Monday, March 31, 2008

Botched Math

The following is from the Scion forum. I don't ever want to have to do this math again, so I'm copying it here where it's easy for me to find the next time this topic comes up over there.

Zeev wrote:
My point is, your house rule makes botching more likely than failure.

Let me translate Zeev's point to math...

On 1 die you (obviously) have a:
10% chance of botching,
50% chance of failing without a botch
40% chance of getting one or more successes

On 2 dice you have:
11% chance of botching,
25% chance of failing without a botch
64% chance of getting one or more successes

Now if a person just looked at that you'd get the mistaken impression that botch chance just keeps going up as more dice are rolled. It doesn't though - that's just a fluke of how the equations work that define the odds.

What's happening is that each extra die causes the chance to "normal" fail to drop in half, but the chance to botch is equal to (60%x(chance to botch on one less die)+10%x(chance to fail on one less die)).

As a result, the odds of botching increase when you go to the second die, and thereafter decrease at a slower rate than the decrease of straight failures.

On 3 dice you have:
9.1% chance of botching
12.5% chance of failing without a botch
78.4% chance of getting one or more successes

On 4 dice you have:
6.71% chance of botching
6.25% chance of failing without a botch
87.04% chance of 1+ success

Already you've become more likely to botch than to just fail the roll.

It certain sounds horrible to think that at a paltry four dice you're already more likely to botch than merely fail. And it gets worse...

On 5 dice you have:
4.651% chance of botching
3.125% chance of failing without a botch
92.224% chance of 1+ success

Zeev is correct that at difficulty 1 you have a better chance of botching than failing if rolling 3 or more dice. One could see that as evidence arguing against houserules the preserve botches when Epics are being used. I believe that's where he's coming from on this point, and I'm sure he'll correct me if I'm misunderstanding him.


However, there's where Zeev's concept is a little flawed: Raising the difficulty of the roll beyond a simple success increases the chance of normal failures, but doesn't increase the chance of botches.

For example, raise the difficulty to 5, and roll 3 dice:
Botch rate remains at 9.1%
Normal failure skyrockets to 89.9%
Success rate drops to 1%

Since the vast majority of die rolls in Scion involve a difficulty above 1, it's a non-issue.

For example, if rolling 10 dice against difficulty 2, you have a less than 0.6% chance of botching, and around a 3% chance of failing without a botch.

Only on difficulty 1 rolls can botches occur more often than normal failure.


digital_sextant said...

And not that I know anything about Scion, but perhaps this could be explained as the "careless factor" that occurs whenever people do something they perceive to be very easy. I'm much more likely to "botch" something I don't think I have to try very hard at than something that takes my full concentration.

r_b_bergstrom said...

I may bring that point up the next time this topic becomes hotly debated on the forum - which is about every 2.5 months or so.