rest of my deck

50% Symbolizing the cards that give you a draw a D, and the cards that do not as N, there are 6 possible permutations of draw vs no-draw. In each permutation, the second N card represents the ...
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How do I determine a character's probability to escape from combat in DungeonQuest (third edition)?

The decks here are small enough that we can explicitly list out all the possibilities: your draw, opponent's draw 1, 2 1, 3 1, 4 1, 6 2, 2 * 2, 3 2, 4 2, 6 3, 2 * 3, 3 * 3, 4 3, 6 4, 2 * 4, 3 * 4, 4 *...
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How likely is Yusri to get me an omniscience effect, if I have Krark's Thumb out?

The probability to win all 5 coin flips with Krark's Thumb is 0.75^5, or 23.7%, up from 3.1% without Krark's Thumb. Normally, you would have a 50% chance to win a single coin flip. Since multiple coin ...
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If I really need a card on my start hand, how many mulligans make sense?

If you “need” the card in your starting hand and there is no other option—you take as many as you need until you have it, then stop. If you don't have it when you've mulliganed to 1, you concede and ...
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Does a maximally clumped deck or a mana woven deck require require more shuffling to sufficiently randomize?

It doesn't make a difference. No deck in a known order is more or less random than any other deck in a known order. A deck with all of the lands together requires just as many shuffles for ...
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Calculating the probability of NOT drawing a traitor card in Dead of Winter

The statistic theory presented by the question is best described Hypergeometric Distribution. During the game set up, 2 Non-betrayal objectives per player, and 1 Betrayal objective, are combined in ...
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Magic The Gathering: probability of casting a 3 drop on turn 3

The thing is, having none of those cards is not the only way you'd be unable to cast the Geist on turn 3. You'd have to account for a whole bunch of possibilities: Zero islands, zero plains, and zero ...
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Fairness of Cowry Shells as Dice

Answer found through personal research. Cowry shells have a roughly 30% chance of rolling a 0. Depending on the shell, that can get as low as 18.65% and as high as 39.11%. (At least, with the test ...
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Probability of being dealt a 7-card flush in Rummy

What are the odds of being dealt all 7 cards in the same suit? The odds of pulling 7 spades in 7 cards the number of ways to pull 7 spades (from among the 13 spades) divided by the number of ways to ...
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Is there a method that gets beneficial diminishing returns when adding more dice, yet stays random?

There's a lot of ways to weaken the effects of the dice. Only the best X dice count - this means the more dice you have the more likely you will get a better overall result, but you are still stuck ...
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If I really need a card on my start hand, how many mulligans make sense?

If you really need a card on your starting hand to win, and otherwise you lose, then you have to take mulligans until the card shows up or you run out of mulligans. The chance to have Treasure Hunt in ...
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Magic The Gathering: probability of casting a 3 drop on turn 3

There is a legit mathematical answer to this question, but it's a bit outside my ability. What I can offer you, however, is an example of how Magic players of various levels of mathematical knowledge ...
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Fairness of Cowry Shells as Dice

I think that fairness in the game is more dependent upon everyone using the same cowry shells, and less on having a 50/50 probability. The game would not be fair if one player used coins, and another ...
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What is the optimal strategy for deciding dice to reroll when trying to reach a target value?

You want to go through each of your dice from lowest to highest and, if your chance of success is at least as good if you re-roll that die than if you do not re-roll that die, add that die to your ...
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What is the optimal way to roll for a two pair in Yahtzee?

This is probably closer to being suited to the mathematics SE, but let's go through it. Also, funnily enough as far as I can tell there is no scoring option for "Two Pairs" in Yahtzee as ...
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Magic The Gathering: probability of casting a 3 drop on turn 3

27.6% if you go first, or 34.5% if you go second. This does not account for mulligans. Those numbers were obtained experimentally using the code posted below. There's an error in the math presented ...
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Token Drawing [All At Once (per player) vs Round Robin] -- Mathematician / Statistician needed for a question

Your sampling sequence is an example of an exchangeable sampling sequence. Exchangeable sequences are not independent. The outcome of earlier samples does affect the outcome of later samples. ...
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Calculate Mind Funeral efficiency

This situation can be approximated by the following model: you have an infinite sequence of lands and non-lands, where the probability P that each is a non-land is equal to the fraction of the deck ...
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In Spades, what is the probability a suit is distributed in a way, given I hold n cards from that suit?

TLDR: The probability is about 8%. Distribution of diamonds for any deal With no restrictions at all then dealing a pack between four hands results in the following probabilities for distributions ...
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What are the odds in Scrabble of not being able to make a legal move on opening turn

I'm going to make an estimate from the following assumption: Any hand that contains at least one vowel, y, or a blank allows a valid move. Any hand that contains entirely consonants does not. ...

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

TLDR: This will happen roughly once every 1.7 million hands. Rules Background Double-deck Pinochle is a four player game using a deck of 80 cards (four lots of ATKQJ in each of ♣♦♥♠) (cf. Wikipedia ...
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