Math / Probability Help

[Darkness]

Pb, that is not correct. I'm not entirely sure where the error lies, but there is one.

The error lies in this conclusion (emphasis mine):

Now a difference of +20 (1+20=21, better than 20) between the two players guarantees success for the primary roller, and a difference -20 (20-20=0, worse than 1) guarantees failure. So each +1 bonus (or -1 penalty) accounts for 2.5% of the overall probability.

I'll try to show that that's not the case (making the formula useful only for the subset of cases in which it is true, if any, but not for the general case).
If my math is off somewhere, it's probably because I'm exhausted and my head hurts. (Yesterday was very taxing and I didn't get much sleep.) If so, please correct it; I'll make sure that my reasoning is transparent and thus mere numbers are easy to correct if they are wrong.

50% + ([Player A modifiers] - [Player B modifiers]) * 2.5% = Probability of Player A Success.

A simple example using Pbartender's formula follows.

Player A: +18
Player B: +0

50% + 18*2.5 = 95%

Pbartender also states that ties make up 5% of the remaining cases.
Thus, the chance that Player B wins (unless we count a tie in his favor) is 0%.

So far, so good. Now let's look if that's correct.

If Player A rolls a 3 or more, he wins regardless of what Player B rolls.
If Player A rolls a 2 or less, he still wins if Player B doesn't roll a number higher than his by at least 18. Otherwise: A2-B20 = tie, A1-B19 = tie, A1-B20 = B wins.

Thus, A wins in all of the following cases (i.e., combinations):
A3-20/B1-20
A2/B1-19
A1/B1-18

Tie:
A2/B20
A1/B19

B wins:
A1/B20

Now, what are the odds?
A has a 90% chance of rolling a 3 or more, in which case he wins and B's roll is irrelevant. (I.e., 20/20 of what B can possibly roll will fail to beat or tie A's roll.)

A has a 5% chance of rolling a 2.
If B then rolls a 19 or less, A wins. (19/20 of 5% = 4.75%)
Otherwise, it's a tie. (the remaining 0.25% of the 5%)

A has a 5% chance of rolling a 1.
If B then rolls an 18 or less, A wins (18/20 of 5% = 4.5%)
If B then rolls a 19, it's a tie (1/20 of 5% = 0.25%)
If B then rolls a 20, he wins (also 0.25%)

A wins: 90+4.75+4.5 = 99.25%
Tie: 0.25+0.25 = 0.5%
B wins: 0.25%

(Ties go to the character with the higher bonus by the RAW, but that's tangential to my point.)

95% /= 99.25%

As for ties: 5% /= 0.5% ('cause a tie can not happen on the numbers A rolls on his d20 that make him win 'automatically.')

QED.

 

[Brennin Magalus]

The probability that the character with the +10 bonus wins is .7. This is easy to see if you start to write out all the possible outcomes in this (discrete) probability space. If you write the numbers 6 to 25 on the top of your paper and 11 to 30 on the left hand side you will see that there are 5 successes (for the character with the +10 bonus, assuming a tie is a failure) in the first row, 6 successes in the 2nd, 7 in the 3rd and so on until you reach the 16th row has 20 successes which is also the case for the subsequent rows (up to row 20, which is the last row). Adding the successes and dividing by 400 (the total number of outcomes, each of which are equally likely) you get .7.

 

[Thanee]

...there are 5 successes (for the character with the +10 bonus, assuming a tie is a failure) in the first row, 6 successes in the 2nd, 7 in the 3rd and so on until you reach the 16th row has 20 successes which is also the case for the subsequent rows (up to row 20, which is the last row). Adding the successes and dividing by 400 (the total number of outcomes, each of which are equally likely) you get .7.

Just look two posts up, there's the table you are talking about (it's just flipped diagonally, your first row is my first column).

Bye
Thanee

 

[Elric]

I agree with Thanee. His initial post lays it out in a (to me) very clear way. Because you have two die rolls (not just one) there is a quadratic element to your solution.

To take the really general form of this problem suppose you generate a random number on (0,1) and add X. Then your opponent generates a random number on (0,1). For x positive, your chance to have a higher number is 1/2 + x - 0.5* x^2.

While the function involved here is not linear, its derivative is linear. The statement "I am up by +5 (out of 20) so I win 1/2 + 5/20= 75% of the time" is just the linear approximation of this function around x= 1/2. However, the derivative is decreasing. It is easy to see that its derivative must fall to 0 as x-->1 because you already win essentially all of the points so an increase in x has almost no effect on win percentage.

 

[orsal]

I agree with Darkness about the error in Pbartender's calculation. If you are doing anything with a DC which you will make with a 21 and fail with a 1, then an extra +1 to your bonus will add 5% to your chance of success. If the DC is out of that range, then an extra +1 will do nothing.

As Thanee noted, there are 20x20=400 possible outcomes, all with the same probability. That means the probability of any event is (# of outcomes)/400. Let's call the player with the higher bonus A, the player with the lower bonus B, and the difference in their bonuses x. Now A will lose iff (B's roll)-(A's roll) is greater than x. Count up the number of ways that can happen. There is 1 way to get a difference of 19, namely A gets 1 and B gets 20. There are 2 ways to get a difference of 18, namely (1,19) and (2,20). And so on -- there are (20-t) ways for B to exceed A's roll by exactly t. So there are

1 + 2 + 3 + . . . + (20-(x+1))

ways for B to exceed A's roll by x+1 or more, and

1 + 2 + 3 + . . . + (20-x)

ways for B to exceed A's roll by x or more. The former is the number to use of A will win ties (official rules); the latter, if B will win ties (variant rule).

Now, 1 + 2 + . . . + k = k(k+1)/2, so:

If A wins ties, there are (19-x)(20-x)/2 outcomes which have x losing, with a total probability of (19-x)(20-x)/800.

If B wins ties, there are (20-x)(21-x)/2 outcomes which have x losing, with a total probability of (20-x)(21-x)/800.

 

[Pbartender]

Yeah, Darkness, I suspected I got that part wrong as soon as I'd gotten off work. Serves me right for trying to do probability mathematics in my head at the tail end of a 12-hour midnight shift.

Oh well. My basic premise for the unmodified roll is correct, but I did get the adjustment for bonuses/penalties wrong. Sorry... My bad.

 

[Thanee]

That adjustment must still be non-linear, tho, if you do it that way.


Difference between A's and B's bonus / A's probability of success

+1 / 57.25 %
+2 / 61.75 %

+17 / 99.25 %
+18 / 99.75 %

Difference between "+1" and "+2" : 4.5 %
Difference between "+17" and "+18" : 0.5 %


The result is symmetric around the 50% mark (which is the center, when both have the same modifier as written in my original post), so whether you step up or down from there (in the same distance to the center) results in the same shift in probability, but the shifts into one particular direction are getting smaller and smaller the more you move away from the center.

And that is, because the result depends on both dice rolls.

Bye
Thanee

 

[johnsemlak]

That adjustment must still be non-linear, tho, if you do it that way.


Difference between A's and B's bonus / A's probability of success

+1 / 57.25 %
+2 / 61.75 %

+17 / 99.25 %
+18 / 99.75 %

Difference between "+1" and "+2" : 4.5 %
Difference between "+17" and "+18" : 0.5 %


The result is symmetric around the 50% mark (which is the center, when both have the same modifier as written in my original post), so whether you step up or down from there (in the same distance to the center) results in the same shift in probability, but the shifts into one particular direction are getting smaller and smaller the more you move away from the center.

And that is, because the result depends on both dice rolls.

Bye
Thanee

Complex stuff

So what ramifications does this have for higher level characters spending skill points on skills that are normally opposed (like Move Silently and Hide)? Can spending points on improving those abilities become less valuable?

 

[Thanee]

It means, that if you expect your opponents to have similar skill levels, you're well advised to raise your skill higher, but if you expect your opponents to have much lower skill levels, there is not much point in going farther. If you expect your opponents to have much higher skill levels, then there is likewise no point in raising your skill, unless you can raise it by a lot.

If the difference between the two modifiers from the opposing people is already high, the gain or loss of a single point is quite low, however, if the two modifiers are close together, every additional point adds/takes quite a bit to/from your total chance of success.

Or in other words... only big changes affect big differences much, while small changes affect small differences noticeably already.

Bye
Thanee