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DiceLab: Free Dice Roller and Probability Calculator

Contents

• An article on mathematics and probability: This article tells you about the history of dice, and how the fundamental Rule of Laplace was discovered. Easy for us, but it wasn't for them!
• 10000 dice game rule(s): Probability calculations for all kinds of games stem from the very simple Laplace rule you already discovered in the first section. But beware: games often use draw without replacement!
• Solve this math riddle: If you're in to mathematics, you can solve the equation that is behind DiceLab.
• Special dungeon and dragon dice are no longer needed: If you're into role playing games such as Dungeons & Dragons or Warhammer, you'll find DiceLab an invaluable tool. It can simulate dice throws, calculate the probabilities of the outcomes, and thus offer a way of replacing odd dice combinations with simpler ones (such as D%).
• A lesson of probability: did you discover the basic fact of Laplace's Rule?
• Download the free calculator of combination(s) with dice: Proceed to this section if you don't want to calculate these probabilities by hand.

An article on mathematics and probability

Dice do well as an example of a probability problem, because one generally accepts they are not loaded and thus fair. However, early dice were far from perfect.

It is generally accepted that animal bones, fruit pits, seeds, and pottery all served as materials for dice with different probability for each side. To make matters worse, no two dice were nearly equal.

The history of probability is remarkable. Even mathematical geniuses like Roberval and d'Alembert didn't quite get it, not even when fair dice were invented.

However, the fundaments of probability, combinatorics, was established around 1150 already, by the indian mathematician Bhascara Acharia (1114-1185) in his work the Lilavati.

What's the role of gambling in probability theory history? The basic formula of probability was rediscovered by Laplace. (Rediscovered, because we know that Gerolimo Cardano already described the "Rule of Laplace) The Rule of Laplace says, that the probability of "success" is equal to "the number of successes" divided by "the total number of outcomes", at least when all outcomes are equally probable.

Illustration 1: Gerolimo Cardano, discoverer of the Rule of Laplace.

For example: what's the chance of throwing less than 3 with a normal, fair die? Less than 3 means: 1 or 2. So there are 2 "successes" in a total of 6 possible outcomes (1,2,3,4,5 and 6). The probability of throwing less than 3 with a fair die is thus 2/6 or 1/3.

What happens if you throw 2 dice, totalling the dots that show up? This is where things become interesting, and where early mathematicians failed to see the solution.

In what follows, we will abbreviate a six-sided die as D6. 2D6 then means: two six-sided dice. If you throw 2 fair dice, there are 36 possible outcomes (6 on each die, independent of the other die, so 6x6=36 outcomes). There is only one of these 36 that has 2 dots facing up: 1 one each die. So the probability of 2 on 2D6 is 1/36. But there are 6 outcomes that have seven dots up: (1,6) (2,5) (3,4) (4,3) (2,5) and (1,6). So the probability of having 7 dots up is 6/36 or 1/6.

Now if you add dice, the dice graphic indicating chances of throwing a specific result will resemble a normal curve, if the number of dice is high.

Table 1: Probability of the sum of dots for 2 six-sided dice (2D6).

Figure 1: The data from Table 1 as a bar chart.

A remark you often hear is "The distribution is normal for probability numbers." This is not fully true. The sum of dots actually follows a discrete distribution of probability. Indeed, it is impossible to throw 7.5 with any number of fair dice.

The chart of probability for a large enough number of dice will be bell-shaped, yet discrete. Calculate the combination & permutation probability as follows. Start with 2 dice and make a table. List the result, then add a third die. Keep on adding dies and calculating the probabilities to get any combination number possible.

So: when you want to know the probabilities for throwing with 3D6, you start with this table:

Table 2: Constructing the probabilities for sums of dots for 3D6. Each light blue cell contains a sum.

There are 216 (=36 x 6) possible outcomes. To know the probability of one outcome, count its occurences in this table, multiply each occurence by the number of combinations it stands for, and divide this total by 216.

For example: the probability of having 8 dots facing up when throwing 3D6 is equal to (1x2+1x3+1x4+1x5+1x6+1x7)/216 = 21/216 =~ 0.0972

If you don't want to calculate these probabilities by hand, download DiceLab below. You can also calculate probabilities for other combinations of dice, for example 2D8 or 10D20.

If you're interested in statistics, you might want to download Sample Size Calculator, a free tool to calculate required sample sizes for different kinds of research.

Note: how to interpret "D2"? A two-sided die? The solution is simple: use a coin for a probability toss.

10000 dice game rule(s)

• What's the dice combination game table(s) face most often, if two dice are rolled?
• What's the chance of lottery winning, given you have to cross 5 correct numbers out of 50?
• Suppose you want to calculate lottery permutation(s), which rule to use?

The lottery probability is explained as follows.

Most lotto games do not replace a ball which has been selected. So you need to adjust the number of successes and the total number of events in the Laplace Rule after each drawing.

For example, suppose we have a simple lotto game with 5 balls, two of which are drawn. For the first ball, all are equal. Suppose we draw ball 1. For the second ball, it is impossible to draw ball 1, because it is not replaced.

The total number of balls is 4. So there are 5 x 4 possible outcomes (five for the first ball, four for the second).

The chance of obtaining (1,2) is thus 1/20.

So you can use probability theory for your ten thousand dice game.

The black jack probability can be calculated with the formula of permutation and by determining the card probability in the same way.

Just make sure you know how many "succesful" outcomes there are, and how many in total. And don't forget that these must be independent. Then use Laplace's Rule and you know the probability.

Solve this math riddle

What's the probability of rolling exactly Z with X dice, each of which has Y sides? This math game (free for all) is not that easy!

Luckily, there is DiceLab to help you out.

Special dungeon and dragon dice are no longer needed!

You can calculate the chance of a random encounter with a dungeon dice monster(s).

For example, if the chance of a random encounter with an Orc is 3-5 on 3D4, DiceLab indicates that the actual chance is 6%.

You can then replace 3D4 by a D100 (simply throw 2D10 of different colors, one of which is interpreted as 10 times its value).

So roll a dice & dungen monster(s) will show up in your RPG! Throw any combination of dice & dungeon monster rule(s) will tell you what monster shows up around the corner!

Dice picture(s) of different sizes can be found on the manufacturer's websites. Dice manufacturer(s) try to make their dice perfect, yet no one beats this simple and easy computer-based dice roller!

A lesson of probability

Teach any kid mathematics with this dice probability calculator.

The short probability tutorial above has already explained how to solve math probability problem(s) on statistics.

So you now know that one of the easiest mathematics equation(s) is the one which governs simple dice probability: the Laplace Rule.

You can calculate it by hand, of course, but as the number of dice grows, this becomes cumbersome.

Exercises: What's the probability of

• throwing exactly 1 with 1D6?
• throwing exactly 1 with 2D6?
• throwing exactly 5 with 2D6?
• throwing at least 5 with 2D6?
• throwing at most 5 with 2D6?
• compare the results from the previous three exercises. What do you remark? Why is this so?
• Only if you know how to work with the Normal Distribution as an approximation of high number of dice: What's the probability of throwing more than 100 dots with 10D20?

Get help with math probability theory, and download the free DiceLab program below.

Download the free calculator of combination(s) with dice

Download this calculator of probability here.

This calculator of permutation(s) with dice actually shows the chance of getting a specific result, for example: how big is the probability of getting at least 10 dots facing upwards if you throw 2D12? (It turns out to be 75%)

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