Numerology: 20 Faces of Evil

Behold, the d20, wonder of the modern age.  Since the coming of the prophet Gygax, this icosahedron of polymers has determined the fate of many an unwary traveller.

Lets figure out how the darn thing works…

To start, lets lay out the Statistics:

20 Possible results: 1, 2, 3, skip a few, 19, 20

Mean:

Remember that the Mean is the average of the possible results, so 1+2+3+…+20 = 210, and 210 divided by 20 is 10.5

Median:

We find the median by counting in an equal distance from each end of the ordered possible results.  So we count 10 up from 1 and 10 down from 20 and we finds the Median is smack dab between 10 and 11, or 10.5

Fun Fact #1:  For normal dice, the Median is always equal to the Mean.

Mode:

For a single, normal die, each face occurs one time, so there is no Mode.

Fun Fact #2: Since there is no mode for a single die, the results tend to feel very random or “swingy”.  It is difficult to make hard predictions since no 1 result is any more likely than any other.

And now the Probabilities:

Chance of any particular face coming up: 1 in 20, or 5%.

Usually in a game, you want to know your chance of rolling some number or better.  There are two ways of figuring this out, you can look at the results you want or that you don’t want.  For example, if you wanted a 17 or better on a roll, then 17, 18, 19, and 20 are the results you want, or 4 in 20, or 20%.  Alternatively, you could say 1 to 16 are bad results, 16 in 20 will fail, or 80%.  Since the total needs to be 100%, then the chance of success is 100-80, or 20%.  Both methods give you the same result, so it becomes a matter of which is easier in a particular situation.

So the chance of:

5 or better 80%
10 or better 55%
15 or better 30%
natural 20, only 5%

So a regular d20 is kind of boring statistically…  how do we jazz it up?

Lets start with a reroll.

If you roll, and get a 1, there is a 19 in 20 chance of doing better.
If you roll, and get a 2, there is an 18 in 20 chance of getting better, and a 1 in 20 chance of getting worse.
And so on…

Although the results are easy to calculate, the value of a decision to reroll is hard to determine, since you may or may not know what you need to roll.  If you roll a 14, and reroll it hoping to get better, that was only a bad choice if 14 was good enough.

So lets try looking at best of two dice, instead of a simple reroll. The distribution is very different here.

This chart shows there are 400 possible combinations of 2d20:

The only way to get a 1 is if both dice come up 1, which is 1 in 400 results, or .25%.
The only ways to get a 2 is if both dice are 2, or if one die is a 1 and the other a 2. 3 in 400 or .75%

Similarly there are five ways to get a 3, or 1.25%.
Seven ways to get a 4, 1.75%.

And so on up to thirty-nine ways to get a 20, or 9.75%.  The chances of rolling a 20 almost double.  More importantly, the mode of this distribution is 20, so a 20 is the most likely result.

Even more interestingly, the chances of rolling a 19 or better is now 9.75%+9.25%, or 19%, almost 4 times normal.  18 or better is 9.75 + 9.25 + 8.75, or 27.75%.

17+ = 36%
16+ = 43.75%
15+ = 51% so more than half your rolls will be 15 or better.
14+ = 57.75%
13+ = 64%
12+ = 69.75%
11+ = 75% compare this to the 50% chance just rolling 1d20 normally.

And that, Ladies and Gentlemen, is how the common, ordinary d20 works, at least according to the math.  We all know, though, that the more dire the circumstances, the more likely we are to roll a ’1′.

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