Probability concerning unfair dice with N sides

Question

Fair warning: I am not a math expert (that's why I'm here).

I would like to be able to calculate the probability of rolling a certain side on a die with n sides where any number of those sides has an unfair advantage over the others.

For example: On a 6-sided die, we could assume that the 1 - 6 sides are longer than the other sides due to wear and tear (or someone shaved the 1 or 6) and, therefore, more likely to come face up on a roll. I know that, on a perfect die, the roll is uniformly random and each side has a 1/side chance of coming up.

How do I calculate the probability of each side if sides 1 and 6 are longer?

What if sides 1 and 6 are longest, 2 and 5 are second longest, and 3 and 4 are shortest?

What if I'm using a 12-sided die? 10-sided? 20-sided?

I'm looking for a formula that I can plug numbers into and I'd really, really like an explanation of how it all works, if possible.

I found this link that talks about using Mathematica to calculate one side shaved on a 6-sided die, but I don't know how this changes when you increase the number of dice, or the syntax being used. I feel like the equation below (from the link) makes sense somewhat (in that σ represents the increase in probability of a 1 or 6, but I would like to know how to calculate σ.

f = { 1/6 + σ, 1/6 - σ/6, 1/6 - σ/6, 1/6 - σ/6, 1/6 - σ/6, 1/6 + σ };

Could I use the same formula to represent a 20 sided die?

f = {1/20 + σ, 1/20 - σ/20, ... ... ... ..., 1/20 - σ/20, 1/20 + σ}

Note: I took an intro to Statistics in college, but that's all the exposure I've had. I would like to understand more, but this is where I am now. I appreciate any help you can give me and any resources you can point me to.

Answer 1

Answering to:

I initially started writing a software program that would simulate uniformly random dice rolls, but then thought about how that wouldn't be realistic because dice get tumbled and smoothed before reaching the consumer (which is why some people have "favorite dice" because they roll 20s often). I realized that I could approximate and fake out the realism, but I got ridiculously curious about how it all actually worked. So, to answer, I'm not measuring real physical objects, but trying to determine how to realistically simulate the rolling of virtual dice.

What you could do is the following. Start with a perfect dice ($p_1 = p_2 = \dots = p_6$). Then, each time you roll it, you can alter the probabilities $p_i$ with an error $\varepsilon_i$. Thus, the probabilities would not be the same in the future. This would simulate the damaging of the dice.

Algorithm:

Start with $p_1 = p_2 = \dots = p_6 = \frac{1}{6}$.

Generate 5 random errors $\varepsilon_1, \varepsilon_2, \dots, \varepsilon_5$.

Compute $ \varepsilon_6 = - \varepsilon_1 - \varepsilon_2 - \varepsilon_3 - \varepsilon_4 - \varepsilon_5$

Set new probabilities to $p_1 + \varepsilon_1, p_2 + \varepsilon_2, \dots, p_5 + \varepsilon_5, p_6 + \varepsilon_6$.

Roll the dice with this new probabilities.

Compute new errors...

I strongly suggest that you choose the way of generating the $\varepsilon_i$ so that they will always be 0 or as near to 0 as possible.

For fine tuning purpose, it would be good if it was not always $p_6$ that gets the difference of all other errors.

This method can easily be adjusted to dices with any number of faces.

Answer 2

There are 2 ways to consider your question.

The probabilistic way is just to say we have an unfair die. This means we give probabilites $p_1$, $p_2$, ..., $p_6$. The conditions are that $p_1 + p_2 + \dots + p_6 = 1$ and that $p_i \geq 0$ for $i = 1, 2, \ldots,6$.

Now we assign the "probability that the dice will show face number $i$" to $p_i$. For example, a fair dice would have $p_1 = p_2 = \dots = p_6 = \frac{1}{6}$. An unfair die like the one you want (faces $1$ and $6$ bigger, $2$ and $5$ normal, $3$ and $4$ smaller) could have $p_1 = \frac{1}{4}$, $p_2 = \frac{1}{6}$, $p_3 = \frac{1}{12}$, $p_4 = \frac{1}{12}$, $p_5 = \frac{1}{6}$, $p_6 = \frac{1}{4}$. This is the probabilistic approach. You decide the probabilities you want, and then you do some calculations (for example, you could be interested what is the distribution of the sum of 2 such dices, and so on).

The statistician has a die and knows/supposes that it is unfair. He cannot know the exact value $p_i$ for each face. However, he can try to guess those values and create tests to see if these values are probable or not. The usual way to guess the values is to throw the dice many times (say 1000 times) and to count of much $1$'s, $2$'s, ... we get. For example, if you count $100$ $1$'s, you would say that the probability of having a 1 could be $\frac{100}{1000} = \frac{1}{10}$. After that, you can build tests to verify this guess.

I don't know if I answered your question. But it truly depends on what you want to do. According to your question, you have the unfair die, and you want to guess the probabilities. This would more be the statistical approach.

This means you have a model (like the one you give with $p_i = \frac{1}{n} + \sigma$) and you must create a test to estimate $\sigma$. Basically, you will have to throw the die several times anyway.

Answer 3

There are two distinct questions posed here.

1. How to work with probabilities of dice that are biased in some way. For an $n$ sided die this means that there are some numbers $p_1, p_2, \dots p_n$ with $p_1 + p_2 + \dots p_n = 1$, where $p_i$ is interpreted as the probability of landing on face $i$. Then the standard methods of probability theory give recipes for calculating the probabilities of outcomes of random processes based on such dice: sums of several dice tosses, results of games, etc. Calculations are easier for the case where all $p_i$ are equal -- known as a "uniform distribution" or "fair dice" -- but there is nothing fundamentally different about this case compared to that of unequal probabilities.

2. How to compute the probabilities $p_i$ from the physics of a die and the dice-tossing process. For coins there is some recent analysis of "dynamical bias in coin tossing" starting from the article with that title by Diaconis, Holmes and Montgomery. For rectangular dice with uniform mass density (not loaded toward some faces), a simple and not necessarily physically accurate prescription is to assign each face a probability proportional to its surface area, or equivalently, probability equal to its share of the surface area of the entire die. This can be viewed as the $k=0$ approximation to the probabilities for a thrown die that rolls onto at most $k$ additional faces after landing.