# Dice Probability

**URL:** <https://boards.straightdope.com/t/dice-probability/438909>\
**Category:** Factual Questions\
**Created:** [February 24, 2008, 2:16am UTC](https://boards.straightdope.com/t/dice-probability/438909 "2008-02-24T02:16:19Z")\
**Posts on this page:** 5\
**Page:** 1

<div class="post-metadata">

**Author:** ![Garula](https://avatars.discourse-cdn.com/v4/letter/g/ea666f/32.png) [@Garula](https://boards.straightdope.com/u/Garula)\
**Post date:** [February 24, 2008, 2:16am UTC](https://boards.straightdope.com/t/dice-probability/438909/1 "2008-02-24T02:16:19Z")

</div>

I have some questions about how to go about figuring out probabilities when rolling dice. Suppose I roll three six-sided dice, take the two highest and add them together, dropping the lowest one. How do I figure out the probability of arriving at any given possible value? Such as, what would be the probability of rolling seven under this setup? What if I used four dice instead, dropping the two lowest? What if I rolled four dice and dropped only the lowest, how about then? How about if I used larger dice, say twelve sided ones? How about if I mixed different types of dice, say two six-sided dice and two eight-sided dice, dropping the two lowest?

I’m hoping there’s some relatively simple methodology I can use to solve these sorts of problems. Thanks!

---

<div class="post-metadata">

**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [February 24, 2008, 3:46am UTC](https://boards.straightdope.com/t/dice-probability/438909/2 "2008-02-24T03:46:08Z")

</div>

There are analytical techniques for this sort of problem, but it’s much easier to write a program to consider all possible outcomes of rolling the dice and compute the probabilities from there.

---

<div class="post-metadata">

**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\
**Post date:** [February 24, 2008, 5:34am UTC](https://boards.straightdope.com/t/dice-probability/438909/3 "2008-02-24T05:34:55Z")

</div>

The simple—as opposed to easy—is to list all possible outcomes of rolling the three six-sided dice (or whatever you’re doing) and counting up how many of them give you each possible result.

As **ultrafilter** suggests, this is much more easily done with the help of a computer program. With three six-sided dice, the number of possible outcomes is 216, and this number goes up rapidly with more dice or more sides.

---

<div class="post-metadata">

**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\
**Post date:** [February 24, 2008, 5:51am UTC](https://boards.straightdope.com/t/dice-probability/438909/4 "2008-02-24T05:51:03Z")

</div>

At a guess, the total would tend toward 3.5 per die, since dropping the lowest one, or two, is of dimishing significance as to total number of dice grows.

---

<div class="post-metadata">

**Author:** ![cerberus](https://avatars.discourse-cdn.com/v4/letter/c/71e660/32.png) [@cerberus](https://boards.straightdope.com/u/cerberus)\
**Post date:** [February 24, 2008, 9:56pm UTC](https://boards.straightdope.com/t/dice-probability/438909/5 "2008-02-24T21:56:15Z")

</div>

You can do this sort of thing on paper:  
[my course](http://www.mindspring.com/~cjalverson/_slides_spring_2008.htm) has a probability unit, including dice-based models. See sessions 1.3, 1.4 and 1.5 for examples.