What is the statistical variance of a +1 on opposed rolls?

[Cielingcat]

I'm really bad at statistics, and as such, for the life of me, I can't figure out how, say, a +1 bonus on an opposed d20 roll (like Trip) changes your chance of success. I know it's more than 5%, but I can't figure if it's 10% or 7.5% or whatever. Could someone help me with this?

[Judging__Eagle]

I thought it was 5%?

The reason it would be more or less would be that you have a rounded or tumbled dice.

Also, you have to take into consideration what you're rolling against.

[Cielingcat]

Assuming that you and your opponent both start at +0, and you get +1 while your opponent remains at +0. I know that against a static DC the +1 is 5%, but I believe that on an opposed roll it increases the likelihood of rolling higher than your opponent by more than 5%.

[bitnine]

It will indeed vary with the bonuses. Consider the edge case where an opponent's roll is so high they just barely beat you with a 1 and you a 20. That +1 increases your odds of winning by (0.05)(0.05), but from 0%, so any way you dice it that's not 5%.

[AlphaNerd]

Rolling an opposed check is statistically the same as rolling 2d20 for a fixed DC, if that helps.

Obviously, it depends on to whom ties go to, but here's a chart of your probability to exceeding, tieing and losing to someone if your relative scores are in the first column.


-19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

	Beating	Tieing	Losing
0.0000	0.0025	0.9975
0.0025	0.0050	0.9925
0.0075	0.0075	0.9850
0.0150	0.0100	0.9750
0.0250	0.0125	0.9625
0.0375	0.0150	0.9475
0.0525	0.0175	0.9300
0.0700	0.0200	0.9100
0.0900	0.0225	0.8875
0.1125	0.0250	0.8625
0.1375	0.0275	0.8350
0.1650	0.0300	0.8050
0.1950	0.0325	0.7725
0.2275	0.0350	0.7375
0.2625	0.0375	0.7000
0.3000	0.0400	0.6600
0.3400	0.0425	0.6175
0.3825	0.0450	0.5725
0.4275	0.0475	0.5250
0.4750	0.0500	0.4750
0.5250	0.0475	0.4275
0.5725	0.0450	0.3825
0.6175	0.0425	0.3400
0.6600	0.0400	0.3000
0.7000	0.0375	0.2625
0.7375	0.0350	0.2275
0.7725	0.0325	0.1950
0.8050	0.0300	0.1650
0.8350	0.0275	0.1375
0.8625	0.0250	0.1125
0.8875	0.0225	0.0900
0.9100	0.0200	0.0700
0.9300	0.0175	0.0525
0.9475	0.0150	0.0375
0.9625	0.0125	0.0250
0.9750	0.0100	0.0150
0.9850	0.0075	0.0075
0.9925	0.0050	0.0025
0.9975	0.0025	0.0000