# 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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### 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 ...
108 views

### Probability to obtain more than X with 3 dice.

This is a dice problem 1) I want to calculate the probability to have more than X throwing 3 dice of 6 faces. X = addition of the result of the 3 dice. 2) This is the first step but if you can also ...
2k 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 ...
4k views

### 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 ...
4k views

### What is the average of rolling two dice and only taking the value of the higher dice roll?

What is the average result of rolling two dice, and only taking the value of the higher dice roll? To make sure the situation I am asking about is clear, here is an example: I roll two dice and one ...
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### Probability of consecutive row of at least $m$ by $n\leq2m$ events. Is there a shorter route?

Inspired by this I started to generalize it. Go out from independent events $E_{1},E_{2},\cdots$ that succeed or fail. This with $p$ as probability of succes, and $p+q=1$. Event $A_{m,n}$ occurs if ...
34k views

### 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?
16k 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?
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### Probability that the sum of all values of 5 pairs of dice will be between 30 and 40

I'm trying to solve a question that asks: If 5 pairs of fair dice are rolled, approximate the probability that the sum of the values obtained is between 30 and 40 inclusive. My approach so ...
35k 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 ...
491 views

### 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 ...
2k views

### dice problem - throws necessary for sum multiple of n

A die (with six sides) is rolled repeatedly and summed. Show that the expected number of rolls until the sum is a multiple of $n$ is $n$. In http://www.madandmoonly.com/doctormatt/mathematics/...
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### A dynamic dice game

I learned the following dice game from another forum. It was not solved there. The dice game is as follows. Toss 6 dice. Each toss you can keep one or more up to the total number of dice tossed. You ...
815 views

### 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 ...
7k views

### 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 ...
3k views

### 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 ...
2k views

### Probability that the sum of 6 dice rolls is even

Question: 6 unbiased dice are tossed together. What is the probability that the sum of all the dice is an even number? I think the answer would be 50%, purely by intuition. However, not sure if ...
4k views

### 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 ...
11k 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 ...
210 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 ...
273 views

### Dice probability puzzle

What is the probability of a run of at least 3 sixes when a die is thrown 5 times? I think I have the answer but from what I have been told its not the correct answer. Would someone like to help?
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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 ...
2k 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 ...
67k views

### If I roll two fair dice, the probability that I would get at least one 6 would be…

11 out of 36? I got this by writing down the number of possible outcomes (36) and then counting how many of the pairs had a 6 in them (1,6) (2,6) (3,6) (4,6) (5,6) (6,6) (6,5) (6,4) (6,3) (6,2) (6,1). ...
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### What's the probability of rolling at least $k$ on $n$ dice with $s_1,\ldots,s_n$ sides?

Is there a way to determine the chance of rolling at least $k$ on $n$ dice with $s_1,\ldots,s_n$ sides? Example: What is the chance of rolling a sum of at least 13 on 3 dice with 6, 8, and 10 sides? ...
624 views

### Probability of 1x six from seven dice?

Could someone help me with how the following is calculated: What is the probability of rolling a die seven times and getting at least one six? My instinct told me it would be $1/6+\cdots + 1/6$ but ...
263 views

### probability of rolling at least $n$ on $k$ 6-sided dice

Is there a simple form for the probability of rolling at least $n$ on $k$ 6-sided dice? Of course you can do it by recursion (see here). But is there a way to do it with just a few binomial ...
292 views

### Number of die rolls needed for average to converge to roll expectancy?

I'm aware there are similar questions, but I haven't been able to find what I'm asking. Say we have an $n$-sided die, labeled with 1 thru $n$ and roll it $N$ times. We take the average, call it $m$. ...
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### Probability Dice Game Follow Up

Original Question: Paul, Dave and Sarah are rolling a fair six sided die. Paul will go first, always followed by Dave, who is always followed by Sarah, who is always followed by Paul, and so on... ...
158 views

### Chance of winning simple dice game

Tossing two fair dice, if the sum is 7 or 11, then I win; if the sum is 2, 3 or 12, then I lose; if the sum is one of rest of numbers then I toss the two dices again. What is probability of winning? ...
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### 100-Sided Dice “Blackjack” Game

I am attempting to determine two variables in this game: The optimum strategy: (What number the bettor should stay at) The expected value given perfect play: (The percent return on a bet when using ...
15k views

### 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 ...
535 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 ...
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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 do I calculate the odds of a given set of dice results occurring before another given set?

Dice odds seem simple at first glance, but I've never taken a Calculus based statistics course or game theory, and I think I may need to in order to solve some of the things I'm trying to solve. I can ...
3k 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 ...
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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 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 ...
446 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 ...
9k views

### Probability Distribution of Rolling Multiple Dice

What is the function for the probability distrabution of rolling multiple (3+) dice. The function is a bell curve but I can't find the actual function for the situation. Example, what is the function ...
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### What is a good strategy for this dice game? [duplicate]

I learned the following dice game from another forum. It was not solved there. The dice game is as follows. You start tossing six dice. After each toss you must put aside at least one of the dice ...
1k 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: ...
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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 ...
9k views

### Probability of All Distinct Faces When Six Dice Are Rolled

If six fair dice are rolled what is probability that each of the six numbers will appear exactly once?
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### 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 $S$...
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### Expected Value of the Difference between 2 Dice

What is the expected value of the absolute difference between 2 N faced dice? What about the difference between 2 dice one with N faces and one with M faces? While finding the expected value of ...
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### What is the pmf of rolling a die until obtaining three consecutive $6$s?

I am trying to find the pmf of rolling a die until 3 consecutive 6s turn up. I am able to find the expected value using a tree diagram, but the pmf is not obvious to me. Let A be the event of not ...
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### Probability of being able to make a given number with two out of xd6

I'm an amateur games designer, and am working on a mechanic which involves rolling on a table with numbers running from 2-12 - the full range of possibilities from adding together two six-sided dice. ...
I am having trouble with solving the following problem: The probability that a $d$-sided dice lands on its $k$th side is equal to $p_k$ for $k\in \{k\in\mathbb{N},k≤d\}$ and $p_1+p_2+p_3+...+p_d=1$. ...
### Is there a way to simulate any $n$-sided die using a fixed set of die types for all $n$?
I am assuming that we can increase the number of dice based on $n$, but they have to be $k$-sided, $k\ge3$. When I say die types, I mean that we are allowed to use non-standard dice such as non-...