Probablities for role-playing games

GURPS: skill rolls

GURPS uses three six-sided dice for skill rolls. This table shows the chance of making the various rolls:

Probabilities when rolling 3 six-sided dice
Roll	chance	equal	greater	greater or equal	less	less or equal
3	1/216	0.5%	99.5%	100.0%	0.0%	0.5%
4	3/216	1.4%	98.2%	99.5%	0.5%	1.9%
5	6/216	2.8%	95.4%	98.2%	1.9%	4.6%
6	10/216	4.6%	90.7%	95.4%	4.6%	9.3%
7	15/216	6.9%	83.8%	90.7%	9.3%	16.2%
8	21/216	9.7%	74.1%	83.8%	16.2%	25.9%
9	25/216	11.6%	62.5%	74.1%	25.9%	37.5%
10	27/216	12.5%	50.0%	62.5%	37.5%	50.0%
11	27/216	12.5%	37.5%	50.0%	50.0%	62.5%
12	25/216	11.6%	25.9%	37.5%	62.5%	74.1%
13	21/216	9.7%	16.2%	25.9%	74.1%	83.8%
14	15/216	6.9%	9.3%	16.2%	83.8%	90.7%
15	10/216	4.6%	4.6%	9.3%	90.7%	95.4%
16	6/216	2.8%	1.9%	4.6%	95.4%	98.2%
17	3/216	1.4%	0.5%	1.9%	98.2%	99.5%
18	1/216	0.5%	0.0%	0.5%	99.5%	100.0%

Some observations:

There is a 50% chance of rolling lower or equal to 10. Thus, a skill of 10 is sucessful 1/2 the time.
There is a ~50% (48.1%) chance of rolling a 9, 10, 11, or 12.
There is an ~80% chance of rolling 7-14.
A skill of 14 gives you a 90% chance of success.
Critical successes are 3 and 4 (~2%). If your skill is 15, 5 is also a critical (~4.5%). If your skill is 16+, 6 is critical (~9%).

Contest of Skills
Bonus	win	lose	Tie
0	45.4%	45.4%	9.3%
+1	54.6%	36.3%	9.1%
+2	63.7%	27.9%	8.4%
+3	72.1%	20.6%	7.4%
+4	79.4%	14.5%	6.1%
+5	85.5%	9.7%	4.8%
+6	90.4%	6.1%	3.6%
+7	93.9%	3.6%	2.5%
+8	96.4%	2.0%	1.6%

GURPS contests of skills

A common roll in GURPS is a contest of skills where two people roll versus a skill. The person who wins by the most (or loses by the least) is the winner. For example, a thief creeping past a guard would roll versus her Stealth skill while the guard rolls versus his Vision. The important variable in these rolls is the amount by which the skills differ. The case where Stealth is 14 and Vision is 11 (difference of 3) is identical to a Stealth of 16 and Vision of 13.

On this table the bonus is always positive. For the negative case reverse the win/lose columns.

Note: This table does not take into account critical success and failure. It would change the numbers by a very minimal amount.

Some observations:

When you have a +5 you have a ~85% chance of winning. This becomes ~90% if you re-roll ties.

GURPS luck probabilities
Skill	No Luck	Luck (3 rolls)	Luck (2 rolls)
3	0.5%	1.4%	0.9%
4	1.9%	5.5%	3.7%
5	4.6%	13.3%	9.0%
6	9.3%	25.3%	17.7%
7	16.2%	41.2%	29.8%
8	25.9%	59.4%	45.1%
9	37.5%	75.6%	60.9%
10	50.0%	87.5%	75.0%
11	62.5%	94.7%	85.9%
12	74.1%	98.3%	93.3%
13	83.8%	99.6%	97.4%
14	90.7%	99.9%	99.1%
15	95.4%	~100.0%	99.8%
16	98.1%	~100.0%	~100.0%
17	99.5%	~100.0%	~100.0%
18	~100.0%	~100.0%	~100.0%

GURPS Luck Rolls

The Luck advantage allows you to roll three times for a given die roll, taking the most favorable of the three rolls. The standard rules are unclear about when you declare that you are using Luck; before any rolls (getting three attempts) or after you fail (getting two additional attempts for a total of three). There is a ruling in the GURPS FAQ which says that you may use the second form.

"No Luck" is the standard chance without using Luck. The "Luck (3 rolls)" column is the chance if you decide to use Luck before you roll. The "Luck (2 rolls)" is the chance if you use Luck after failing once.