# 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%.

#### 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.

#### diaglo

roughly....7-3 = 4

4 * 5% = 20%

you have a 20% greater chance of fooling your opponent.

#### Altamont Ravenard

##### Explorer
(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
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``````

#### Rodrigo Istalindir

##### 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?

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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