I have made some progress in AnyDice showing the probability for dice resolution system in the RPG system I've been working on. Here is the link to what I have so far:   I need particular help on when a roll gets a 2 or 3 that that die can be rerolled once. I am interested to see how this alters the probability of success.
121806
I am joining a D&amp;D group. To generate each ability score, the DM has us roll 4d6, reroll any 1 or 2, and then drop the lowest die. If the reroll is a 1 or 2, then reroll again and again until the result is 3 or higher.  I am curious about the distribution, and wonder if AnyDice could do it. How can I calculate the distribution of this ability score generation method in AnyDice?
34029
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 this is correct. How should I go about solving such a problem?
58384
I wish to draw integers from 1 to some specific $N$ by rolling some number of fair six-sided dice (d6). A good answer will explain why its method produces uniform and independent integers.  As an illustrative example, it would be helpful to explain how a solution works for the case of $N=150$.  Furthermore, I wish for the procedure to be as efficient as possible: roll the least number of d6 on average for each number generated.  Conversions from  to decimal are permissible.    .
113008
Four dice are thrown, what's the probability that:  a) None of them fall higher than three?  b) None of them fall higher than four?  c) That four is the highest number thrown?  So for a, I wanna think about the denominator first. There are 6 possible outcomes for each dice, and we have four dice. So we're technically picking r out of n objects, aka picking 4 possible values out of 6. So the denominator/sample space should be $6^4$ right?  Now this is where I'm getting tripped up. Order technically shouldn't count, because we only care about the quadruplets that don't have a value higher than 3. I might be wrong, maybe I am, but could someone explain a bit more? My professor stated that typically in the sample space/denominator, we want order to count.   So I'm trying to think of the numerator now, so we want to find the probability that no values appear higher than 3, so some events that can occur are: $(1, 1, 2, 3), (1, 2, 2, 3), (1, 1, 1, 1)$ etc. Again, I can't see why order should count here, because if we only are concerned about what appears rather than how they appear, then we can say $(1, 1, 2, 3)$ is equal to $(1, 2, 1, 3)$? So if order does not count, and we have replacement, then does this lead the case where we use $$\binom{n-1+r}{r}$$to find out the probability?   I'm pretty sure if I can do a), I could prob do b), but if someone could maybe lead me in the right direction for c) I would appreciate that too!
36835
I have a d20 that seems to be, well, remarkably lucky.*  How can I determine whether it's really just luck, or whether the die is in fact unfairly biased?  *) Well, I don't, really.  This is actually a spin-off from , which is specifically about determining whether a die is loaded.  This one is intended as a more general question about how to detect any kind of bias in dice, since we apparently don't have one yet.  I've posted my own answer below, but feel free to add more.
50454
Several people are playing a game with random events and require a way to produce a random number. (Such as dice rolls or a lottery.)  Can this be done such that each player has the power to be reasonably sure that the random number was fairly selected, without having to trust anyone else?  This question stems from a discussion in comments to my answer on the . I came up with a protocol but someone else spotted a flaw. This site seems more appropriate. Please feel free to make use of my answer to that question.
120292
We roll a die until we obtain all numbers from $1$ to $6$. I found the expected value of rolls computing it like $X = X_1 + \dots + X_6$ where $X_i$ is number of rolls needed to obtain a result different from previous $i-1$ and using a geometric distribution. And my result is correct. But then I wanted to find a variance. Firstly I thought of doing it this way: $$\text{Var} (X_1 + \dots + X_6) = \text{Var}(X_1) + \dots + \text{Var}(X_6) + 2 \sum_{1\le i&lt;j\le6} \text{Cov}(X_i, X_j),$$ but covariance is not easy to find here. Can somebody please show me how to find a variance of number of dice rolls?
83383
This is not homework. I am interested in understanding if my logic is correct with this simple stats problem.  Let's say I have a 2 sided coin where the probability of flipping a head is $P(H)$ and the probability of flipping a tail is $1-P(H)$. Let's assume all flips have independent probabilities. Now, let's say I want to maximize my chances of predicting whether the coin will be a head or tail on the next flip. If $P(H) = 0.5$, I can guess heads or tails at random and the probability of me being correct is $0.5$.  Now, suppose that $P(H) = 0.2$, if I want to maximize my chances of guessing correctly, should I always guess tails where the probability is $0.8$?  Taking this one step further, if I had a 3-sided die, and the probability of rolling a 1, 2, or, 3 was $P(1)=0.1$, $P(2)=0.5$, and $P(3)=0.4$, should I always guess 2 to maximize my chances of guessing correctly? Is there another approach that would allow me to guess more accurately?
58836
Suppose, we roll a fair die $n$ times. The expected value of how many times each side of the die will appear is $\frac{n}{6}$. The question is, what is the probability that any side of the die will come up more than $k$ times, where $n&gt;k&gt;\frac{n}{6}$? Or you could put it this way: what is the probability that no side will come up more than $k$ times? Clearly these two probabilities add up to 1.
18723
Happy Sunday, I was wondering how to calculate how many combinations could be rolled in the game of Yahtzee in getting a large straight. A large straight is when you roll and the dice com out 1,2,3,4,5 or 2,3,4,5,6. Can any one give me some guidance???
114066
Say I want to roll a six-sided-die until all the rolls I have made added together equal X (for example a number like 812). How do I calculate how many different possible combinations of dice rolls add up to X?  for example: say I want to find the different roll combinations that equal 6: 6 51 15 42 24 33 123 312 213 114 411 141 1113 3111 1311 1131 11112 21111 12111 11211 11121 and so on...  I have found that this approach does not work for larger numbers. Thanks  Thanks
42879
A fair $6$-sided die is rolled $6$ times independently. For any outcome, this is the set of numbers that showed up at least once in the different rolls. For example, the outcome is $(2,3,3,3,5,5)$, the element set is $\{2,3,5\}$. What is the probability the element set has exactly $2$ elements? how about $3$ elements?   I know the sample space is $6^6$. The counting for $2$ elements is $6C2 \cdot 6!$? I would really appreciate the help! :)
119883
Old 1E player now getting back into 5E.  What are HIT DICE?  I understand hit points and a 3rd level player has rolled 3 times for HP ( I do like the MAX at first level and then roll option)  but what are Hit dice and why do they matter?
42365
OK so if your game rolls lots of dice you can just call a random number generator in a loop. But for any set of dice being rolled often enough you will get a distribution curve/histogram. So my question is there a nice simple calculation I can run that will give me a number that fits that distribution?  E.g. 2D6 - Score - % Probability  2 - 2.77%  3 - 5.55%  4 - 8.33%  5 - 11.11%  6 - 13.88%  7 - 16.66%  8 - 13.88%  9 - 11.11%  10 - 8.33%  11 - 5.55%  12 - 2.77%  So knowing the above you could roll a single d100 and work out an accurate 2D6 value. But once we start with 10D6, 50D6, 100D6, 1000D6 this could save a lot of processing time. So there must be a tutorial / method / algorithm that can do this fast? It is probably handy for stock markets, casinos, strategy games, dwarf fortress etc. What if you could simulate the outcomes of complete strategic battle that would take hours to play with a few calls to this function and some basic maths?
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First some context, I'm not a mathematician, not even close (as you will soon see) I do grasp some things about it but in a need to know basis, so plain english answers are appreciated (too).  I can't figure out this one even though some have tried very hard to explain/convince me.  So let's say I'm on a roulette (let's leave out the zero, or make it a coin for that matter).  The roulette hits 5 (or any amount of) times red on a row.  My question is why should't I bet to black (or better said why it is the same the one I pick).  I DO UNDERSTAND THE BASIC PRINCIPLE, that there is a material reality which makes the ball randomly fall on any of the two colors (which gives the options a 50-50 chance).  But, I also understand that if you "say": I bet red will come out six times on a row, you do have a very low probability of that happening so:  How is it that if you have seen the ball fall 5 times on a row on red, and you bet on red, you are NOT betting on 6 times on a row on red (that does have lower probability).  Thanks in advance!    I hope I was clear enough with my question, if not, please ask for any clarifications needed.  Sorry for the VERY plain English, feel free to modify or suggest a change to anything that may be misleading.
43742
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 dice, this isn't just a textbook exercise. I'd like to minimize how long it takes me to perform this experiment with each die :)   I know this depends on my expected "confidence level" (95%? 99%?) If I choose a 95% confidence, for example, does that imply that 1 out of 20 fair dice will fail this test? Or that a single fair dice would fail the test 1 out of 20 times? If so, that sounds fairly high.   Are there standard techniques for doing this kind of a test?  Edit: It is beyond the scope of the math-focused question I've asked here, but I've explained more about the overall testing scenario over on the stats site here:
66299
Consider a six-sided die with numbers from $1$ to $6$. Imagine you have a jar with $99$ of such dice. You throw all dice on the floor randomly. You look at one of the dice on the floor at a time. For each die, you do the following:   If it landed at an even number $(2,4,6)$, you turn the die so that it lands on the number $1$. If the die landed on an odd number $(1,3,5)$, you throw the die up in the air, so it can land on any number.   After you finish doing the above for all dice on the floor, you come back to the first die and repeat the entire process again. You keep on doing this until eternity (for a billion years, let’s say). If I come into the room after a billion years, how many dice on the floor will have even numbers up?