Questions tagged [dice]

For questions on dice, small throwable objects with multiple resting positions, used for generating random numbers. This makes dice suitable as gambling devices for games like craps or for use in non-gambling tabletop games.

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1answer
34 views

Finding the threshold at which the second player should re-roll (expected value game)

Player 1 rolls a 10 sided fair die. Player 2 rolls a 20 sided fair die. Player 2 is allowed to reroll a second time, but is not allowed to look at player 1's roll before deciding whether they want to ...
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0answers
23 views

help me resolve this probability case [closed]

1 A box contains 50 dices, half of them are fair, the others are not. Among the latter ones the face 1 appears with probability 1/2, while the other faces appear with probability 1/10. We randomly ...
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0answers
36 views

Best strategy in dice game if you can keep more than one number

The question is similar to the original one, The expected payoff of a dice game, but now we can keep $m>1$ numbers (dice). Formally, let $X_1,...,X_n\sim \text{Unif }[0,1]$ be a sequence of i.i.d ...
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2answers
243 views

Getting a negative variance for the sum of dice rolling

I'm trying to find what I did wrong. If $X$ signifies the sum of what you get from rolling a regular die (1-6), 100 times and $X_i$ for a single roll. Then: $$E\left[X\right]=\sum_{i=1}^{100}\frac{7}{...
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2answers
72 views

Calculate the probability of a “good” dice roll

As part of a bigger game-theory problem, I've been trying to solve a rather simple probability question, and I seem to be getting the wrong answers. Here's the problem: Dice are rolled to determine a ...
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0answers
12 views

Expected number of draws required to identify certain balls out of a pocket holding 3 colors of balls.

A pocket contains in total 12 balls. Two of them are black balls, $k$ $(k<10)$ are white balls, and $10-k$ of them are of other colors. You are asked to identify the two black balls and $m$ $(m<...
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1answer
22 views

Expected payoff of dice game score accumulation for $1,2,3$

Roll a fair die until game stops. Game stops when you roll 4,5, or 6. When you roll 1,2,3, your score increases by 1 and you can roll again. When you roll 4,5, you get paid your accumulated score. ...
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1answer
30 views

Throwing $n$ 6-sided dice, what is the probability that the greatest thrown number equals $k=1,…,6$?

Let $n$ be the number of 6-sided dice thrown and $k=1,...,6$. Further, let $X =$ "Greatest thrown number". What are the probabilities $P[X=k]$ for every $k=1,...,6$? Obviously, $P[X=1] = \...
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1answer
46 views

Expected value of dice game — Can Wald's equality be used?

This is somewhat related to Expected payoff of dice game, where the problem is: You roll a fair $6$-sided die. For each roll, you're paid the face value. The game stops when you roll a $1,2,3$. If ...
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3answers
49 views

Expected payoff of dice game

You roll a fair $6$-sided die. For each roll, you're paid the face value. The game stops when you roll a $1,2,3$. If you roll a $4,5,6$, you can roll again and keep accumulating payments. There are ...
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2answers
129 views

When should I stop playing this dice game?

The rules are as follows: You start with \$1 and roll a six-sided die. If you roll anything but a 1, you double your money (so \$2 for the first roll, $4 for the second, and so on). If you roll a 1, ...
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Using poker hands in a six dice game? [closed]

If I wanted to use the traditional poker hands in a game using six dice instead of five (not necessarily poker), how would the hands be ranked? Using regular six-sided dice. Would three pairs (2,2,3,...
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0answers
29 views

Closed form solution for sum of dice rolls > x

I'm trying to figure out a good closed form solution for a problem which I have already simulated and have a rough answer to. The problem is basically rolling 9 dice- each of different face numbers- ...
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1answer
44 views

Optimal strategy to maximize cumulative sum of dice rolls but the sum cannot be a square number [duplicate]

The rule of the game is as follows: The player rolls a fair six-sided dice repetitively until the game is over. After each roll, if the cumulative sum of all the rolls so far is a square number (1, 4, ...
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1answer
36 views

Probability of getting die roll match (e.g roll a $5$ on roll $5$) at least once on an $n$-dimensional die in $n$ rolls as $n$ approaches infinity

The title pretty much says it all. If you have some n-dimensional die, and you roll the die n times, what is the probability you'll get at least one "roll match" - that you roll the number i ...
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2answers
69 views

Probability of getting a total sum of n when rolling 3 dice

I'm having a bit of trouble understanding the probability of getting a number n when rolling multiple dice. When rolling $2$ dice, I noticed that the probability of getting the numbers $2-12$ goes ...
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0answers
38 views

Dice game probability

Suppose you roll a regular $6$-faced dice. For each roll you are paid the face value. If you get $4, 5$ or $6$ you may roll again. If you get $1,2$ or $3$ the game stops. Find the expected payoff I ...
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2answers
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Expected number of tosses to observe a fixed number $k$ times in a row = 6 * expected number of tosses to observe any number $k$ times in a row.

By "fixed," I just mean, say, observing $k$ consecutive 6s when tossing a fair die. The latter case has the solution $E[N] = \frac{6^k - 1}{5}$, shown Expected number of rolls until a number ...
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3answers
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Is there a general procedure for inducing a formula / recognizing a pattern?

For some problems, it's pretty simple to deduce what a formula should look like after you enumerate a few examples, but for some problems, it's not so clear. For these less clear examples, is there ...
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1answer
28 views

Finding the distribution of the number of tosses it takes to observe all 6 sides of a fair die

I found this question on http://www.cis.jhu.edu/~xye/papers_and_ppts/ppts/SolutionsToFourProblemsOfRollingADie.pdf. It is Q(d): Roll x times until getting all the faces from “1” to “6”, what’s the ...
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1answer
75 views

How to derive the expected number of rolls until a number appears $k$ consecutive times

In Expected number of rolls until a number appears $k$ consecutive times, the formula was given to be $E[N] = \frac{6^k - 1}{5}$. You can prove this formula using induction like in the accepted answer....
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2answers
103 views

A very confusing statistics problem from a high school math student

I have just made an account to ask this question, since nobody has been able to clear up my confusion as of yet. I am a high school student so if I am using incorrect terminology or notation I do ...
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1answer
52 views

Probability of dice roll between values

Context: In calculating the optimal policy for MDP's an algorithm called Value Iteration is used. I am using this algorithm to calculate the optimal policy for a small game and test my knowledge in ...
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5answers
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Is it possible to get all possible sums with the same probability if I throw two unfair dice together?

I throw 2 unfair dice, suppose that $p_i$ is the probability that the first die can give an $i$ if I throw it, for $i =1,2,3,..6$ and $q_i$ the probability that the second die can give an $i$. If I ...
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0answers
59 views

Guessing the number of loaded dice from their sum

I need help answering this mathematical riddle: You have $b$ bags with an infinite number of loaded $k$-sided dice. Their "loadings" (probability vectors of length $k$) are different for ...
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2answers
34 views

Given an N-sided die, what is the probability that the second roll of a greater than N-sided die will be greater than the first roll?

First off, I am not a math guy - so please accept my apologies. My 4th-grade daughter is making up a game to bring to school (dice wars), in the game two people roll a d6 and the higher number wins, ...
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0answers
32 views

Chances of at least one each of two values on X dice

I'm trying to work out the chance of rolling at least one each of 2 different values (say for example at least one 1 and at least one 2) on x dice. I'm not concerned about the totals, it's just the ...
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1answer
50 views

A 4-sided fair die is independently rolled two times

A 4-sided fair die is independently rolled two times. a) Define two events 𝐷 = “at least one of the rolls is 3” and 𝐸 = “at least one of the rolls is 2”. What are their probabilities? b) Given 𝐷 ...
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2answers
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12 sided dice probability question

I need help with determining the probabilities of rolling four 12-sided dice: Red, White, Blue, and Yellow. I tried to program a simulation in BASIC on Windows, and the language on Windows 10 isn't ...
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0answers
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two people take turns throwing a die, what is the probability the second person beat the first one?

Two people take turns throwing a fair die, let $M_n$ denote the number of times during the first $n$ throws ($n$ is the total number of throws for both people) that the second person throw beats the ...
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1answer
33 views

Probability of not choosing somethings

I am a newbie and I have spent a few days solving my following problem but I can't. So I would like to ask the help from all members here. My problem is the following: I have $N$ people and $M$ ($M$ &...
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0answers
29 views

Roll four 6D. Ignore the largest result. Will the sum of the remaining results be less than 8?

I thought I could make an overview of possible outcomes when rolling four dice. So look at possible sums when rolling four dice, 4-24, and which combinations that give these sums. Then one can exclude ...
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2answers
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How to calculate the probability of dice outcomes of a D$30$?

One D$30$ dice is rolled three times, will the difference between the maximum and minimum value be greater than $15$? So I guess this is the same as just picking three random numbers between $1-30$, ...
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2answers
86 views

Probability of some dice rolls

Question A fair six-sided die is rolled $12$ times. Find the probability that each of the six possible outcomes ($1$, $2$, $3$, $4$, $5$ and $6$) come up at least once. My working Let $X_n$ be the ...
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5answers
311 views

Probability of rolling a 6 immediately after a 1 is rolled

Question Ann and Bob take turns to roll a fair six-sided die. The winner is the first person to roll a six immediately after the other person has rolled a one. Ann will go first. Find the probability ...
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2answers
166 views

Combinatorial identity on decreasing dice throws

Suppose I repeatedly throw fair $n$-sided dice until I throw a $1$, at which point I stop. I want to know the probability $p(n)$ that my sequence of throws will be decreasing, such as $5-4-2-1$ or ...
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0answers
24 views

Odds of K of the Same Number Appearing on a D sided dice in N throws

Say I have a D sided dice and I toss it N times. Given a value K where K >=1 and <= N, what are the odds that K of the rolls have the same number? What are the odds that >= K of the rolls ...
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6answers
142 views

Probability question from Brilliant

Which is more likely? You roll two dice 5 times and, every time, one of the two comes up as 1 and the other as 6. You roll 10 dice all at once. 5 come up as 1s and the other 5 come up as 6s. The ...
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2answers
41 views

Dice probability, Exemple, having at least three 5 or more, on seven dice throw.

I'm having trouble calculating probability on dice rolls, here's the setup : (All dices are fair 6 faces classic dice) It's a D&D Roleplaying kind of game, the rules say this : For each action, ...
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1answer
32 views

Fair die - probability - geometric

Throw a fair die. X is the number of throws until the first appearance of 6, Y is the number of throws until the first appearance of 5. I need to calculate E(x), so I used geometric expectency (1/1/6) ...
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1answer
29 views

How does the probability of reaching a threshold of D6 scale with the size of the dice pool?

Assume the following: you have a pool of $N$ d6s. You roll the dice, tally up the number of 6s rolled, and tally up the number of 1s rolled. Then, subtract the number of 1s from the number of 6s. Call ...
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2answers
122 views

Expected value of the sum of two dice.

Came across this question: We roll two dice. Let $X$ be the sum of the two numbers appearing on the dice. Find the expected value of $X$. Find the variance of $X$. I'm not sure how to do either, ...
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2answers
69 views

Probability using a coin and a six-sided die

Assume a coin and a six-sided die. At the first move, you flip a coin. If you get a heads, you get to roll the die. If you get a tails, you have to flip the coin again. When you roll the die, if you ...
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1answer
33 views

How to calculate probabilties of specific dice showing up while having a minimum total sum when rolling multiple dice.

I am trying to design a dice game (up to 5 dice) and here is the problem I am looking to solve: I am rolling $N$ dice, and I want to know the probability of rolling exactly $M$ ones (1's) & the ...
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2answers
54 views

Probability of sequences of dice rolls

I would really appreciate help with a calculation of probabilities. Let's say I have a die with six sides. I roll the die 7 times and get the following sequence: 5, 1, 2, 2, 4, 5, 1 Given some random ...
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0answers
15 views

Expected number of transfers/move in a 6 round dice game

In a dice game, the game piece moves the number of steps that the dice show, except when it shows 4, then 1 step is moved instead. calculate the expected number of transfers after 6 rounds. Is the ...
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0answers
22 views

Expected value of a dice game

Consider a fair 6-sided die. If you roll a 1,2,3 you increment your score by 1 and roll again. If you roll a 4,5, the game terminates and your payoff for the game is your accumulated score. If you ...
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1answer
44 views

Probability with multiple dice rolls

A friend of mine recently created a game where your movements on the plateau are determined by series of dice throws. The rules are simple: you have exactly $6$ dice that you should throw ...
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2answers
50 views

What does getting a number on 2 dice means?

In chapter 13 (beginning of probability theory) of the book "challenge and thrill of pre college mathematics", after theorem 3 (i.e. for any 2 events E and F, P(E union F)=P(E)+P(F)+P(E ...
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3answers
75 views

Rolling a die until the sum of all numbers rolled is greater than 6

I'm trying to solve a problem in which I'm asked the probability mass function of $X$ where $X$ is the number of times a dice is rolled until a sum greater than six is encountered. I was able to ...

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