A lot of people think that if you roll three six sided dice you have the same chance of rolling a three as you do a ten.This is not the case and this article will show you how to calculate the mean and standard deviation of a dice pool.You can learn the terminology of dice mechanics.The 6 sided variety of dice are usually found in d2( Coins), d4(3 sided pyramids) and d8(Octahedra).2d6 would be a roll of two six sided dice.Some formulas assume that the number of sides on each die is the combination value.There are many methods for determining the likelihood of a sum.
Step 1: The number of dice, their sides and the desired sum should be noted.
Step 2: Take a look at all the ways that sum can be reached.
This can be difficult for large numbers of dice.This is the same as finding the entire partition of k into exactly n parts with no part larger than r.The dice are presented in non-decreasing order in order to ensure that no partition is counted twice.
Step 3: Partitions listed in the previous step are not likely to be the same.
This is why they have to be listed.The partition 123 covers 6 possibilities, while the partition 114 only covers 3 and the rest includes itself.The number of ways to permute the digits in each partition is calculated using the multinomial formula.This information has been added to the table.
Step 4: To get the desired sum, add the total number of ways.
Step 5: Divide by the number of outcomes.
Each die has the same probable faces.
Step 6: The outcomes of a single die should be noted.
They should be recorded in a spreadsheet.The 6-sided dice are used in the example.The blank rows for negative sums are treated as zeros and allow the same formula to be used in all rows.
Step 7: Use the formula shown in the column for 2 dice.
The sum of the following events is the probability of 2 dice showing any sum k.For very high or low values of k, some or all of these terms might be zero, but the formula is valid for all k.The first and second die show k-2.The first die shows k-1, and the second shows 3.The first die shows k-4 and the second shows 4.The first and second die show k-5.The first die shows k6 and the second shows 6.
Step 8: For three or more dice, the same formula still applies, using the now known probabilities for each given sum on one die fewer.
The formula can be filled down and across until the table has enough data.
Step 9: It is easy to convert between the number of ways shown in the spreadsheet.
The number of sides on each die is the probability.The probability can be calculated directly from the spreadsheet.
Step 10: Write the answer to the question, (1/r)(x + x + ).
+ xThe generating function is used for a single die.The probability of the die showing k is the x term's coefficients.
Step 11: To get the generating function for the sum shown on n dice, raise the polynomial to the n power.
That is calculated by summing x and x together.You will probably want to do this on a computer if you are larger than 2.
Step 12: Sometimes theoretical results are easier to derive with a generating function, but Computationally this is equivalent to the previous method.
Throwing two regular 6-sided dice has the same distribution of sums as a die labeled (1, 2, 3, 4, 5, 6, 8)This is because of (x+x +x+)x.
Step 13: For a large number of dice, exact computation may be difficult.
As the number of dice increases, the sum of identical dice approaches a normal distribution according to the central limit theorem.
Step 14: The mean and standard variation are determined by the number and type of dice.
The formulas below apply if n dice is numbered 1 to r.The mean is two.The difference is 12.The square root of the variance is the standard deviation.
Step 15: The normal distribution can be used to approximation the sum of the dice.