# 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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### Optimizing a dice roll in Yahtzee

Say the only score boxes you have available in the middle of a Yahtzee game are fives and fours from the upper section and it's your turn. Your initial roll gives you one five and one four, the rest ...
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### Splitting a sequence of items into statistically normal length groups

I have a file, that is made up of X chunks. Each chunk is roughly similar size, but not identical. Each chunk has 3 attributes: the index within the file, the size, and the CRC32 hash of its contents. ...
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### Expected value of sum of number of dice after rolling a die n times

A fair six-sided die is rolled $Y$ times, where $Y\sim \operatorname{Po}(4)$. Let $X_i, \,i=1,\ldots,Y$ be the number obtained on the $i$th roll, and let $$Z=\sum_{i=1}^Y X_i$$ How do I calculate the ...
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### Whether a dice game is profitable in the long run

As part of a course, I came across this question "A player is offered to play against a dealer who rolls 6 dice simultaneously. If there's a 'street' (each of the numbers from $1$ to $6$ appears on ...
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### Is my six sided dice weighted?

I was playing a game with my son, and it seemed that we rolled two or five about half the time. He was wondering if the die was weighted. So I re-rolled it a bunch of times and jotted down the ...
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### Probability of each of following dice rolls [closed]

I am trying to find the probability of each of the following dice outcomes when rolling two dice at the same time: The probability of rolling jneither twos or threes. The probability of rolling one ...
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### Dice rolled 40000 times, let X be number of times 3 is rolled, find P(10000<X<12000)

I've been going through past exam papers and I'm not sure how to do this. Let X be the number of times the outcome 3 comes up in 40000 throws of a fair astragalus. (You can assume that an astragalus ...
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### Dice rolled 6 times. What is the probability to roll 2 twice, 4 twice, and 6 twice? Also, what is the probability to roll 2 thrice, and roll 4 thrice?

Here's what I currently have, however im not sure its correct. a) $\mathbb{P}(2 \text{ twice}, 4 \text{ twice}, 6 \text{ twice}) = \{a, a, b, b, c ,c\}$ where $a$, $b$, $c$ are numbers between $1$ ...
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### Expected number of rolls to win a modified Bingo game

Suppose I have a 5x5 Bingo card with each box numbered sequentially from 1 to 25, and a fair 25-sided die will be rolled repeatedly. How do I find the average number of rolls it would take for a 50% ...
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### Is there a difference between rolling 3d6 and rerolling lowest twice, versus rolling 5d6 and dropping the two lowest values?

I was rolling stats for a set of characters (main + backup) with my DM, and he told me I could choose between 3 sets of two rolls. One rolled by him, one rolled by another player, and one rolled by me....
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### Best re-roll strategy for a large straight in Yathzee, given $\{1,1,1,6,6\}$

Fun question. Today I was struck by this seemingly simple problem: given a roll with 5 dice, the outcome which is $\{1,1,1,6,6\}$, and two rolls remaining, what is the optimal strategy to gain a "...
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### Repeatedly rolling a fair 6 sided die

Suppose you repeatedly roll a fair 6 sided die. What is the probability that you obtain at least one 5 before the first 6? Will this involve an infinite sum?
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### Can you please explain whether this game is a game with multiple events?

There's a game where there are two dices distributed to each player. The player that rolls a higher number than the other wins the game. My teacher tells me that this game is not a game with multiple ...
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### probability of a die rolling 6, N times more often than any other number

I am looking for a way to calculate the possibility of a die rolling 6, at least N rolls more than any other number. Here is an example: Let's say I throw a die 100 times and I want to calculate the ...
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### Finding the expected value of the sum of four dice with only the largest three counting

Below is a problem that I did. I have good reason to believe that the answer is around $12.244599$. Problem: Suppose a person rolls $4$ six sided dice. He does not count the die with the lowest value....
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### Expected number of throws in a custom dice game, difference manual and simulation

There is a difference in my computed expected number of throws by hand and by simulation. The difference is about 1/4, but my question is, which one is wrong? Clarification of the rules of this game: ...
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### Probability of throwing two dice and getting one 3 / 5.

Imagine playing the game Yahtzee and you are going for a large straight ( every dice fits in, example: $1, 2, 3, 4, 5$). On your first roll you get $1, 2$ and $3$ already, but you still need the $4$ &...
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### Dice Sum Probabilities and The Normal Distribution

https://mathworld.wolfram.com/Dice.html How does one find the error of a sum of four or five 6-sided dice probability distribution in relation to the Normal Distribution?
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### Game of dice probability

Let's say I'm playing a game of dice with another 5 people. The game goes like this: each of us has 5 dice (30 dice total) and we roll them, and I can only see what I've rolled (my 5 dice). The ...
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### In a group of people rolling dice, what is the probability that you tie for highest roll multiple times in a row?

This problem was inspired by a Reddit post which you can find here. You are in a group of 40 people. Each of you rolls a 100-sided die. If a single person has the highest roll, then that person ...
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### Rolling dice possible outcome

Can you help me to find the solution to the following question? Fifty identical dice are rolled. How many distinct outcomes with positive even number of 1’s, 2’s, ..., of 6’s are possible? For ...
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### Play rolls 4 dice - Probability

I have a question - If a player rolls $4$ dice, and the maximum result is the highest number he gets (for example he tosses and gets $1$,$2$,$4$,$6$ the maximum result is $6$). His opponent rolls a ...
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### Expected number of rolls needed to get four distinct results

Suppose I have a fair, six-sided die. What is the expected number of rolls it would take me to get four distinct outcomes?
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### Expected number of rolls until all numbers have been rolled an odd number of times (at the same time)

The title is a reformulation of a game that is played as follows: Before you are six light bulbs (all start turned off). You repeatedly roll a fair six-sided die and each time you roll a number you ...
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### After rolling two dice and flipping 12 coins. What is the probability that the # of heads is equal to the sum of the numbers showing on the two dice?

Suppose we roll two fair six-sided dice and flip 12 coins. What is the probability that the number of heads is equal to the sum of the numbers showing on the two dice? My solution is: \frac{\binom{...
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### How to show all possible ordered outcomes of 4 dice

I'm working on a project that requires I add a unique result for each and every possible ordered outcome of a roll of $4$ dice at one time. I'm stuck on how to show this in Excel or Google Sheets. Any ...
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### Is there a mathematical model to work out which face of a dice will appear when you turn it

I written a program to draw a 3D cube [a dice]. And I hard coded which face appears when you move it from one face to another. So imagine I start with a 1 facing me; I have a 5 above, a 4 to the right,...
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### Expected number of turns for getting six 1's in six dice.

You have 6 unbiased dice. What is the expected numbers of turns required to get 1 in all the faces such that whenever you get 1 in any of the dice, you do not roll it for next three turns. For ...
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### Richard Hamming's art of probability solution

I am currently using Richard Hamming's art of probability book to brush some basic concepts regarding the subject. In this book, problem 1.7-17: On the roll of two dice, what is probability at ...
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### Statistic: die manipulate in certain way, probability of rolling a certain score? [closed]

I have a question regarding to an exercise: A dice is manipulated in such a way, that the probability of rolling a certain score is proportional to the corresponding numerical value (e.g. rolling the ...
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### In a small town there are $6$ repairmen and one day $10$ households call for TV repairmen

Question What is the probability that precisely $2$ of the $6$ repairmen will be called? The actual question is more generic but I've used a specific example here in the hope that I can generalise ...
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### Optimal strategy for a dice game with an element of choice

This was given as part of an assignment on dynamic programming using Bellman equations, but I feel like it could also be solved purely in terms of probability. (I've already submitted the assignment, ...
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### Permutation and combination dice question [closed]

Alice is rolling 5 different colored six-sided dice. How many ways can she roll them so there is one pair, but others are different
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### Formula for the probability of rolling the most probable sum of n s-sided dice

The most likely sum of any roll of n unbiased s-sided dice is given by: $P (n, s) = \frac{1}{2} n ( s + 1 )$ This tells us, of course, that for an odd number n, the most likely sum will be ...
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### Dice rolling and conditional probability

We are rolling a six-sided standard dice. We roll until we see an odd number appears. If this happens in one of the rollings, if this number is 5 we win and otherwise we lose - obviously the number is ...
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### Mathematically modeling dice guessing game with probabilities (Victor Shoup's book)

I've been trying to learn probabilities from Victor Shoup's book. But I got stuck at example 8.12 at page 215. I, sadly, do not feel able to reproduce the chain of examples leading to this one, which ...
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### Build a number from $n$ dice, perhaps excluding one

We are rolling $n$ dice and want to be able to make a number $m$, but can only exclude one die. For example, if we roll $4$ dice, want the number $12$, and get $1, 4, 5, 6$, we can get it by ...
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### Probability of getting first consecutive 6 on roll k

Let $X$ be the random variable that counts the number of dice throws (fair and 6-sided) till we throw two consecutive $6$. For example, taking the rolls w = $(1,4,3,6,6) \ \ X(w) = 3$. I am trying to ...
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### The way a loaded dice affects probability

There are 6 dice. The probability of each number is the same with an exception. The number 4 has a 2/7 chance of occurring while everything else has a 1/7 One of my main issues is that I'm having ...
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### Probably smallest number choosen from $X$ numbers with Probability $P$?

TL;DR; I am looking for a formula to do one random draw for "$f_P(X)$ = the smallest number of a set of $X$ numbers, if each number is choosen with Probability $P$?" First of all, I am sorry if I ...
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### Probability of n m-sided dice being x greater than another n m-sided dice rolled

Question: What is the probability of me rolling n m-sided dice (ndm) and it being x value greater than an opponent rolling ndm dice? Example: What is the probability of me rolling 2d6 and getting 3 ...
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### What's the probability of getting three consecutive numbers among five dice throws?

The problem is stated as follows: "Roll a 6-sided die five times. Note the results. The order of appearance is irrelevant. Will there be three consecutive numbers among the results? Example: if you ...
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### How to compute the expected outcome of a blackjack-like game?

There is a gambling game called blackjack, played with a 101-sided die (0-100) A better rolls, and can choose to hit or stay, if the better hits then the dice is rolled again, if the better stays ...
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### Probability Question on a dice roll - Conditional probability

I have a question which I have a hard time wrapping my head around it. If I roll a dice three times, and I know that I got three different numbers, what is the probability that one of the ...
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### Basic Combination: How many unique combinations* of four 3-sided die?

I have four dice. Each die has 3 sides. I want to know the total amount of unique combinations*. ...
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### Mark a six-sided die with the results of six rolls of a previous die. How many iterations until all the faces match?

This is FiveThirtyEight's "Riddler Classic" puzzle for 27 March, 2020: From Chris Nho comes a question of rolling (and re-rolling) a die: You start with a fair 6-sided die and roll it six ...
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### Probability of not get 1 after rolling a dice 20 times?

When rolled a dice 20 times, what is the probability of not getting a 1. My though is since 5/6 ^ 20, but it doesnt seem right.
### What is the probability of getting a Yahtzee within $n$ rolls?
Say there is the standard situation in Yahtzee with $5$ $6$-sided dice. The player tries from the outset to get a Yahtzee. The player is given the conventional liberty of choosing which dice to re-...