Probability Curves - Calling All Mathematicians!

[ggroy]

Essentially, you are recreating coin flips, except the 50/50 odds are shifted one way or another.

Coin flipping is even easier to model. It is a binomial distribution with probability of success p=0.50.

 

[ggroy]

Example: 3d6 vs 4 rolling 1, 4, 5 is two successes.

This is simply a binomial distribution with parameters:

N = 3 trials
p = probability of rolling 3d6 >= 4
n = 2 success


For the same example with a static bonus adjustment of +2, it becomes a binomial distribution with parameters:

N = 3 trials
p = probability of rolling 3d6 >= 2
n = 2 success

 

[ggroy]

I've been able to work out equations so far to calculate the odds of rolling X number of successes on Y dice with Z target value, varying number of dice, size of dice, number of successes and target value.

Are you looking at how many attack rounds it will take on average to kill a particular monster with a particular weapon damage dice pattern?

 

[ggroy]

My 'trials' are dependent on one another though - the first success has a probability of x, but once that occurs, subsequent trials have success probabilities of y where x>y; maybe even a third probability z where y>z.

What would be a concrete example of this. I'm not quite grokking what you are explaining here.

 

[ggroy]

I've done a couple such bonus 'calculations' the hard way by writing out all possible die results and counting successes by hand with the bonus included and know that it flattens the curve out and seriously ups your odds of getting ANY successes, but I'd like to find a function for this rather than writing out dice combos.

What you're after here are the coefficients of the expanded polynomial in my previous post on generating polynomials. It essentially boils down to finding a formula for finding coefficients of polynomials like:

(a_1 + ... + a_X)^k *(b_1 + ... + b_Y)^m *(c_1 + ... + c_Z)^n ....

where X, Y, Z, k, m, n, etc ... are positive integers.

 

[ggroy]

If you're designing your own homebrew rpg, another equation which is useful is related to how many attack rounds it takes to hit an opponent.

With a probability p of hitting an opponent, the average number of attacks it takes to hit an opponent is 1/p with variance (1-p)/(p^2).

For example, a 50% probability of hitting an opponent takes on average around 2 attacks to hit it. A 25% probability of hitting an opponent takes on average around 4 rounds to hit it. A 20% probability of hitting an opponent takes on average around 5 rounds to hit it. More generally:

p    average number of attacks to hit

0.50  2
0.45  2.22
0.40  2.5
0.35  2.86
0.30  3.33
0.25  4
0.20  5
0.15  6.67
0.10  10
0.05  20

Changing the probability of hitting an opponent can drastically affect the average number of attacks it takes to hit the opponent. For example, if the opponent suddenly gets buffed where it's AC is so high that there's only a 10% probability of a player hitting it, on average it will take 10 rounds to hit it. Or even worse, the opponent is so buffed that there's only a 5% probability to hit it (ie. rolling a 20 on a raw d20 roll), on average it will take 20 rounds to hit it.

I generally try to avoid situations where it requires the players to roll a natural 19 or 20 on their d20 rolls to hit an opponent. It typically drags the encounter into a slow grind. I'll usually adjust it so that the hardest is rolling a natural 18 on a d20 for a hit, under normal circumstances.

The number of hit points to allocate to particular monsters will depend on how many rounds one would want an encounter to last. These days I usually just use generic minion-like monsters, and modified minions which take 2 or 3 hits (or more) to kill them. Though for "mini-bosses" or "big-bosses", I'll usually allocate an appropriate number of hit points.

With modified minion type monsters/badguys that take 2 or 3 hits to kill, I usually try to make it such that it takes the party around 7 or 8 rounds to kill most of them. The "mini-bosses" will take slightly more rounds.

In general, I use this framework to check things like whether an encounter is too overpowered on one side, or if a skill challenge or check is too hard or too easy.

 

[Jasper Flick]

RyvenCedrylle, you can count successes in AnyDice without much of a hassle, if you keep it simple.

For example, rolling 6d10 with target value 7 is 6d10c>=7. Checking the graphs, you'll see - for example - that getting at least 3 successes has a 45.57% chance of occuring.

Using this syntax, you can easily vary how many dice you roll, how big they are, and what target to reach.

You can get a lot more complicated, but that requires more complicated input as well.