You may wonder: what is the probability of rolling the SAME value on each and every die from a set of two dice or more (“n” dice, where n larger than 1).

To answer this question we must consider the following:

Let us start by saying that the chance of getting any single given value on a single die is defined as the probability “p”. This probability, of course, is simply p = 1/s , where “s” is the number of sides of the die. I.e. the probability of rolling a given number on a 6-sided die is always p(6) = 1/6 . If we were to throw several dice, one after another, the probability of each one by itself rolling any single given value is still “p”.

If we want every die we roll to have the same value, however, then the probability “P” of this happening is calculated by multiplying “p” by itself as many times as there are dice (“n”).

I.e. the overall probability “P” equals “p” to the power “n”, or

P = pⁿ = (1/s)ⁿ

For example, if we consider three 8-sided dice, the chance of rolling 5 on each of them, one after another, is:

P = p³ = (1/8)³ = 0.0001953125 = 0.01953125%.

Or for example, if we consider three 6-sided dice, the chance of rolling a 3 on each of them, one after another, is:

P = p³ = (1/6)³ = 0.004629629 = 0.04629629%.

Interestingly, the same formula can be used for a coin, which you can simply treat as a 2-sided die. The probability of throwing a coin heads-up three times in a row is thus:

P = p³ = (1/2)³ = 0.125 = 12.5%.