# Tagged Questions

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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### 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?
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### 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 ...
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### Is it possible to create a completely random integer between 1 and 13 using standard dice in a D&D dice kit?

I am asking if it is possible to choose a random integer using a certain set of dice. Assume for these calculations that the dice are perfectly randomly distributed. There are 7 dice in a set, with ...
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### Simulate repeated rolls of a 7-sided die with a 6-sided die

What is the most efficient way to simulate a 7-sided die with a 6-sided die? I've put some thought into it but I'm not sure I get somewhere specifically. To create a 7-sided die we can use a ...
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### 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 ...
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### What are the odds of hitting exactly 100 rolling a fair die

I roll a fair die and sequentially sum the numbers the die shows. What are the odds the summation will hit exactly 100? More generally, what are the odds of hitting an exact target number t while ...
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### Expected outcome for repeated dice rolls with dice fixing

Here is another dice roll question. The rules You start with n dice, and roll all of them. You select one or more dice and fix them, i.e. their value will not ...
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### 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 ...
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### 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 ...
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### 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 ...
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### On average, how many times must I roll a dice until I get a $6$?

On average, how many times must I roll a dice until I get a $6$? I got this question from a book called Fifty Challenging Problems in Probability. The answer is $6$, and I understand the solution ...
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### 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 ...
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### Rolling $n$ $k$-sided dice and discarding the lowest $m$ of them.

In this question I will use the notation $\Bbb{E}(n,k,m)$ to refer to the expected average of rolling $n$ $k$-sided dice and discarding the lowest $m$ of them. The most trivial response happens ...
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### “$n$-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 ...
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### Rolling a die until two rolls sum to seven

Here's the question: You have a standard six-sided die and you roll it repeatedly, writing down the numbers that come up, and you win when two of your rolled numbers add up to $7$. (You will ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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?
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### Independence intuition

Toss two fair dice. There are $36$ outcomes in the sample space $\Omega$, each with probability $\frac{1}{36}$. Let: $A$ be the event '$4$ on first die'. $B$ be the event 'sum of numbers is $7$'. $C$...
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### 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: ...
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### Die that never rolls the same number consecutively

Suppose we have a "magic" die $[1-6]$ that never rolls the same number consecutively. That means you will never find the same number repeated in a row. Now let's suppose that we roll this die $1000$ ...
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### What is the probability on rolling $2n$ dice that the sum of the first $n$ equals the sum of the last $n$?

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 ...
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### Toss a fair die until the cumulative sum is a perfect square-Expected Value

Suppose we keep tossing a fair dice until we want to stop, at which point the game ends and our score is the cumulative sum, or until the cumulative sum is a perfect square, in which case we lose and ...
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### “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 ...
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### What is the probability of my sum reaching exactly 10? [closed]

I throw a 6-sided dice (with values: 0,1,2,3,4,5) multiple times and add each value to a sum, which is 0 in the beginning. What is the probability of my sum reaching exactly 10, 11, 12, 13, 14? After ...
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### 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 ...
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### 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 ...
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### We roll a six-sided die ten times. What is the probability that the total of all ten rolls is divisible by 6?

So the question is really hard I think. I tried using a simple way by calculating the probability of each combination that makes a sum divisible by six, but it would take forever. Does anyone have any ...
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### Probability of three dice falling in the same quadrant of a box

This is surely really basic for most people here but it's tripping me up. You get a box and draw lines to split it up into 4 parts. I got asked what the probability was that when rolling three dice,...
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### 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 ...
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### “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 ...
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### A fair die is rolled $N > 6$ times. What is the probability the last 6 rolls were exactly $1,2,3,4,5,6$?

You rolls a fair die $N > 6$ times and you want to rolls the sequence $1,2,3,4,5,6$ in this order. What is the probability that the last 6 rolls were (in consecutive order) $1,2,3,4,5,6$, (So, on ...
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### 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 ...
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### Roll a fair die until a 6 appears for the third time. What is the chance that all six values have occurred?

The question in the title is a homework question that I have been stumped on for some time. My approach thus far was to treat it as an occupancy problem. From class we derived the following formula ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Exploding (a.k.a open-ended) dice pool

Say we role $n$ identical, fair dice, each with $d$ sides (every side comes up with the same probability $\frac{1}{d}$). On each die, the sides are numbered from $1$ to $d$ with no repeating number, ...
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### 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 ...
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### 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 ...
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### How can I (algorithmically) count the number of ways n m-sided dice can add up to a given number?

I am trying to identify the general case algorithm for counting the different ways dice can add to a given number. For instance, there are six ways to roll a seven with two 6-dice. I've spent quite ...
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### Throwing dice 10 times

A fair dice was thrown 10 times, and it's been registered that all numbers 1-6 have appeared at least once. If this is true, what's the probability that at least 2 sixes have appeared? At first I ...
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### 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 ...