The math of SWD20

[AdInfinitum]

It's been a decade since my last math class, and I'm having trouble figuring something out. I have 18 con and DR 4. What are my odds of surviving a critical hit off an e-web? In other words, what are the chances of 6d6 rolling 21 or less? Considering that the average of 6d6 is 21, that has to be pretty close to 50%, right?

And feel free to make all the "Never tell me the odds" jokes you wish. :smalltongue:

[meschlum]

The exact solution involves compting the odds of getting a 6, adding the odds of getting a 7, and so on. This is long and error prone.

So we cheat!

The likelihood of getting N on 6d6 is equal to the likelihood of getting 42 - N (6 happens as often as 36, 7 as often as 35, etc.)

So since you're after 21 or less, you know the odds X of getting 20 or less are equal to the odds of getting 22 or more. So you have

2 X + probability of getting 21 = 1 (you always get something)


We just need to figure out the odds of getting 21. Which is tedious.

So we cheat again!

The first 3d6 don't matter. Whateevr they roll, you can still get 21 afterward.

In fact, if you roll K on the first 3d6, you need to roll 21 - K on the next 3d6. And those two events have the same odds (3 and 18, 10 and 11, etc.)

So we 'just' need to get the odds on 3d6 and work from there.

And, cheating a bit more, we really only need the odds for 3-10, since the rest is equal.

3: 1 / 216
4: 3 / 216
5: 6 / 216
6: 10 / 216
7: 15 / 216
8: 21 / 216
9: 25 / 216
10: 27 / 216

So we get: 1 + 9 + 36 + 100 + 225 + 441 + 625 + 729 for the 3-10 range, or 2166.

Which doubles to 4332 for the 3-18 range on 3d6, so that's the probability of getting 21 on 6d6 (well 4332 / 46656, as 216^2 = 46656).

We get X = (46656 - 4332) / (2 * 46656) = 21162 / 46656.

So the final odds are (21162 + 4332) / 46656 = 25494 / 46656, which can be simplified to 4249 / 7776.

And that's roughly 54.6%, so slightly better than even odds.

If you want to skip the last bit of math, note that 4332 is almost 10% of 46656, so you get 2 X ~ 90%, which means your odds are around 45% + 10%. Which saves on the last bit for a minor loss of precision.