Probabilities for opposed skill checks

DanMcS

Explorer
Your probability of winning an opposed check can be calculated as follows, assuming the defender wins ties.

D = (your bonus) - (opponent's bonus).

%victory = 0.475 + (41D-D^2)/800.

I could explain the whole formula, but that's the distillation. If your skill is 7 and your opponent's is 3, -> D=4, your probability of victory on any given roll is 66%.
 

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Gnimish88

First Post
If I have a total bluff modifier of 7 and you have a total sense motive modifier of 3, what are the odds that you will realize that I am not being honest to you?

Does anyone know the methodology I should use or where I can find a probability calculator that I could use for opposed skill checks? The ways I have tried to figure it out have produced unsatisfactory results I would prefer to check against something a little more reliable then my memory of math from high school.
 


(lol, the last reply comes before the original post!!!)

DanMcS, I'd still love to see the formula construction.

Gnimish88:

Basically, in probabilities, you have to check the number of times a particular event will occur, and divide by the total number of possibilities.

In this case, let's say the event is "Character B realizes that Character A is not being totally honest". In order for this event to occur, Character B has to roll a Sense Motive check that is equal or higher than the Bluff check (since the Bluff check sets the DC for the Sense Motive check).

Since there is a difference of 4 in the skill modifiers (in favor of Character A), Character B will need a roll, on a d20, that is higher by at least 4.

There are 400 possible d20 vs. d20 roll results. You have to count the number of times where the roll from Character B is at least 4 higher than Character A. In this case, Character A will Bluff Character B 66% of the time, and Character B will realize that Character A is lying to him 34% of the time.

Alas, I can't remember how to construct a mathematical formula to reprocuce this. What I did is I made a little excel sheet detailling all the possibilities.

(and now I can't attach the file... :\ Well doesn't matter since Dan's formula takes a lot less space than my excel file...)

AR
 

DanMcS

Explorer
Altamont Ravenard said:
(lol, the last reply comes before the original post!!!)

DanMcS, I'd still love to see the formula construction.

Yeah, they're messing with the server clock, I was actually driving around in my car yesterday at the time I supposedly posted the answer. Heh.

Formula as follows:

You can roll 1-20. So can your opponent. You will win from 0-20 out of 20 of his rolls. For instance, if you are more skilled by 10, and roll an 11, your result is a 21, you win 20/20 times since he can't roll a 21.

Example: I am more skilled than my opponent by +4 (I have a +7 and he a +3, but only the difference matters).
You roll: you win(out of 20 of his rolls)
20:20
19:20
18:20
17:20
16:19
15:18
14:17
13:16
...
3:6
2:5
1:4

For instance, if I roll a 17-20, my result is above 21+, he cannot win. If I roll a 1, my result is 5, I win only when he rolls a 1-4.

The formula is as follows: There are 400 results possible.

The number of times I win is 20* (D-1): this accounts for my roll of 20,19,or 18.

Then the formula for the sum from when I roll 17 to 1 is the sum from 20 to 4. Turns out, this is the sum from 20 to D, which is

(20+D) * (21-D)/2.

That is, (17-4)/2 * 24, which is actually a really complicated way to do it, now that I think about it. A simpler formula would be the sum from 1 to 20 minus the sum from 1 to (D-1), which is
210 - (D^2-D)/2. Hmph.

Anyway, the whole thing sums up to the number of times I win out of all 400 tries, which is

20 * (D-1) + 210 - (D^2 -D)/2.

Simplified out, that's [190 + (41*D -D^2)/2], all over 400, which becomes
0.475 + [(41*D-D^2)/800].
 

RichCsigs

First Post
I'm just curious (and if I come across as snarky here, I really don't mean to), but why would you want to know this, other than "just cuz". Is there a practical application to this I'm not seeing?
 

DanMcS

Explorer
RichCsigs said:
I'm just curious (and if I come across as snarky here, I really don't mean to), but why would you want to know this, other than "just cuz". Is there a practical application to this I'm not seeing?

Obviously, it's because rolling 2 d20s in opposition to each other is too complicated, he wants to create a simplified table showing what you have to roll on a d400 for your character to succeed at opposed rolls versus opponents at a variety of skill levels in opposition to him. It's a variation on "players make all the rolls", from UA.

:)

Some people like to know the odds. Dunno why. For instance, with this kind of a formula, you can see that being better than someone in an opposed roll by a mere 1 point puts you over 50% to win, but you need 7 points of lead to get 75%, 15 points to get 95%, and you actually need to be 20 better than them to be 100% to win.

Incidentally, the formula posted above is only valid when your skill is greater than or equal to your opponents. To figure the odds if your opponent is better, the formula is:

(380 + 39D + D^2)/800.

Should be a way to generalize those, but two formulae isn't too bad.
 

kamosa

Explorer
Column 1) Advantage for Player
Basically if the player has a 7 enemy 3 the advantage is 4


Column 2) Number of times the Player would win, (win implies having a combined number 1 greater than the enemy. IE if enemy rolls a combined 15 and the player rolls a combined 16 this is a one point win.) Wins by more than 1 are not counted in this column.

Column 3) % Chance of occurring


Column 4) Cumulative chance of success
Obviously a win by 2 or more is still a win for the players, so this column shows the cumulative chance of a win by the player.

Code:
20	1	0.25	100
19	2	0.5	99.75
18	3	0.75	99.25
17	4	1	98.5
16	5	1.25	97.5
15	6	1.5	96.25
14	7	1.75	94.75
13	8	2.0	93
12	9	2.25	91
11	10	2.5	88.75
10	11	2.75	86.25
9	12	3	83.5
8	13	3.25	80.5
7	14	3.5	77.25
6	15	3.75	73.75
5	16	4.0	70
4	17	4.25	66
3	18	4.5	61.75
2	19	4.75	57.25
1	20	5	52.5
Even	19	4.75	47.5
-1	18	4.5	42.75
-2	17	4.25	38.25
-3	16	4	34
-4	15	3.75	30
-5	14	3.5	26.25
-6	13	3.25	22.75
-7	12	3	19.5
-8	11	2.75	16.5
-9	10	2.5	13.75
-10	9	2.25	11.25
-11	8	2.0	9
-12	7	1.75	7
-13	6	1.5	5.25
-14	5	1.25	3.75
-15	4	1	2.5
-16	3	0.75	1.5
-17	2	0.5	.75
-18	1	0.25	.25
 

RichCsigs said:
I'm just curious (and if I come across as snarky here, I really don't mean to), but why would you want to know this, other than "just cuz". Is there a practical application to this I'm not seeing?

These kind of things can be useful to DM's trying to guage how challenging a particular encounter will be, especially if they find themselves making it up on the fly.
 

BASHMAN

Basic Action Games
Even easier way

An easier way to do it, is to make the "defender"'s skill bonus+10 the target number, basically like an AC score. The check is rolled like an attack, with the exception of tie going to the defender.

So a person trying to sense motive +5 (the attacker) against a person with bluff +7, needs to make a s.m. check against DC 17.

This is basically a variation on a rules variation offered in Unearthed Arcana, for players to "defend themselves" by rolling enemie's attack rolls (well, actually, they roll their ACbonus+d20 vs 10+enemies Attack Bonus, but... that is a tale for another time)
 

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