For questions on dice, a small throwable object 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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59
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8answers
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Given a die, what is the probability that the second roll of a die will be less than the first roll?

If you are given a die and asked to roll it twice. What is the probability that the value of the second roll will be less than the value of the first roll?
26
votes
3answers
11k views

Expected time to roll all 1 through 6 on a die

What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die? The order in which the numbers appear does not matter. I had this questions explained ...
24
votes
7answers
2k views

New to probability - Is this true?

I read this somewhere and can't find it to verify: "If you don't have a dice with 12 sides, but do have one with 20 sides and you need to make rolls for 12 sides, you can use the numbers 1-12 on the ...
19
votes
3answers
716 views

How many times do I roll an unfair die to determine its bias?

This question comes from computer security, but I'll distill it into a probability question: I have a biased die with 96 sides. 95 sides are equiprobable, each having a 1% chance of landing up. The ...
17
votes
2answers
1k views

Expected number of rolling a pair of dice to generate all possible sums

A pair of dice is rolled repeatedly until each outcome (2 through 12) has occurred at least once. What is the expected number of rolls necessary for this to occur? Notes: This is not very deep ...
14
votes
2answers
227 views

Generalized nontransitive dice

Let $X_1, \ldots, X_n$ be a collection of random variables. Consider the directed graph with vertex set $\{ 1, 2, \ldots, n \}$ where there is a directed edge $i \to j$ if $\mathbb{P}(X_i > X_j) ...
13
votes
1answer
711 views

“$n$-Yahtzee” question

Suppose that you have $n \geq 1$ standard 6-sided dice. If all the dice display the same number, we call this an $n$-Yahtzee. Follow this algorithm: Roll all the dice. Set aside those dice with ...
12
votes
3answers
545 views

Best Strategy for a die game

You are allowed to roll a die up to six times. Anytime you stop, you get the dollar amount of the face value of your last roll. Question: What is the best strategy? According to my calculation, for ...
11
votes
3answers
1k views

Keep the value of an 8-sided die roll, or gamble by taking a 12-sided die roll. What's the best strategy?

Consider a dice game in which you try to obtain the largest number (e.g. you win a prize based on the final dice roll). You roll an 8-sided die, with numbers 1–8 on the sides. You may either ...
11
votes
1answer
656 views

What's the probability that a sum of dice is prime?

Prompted by today's Minute Math question on the MAA site (http://amc.maa.org/mathclub/5-0,problems/T-problems/T-web,ia/2005web/tb05-12-ia.shtml), I started thinking about the probability that the sum ...
9
votes
3answers
5k views

The expected payoff of a dice game

There's a question in my Olympiad questions book which I can't seem to solve: You have the option to throw a die up to three times. You will earn the face value of the die. You have the option ...
9
votes
2answers
799 views

Most Probable Sum [duplicate]

Possible Duplicate: Probability of dice sum just greater than 100 A fair dice is rolled and the outcome of the face is summed up each time. We stop rolling when the sum becomes greater than ...
9
votes
2answers
192 views

Optimal Strategy for Bar Dice

This game is played in bars in Wisconsin, USA, but I'm sure variations are played many places around the world. The game has practical value, since once mathematicians figure out the best strategy, ...
8
votes
3answers
884 views

“Go-first” dice for $N$ players

I'm interested in sets of dice that can be used to determine who "goes first" (hence the name) in an $N$-player game; more generally, I want to determine a complete ordering of the players with a ...
8
votes
2answers
431 views

“8 Dice arranged as a Cube” Face-Sum Equals 14 Problem

I found this here: Sum Problem Given eight dice. Build a $2\times 2\times2$ cube, so that the sum of the points on each side is the same. $\hskip2.7in$ Here is one of 20 736 ...
8
votes
2answers
119 views

The rotation of dice on a grid

Imagine you have a plane, flat surface with a square grid drawn on it. You have a standard cubic die which is placed flat on the surface. Its length is the same as the length of side of each grid ...
7
votes
3answers
933 views

What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6?

Similar to: "What is the expected number of dice one needs to roll to get 1,2,3,4,5,6 in order?" but we allow repeats so 1,1,2,2,3,4,4,4,4,5,5,6 would count. My answer (or simulation) is flawed as I ...
7
votes
3answers
516 views

Thought experiment for dice game

Two players in a dice game to see who can roll a total of 60 first, taking turns each rolling 2 dice. For one player, one die is 4 sided and one is 6 sided. Therefore the average roll for this ...
7
votes
2answers
958 views

Probability of 3 of a kind with 7 dice

Similar questions: Chance of 7 of a kind with 10 dice Probability of getting exactly $k$ of a kind in $n$ rolls of $m$-sided dice, where $k\leq n/2$. Probability was never my thing, so please bear ...
7
votes
2answers
290 views

What is the most unfair set of three nontransitive dice?

In a set nontransitive dice, each die is superior to another die, but is inferior to a third. It is similar to the game of rock-paper-scissors. Here is one example: ...
7
votes
3answers
266 views

What is the probability that $n$ dice tie on successive rolls?

The Question What is the probability, rolling $n$ six-sided dice twice, that their sum each time totals to the same amount? For example, if $n = 4$, and we roll $1,3,4,6$ and $2,2,5,5$, adding them ...
7
votes
1answer
263 views

What are fair point values for Zombie dice?

Described below is a modified version of Zombie dice ( http://www.sjgames.com/dice/zombiedice/ ) which the kids and I play. As in the original game, the idea is to sequentially roll sets of 3 ...
7
votes
1answer
188 views

Why is the best position for LCR not the last person?

For the uninitiated, LCR is a game in which each player starts with three "tokens" and rolls up to three dice (at most as many as tokens they have). Each die has three sides which indicate that ...
6
votes
3answers
393 views

Probability of rolling a die

I roll a die until it comes up $6$ and add up the numbers of spots I see. For example, I roll $4,1,3,5,6$ and record the number $4+1+3+5+6=19$. Call this sum $S$. Find the standard deviation of ...
6
votes
4answers
181 views

Probability that $ax^2+bx+c$ has no real roots after rolling 3 dice.

Suppose that I roll $3$ dice and write down the outcome as the coefficients $a,b,c$ in the polynomial $ax^2+bx+c$ respectively. What is the probability that this polynomial has no real roots? So, I ...
6
votes
4answers
1k views

Confusion over probability problems

I am stuck with these probability problems, A pair of unbiased dice is rolled together till a sum of either $5$ or $7$ is obtained. Find the probability that $5$ comes before $7$? A letter is ...
6
votes
2answers
884 views

What is the expected number of dice one needs to roll to get 1,2,3,4,5,6 in order?

If I have a fair die and throw it until I get a run of 1,2,3,4,5,6 in order, how many times on average must I throw the dice?
6
votes
1answer
109 views

Bounds for die roll

Given a $n$-sided fair die, show that the probability $P$ that all faces from $1$ to $11$ show when doing $3n$-rolls is bounded by $1 - 11 \cdot \frac{(n-1)^{3n}}{n^{3n}} \leq P \leq 1 - 11 \cdot ...
6
votes
2answers
76 views

Three equal die rolls in a row in a $3\times 3$ array

If 9 six-sided dice were rolled consecutively and displayed in a $3\times3$ format, what is the probability of a line of 3 of the same number occurring? Considering the 8 ways that are possible; 3 ...
6
votes
1answer
330 views

“8 Dice arranged as a Cube” Face-Sum Problem

I found this here: Sum Problem Given eight dice. Build a $2\times 2\times2$ cube, so that the sum of the points on each side is the same. $\hskip2.7in$ Here is one of 20 736 ...
5
votes
4answers
5k views

How many times to roll a die before getting two consecutive sixes?

Basically, on average, how many times do you have to roll a fair six-sided die before getting two consecutive sixes?
5
votes
3answers
5k views

How many rolls do I need to determine if my dice are fair?

Roughly how many times do I need to roll a 6-sided die to feel confident that it's giving "fair" results? What about a 10-sided or 20-sided die? Note that I will be actually manually rolling physical ...
5
votes
3answers
663 views

What equity is necessary to offer the doubling cube in Backgammon (dice game)

Edit: This question is a lot shorter than it is. Don't get intimidated. If you know backgammon, just skip to question 2. In Backgammon, each game is played for one point (or one dollar) between two ...
5
votes
2answers
446 views

Probability for twelve dice

In 'An Introduction to Probability Theory and Applications' by W. Feller I encountered this apparently innocuous problem. A throw of twelve dice can result in $6^{12}$ different outcomes, to all ...
5
votes
2answers
129 views

Choosing between $n$ things using dice?

For which $n$ is there a finite algorithm to choose between $n$ things with the same probability using a die? For example, we can choose between 2 things, 3 things, 4 things, 6 things, and 8 things, ...
5
votes
3answers
199 views

probability of A dice with X faces beating B dice with Y faces

I am looking for a formula to determine the probability of a A Die with X faces rolling higher than B die with Y faces. An example would be: what is the probability of 4d10 rolling greater than 3d12 ...
5
votes
2answers
184 views

Probability of throwing the same multiset twice in a row with six dice

Six dice are thrown. The six dice are thrown a second time. What is the probability of getting the same numbers as in the first throw? If the order of the six numbers matters, the problem is easy, but ...
5
votes
2answers
231 views

Roll a die, pick that many balls from an urn

An urn has $5$ white and $10$ black balls. A die is rolled, and that many balls is drawn from the >urn. What is the probability that all the balls drawn are white? My thinking is that each die ...
5
votes
2answers
975 views

100-sided die probability

The question is as follows: You are given a 100-sided die. After you roll once, you can choose to either get paid the dollar amount of that roll OR pay one dollar for one more roll. What is the ...
5
votes
2answers
272 views

Multiple dice rolls (D&D Skill challenge)

In Dungeons and Dragons there is something called a skill challenge, which involves rolling some dice (no surprise there). An example of a skill challenge: Skill challenge DC 25 thievery. Roll a d20* ...
5
votes
1answer
537 views

Probability of consecutive dice rolls

This is probably quite a simple question but here I go.. Suppose you are going to roll a six-sided (fair) die N times, what is the probability that you will get at least one set of three consecutive ...
5
votes
2answers
64 views

Number of dice rolls taken to reach a certain sum

I was reading about sums of dice rolls and Chernoff bounds, and I thought of a question I couldn't obviously answer with the techniques I know. You're given some number $x$ and told it was generated ...
5
votes
1answer
284 views

What are the probabilities of a tie when rolling to see who goes first in a board games for various numbers of dice and various numbers of players?

When playing many board games, the first step is to have everyone roll a die to see who goes first, with a roll off in the case of a tie. While doing that over the Christmas break, my husband ...
4
votes
2answers
2k views

Probability of either dice showing a specific number

If I throw 4 dice together, what is the probability that either one of them will show the number 3 ? I tried to calculate it and got to $\frac{(4)}{6}$ (which is highly unlikely to be correct).. any ...
4
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3answers
4k views

Probability question (Six dice)

If six fair dice are rolled what is probability that each of the six numbers will appear exactly once?
4
votes
2answers
317 views

a sequential game of dice

consider the following game: 10 dice are tossed and those showing 3 are more are retained. [those showing 2 or less are discarded.] the remaining dice are tossed again and those showing 4 or more are ...
4
votes
2answers
508 views

Rolling a fair die 4 times, what is the probability of getting an increasing sequence of numbers?

Game: I roll a die 4 times. What is the probability that I get a strictly increasing sequence of numbers. My initial thought is as follows: we condition on R1 (the first roll) being a 1, 2, or 3 ...
4
votes
3answers
438 views

Statistical Significance Dice Probability

I think I just need to be pushed to the right formula or algorithm... Imagine you've got a "real" dice which is not an optimal one. And you want to know the confidence intervals for each result. So ...
4
votes
3answers
83 views

Probability Questions! [duplicate]

Alex, Bret, and Chloe repeatedly take turns tossing a fair die. Alex begins; Bret always follows Alex; Chloe always follows Bret; Alex always follows Chloe, and so on. Find the probability that Chloe ...
4
votes
2answers
163 views

3 Die Probability

The number of times you have to roll a set of 3 die to see a 6 on each die appears to be around ~10.555 according to a implementation of this in c++ and c#. How would this be statistically proven as ...