**Christian**and a read post from

**Saxywolf**on essentially the same question:

*What is the probability of rolling a given value on an*

**D**-sided die, if you roll**N**dice and take the highest (or the lowest).Here's the trick:

The probability of rolling a 1 on 1 d6 is 1/6. (regular 6-sided dice)

**N**d6 the probability of rolling a 1 as the maximum is 1/6 times itself

**N**times, or (1/6)^

**N.**

**D**for 6 above, so that the probability of rolling a 1 as the maximum of

**N**

**D**-sided dice is 1/

**D**times itself

**N**times, or (1/

**D**)^

**N.**

**as the maximum. The probability of rolling 2-or-less is 2/**

*2-or less***D**, and the probability of rolling a 2-or-less as the maximum is 2/

**D**times itself

**N**times, or (2/

**D**)^

**N.**

That's 2-or-less, but we really just want the probability of rolling 2,

*not 1 or 2*. BUT we already know the probability of rolling a 1 as the maximum on the same dice, so we can subtract that to get what we want:

The probability of rolling a

**2**as the maximum is 2/

**D**times itself

**N**times, minus the probability of rolling 1 as the maximum, or (2/

**D**)^

**N -**(1/

**D**)^

**N**.

And that's it. Using the same math you can work the complete distribution of the maximum for any number of dice with any number of (equally likely) faces. Just start at

**1**and work up. For minimums, just turn the problem around and find the probability of

*X-or-less*.

Link to Google Docs Spreadsheet

See the

**blue numbers**in the green-shaded cells? Change those to the number of side on your dice and the number you want to roll, and it will calculate the distribution for you.

And a chart to display the results: