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Old 01-20-2016, 07:13 PM   #1
trooper6
 
Join Date: Sep 2004
Location: Medford, MA
Default Gathering Successes Thresholds (Call for Math Wizards)

Hi All You Math Wizards!

There is this thing that I do in my GURPS campaigns for long skill tasks while in combat time that makes everything very exciting. It is sort of inspired by other game systems but also inspired by energy gathering magic from Thaumatology.

Let's say there is a fire fight going on. Some PCs are shooting, trying to hold off the bad guys...while one PC is trying to search a computer for incriminating information. How long does that take?

I like to make the hero roll against the relevant skill (say Computer Hacking), and gather a certain amount of margin of successes before the task is complete. They also get 1 for a success and 2 for a Critical Success. So, maybe the Hacker needs to gather a margin of success of 10. Turn 1 she succeeds by 4. With it 1 for the success she has a total of 5 and she needs to gather another 5 margin of successes before she finds the hidden info. Then the other players shoot, aim, whatever. Next turn she succeeds by 3...she'll need one more turn...and turn 3 she gets what she needs and now the team can start retreating. Failure results in no progress. Crit fail either results in loss of all accumulated successes, or you can no longer do the task depending on what makes sense.

So far this has been really fun and the players have all enjoyed it.

So what am I asking of the math wizards? You who know probabilities?

So on average, given x skill level, how many turns will it take to achieve y success margins?

How would one construct a chart or table that could be made with the x axis being skill level, y axis success margins and the then data in the table the average number of turns it would take?

If I had that I could decide on difficulty levels.

Thanks math wizards!
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Old 01-20-2016, 08:04 PM   #2
Eukie
 
Join Date: Dec 2013
Default Re: Gathering Successes Thresholds (Call for Math Wizards)

I tried to assemble a table with this information, using the assumption that Critical Failures have no effect other than not adding a success (it would complicate things greatly):

Code:
Skill	Average Successes\Threshold	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
3	0.023	44	87	131	174	218	261	305	348	392	435	479	522	566	609	653	696	740	783	827	870
4	0.0414	25	49	73	97	121	145	170	194	218	242	266	290	315	339	363	387	411	435	459	484
5	0.0875	12	23	35	46	58	69	80	92	103	115	126	138	149	160	172	183	195	206	218	229
6	0.1798	6	12	17	23	28	34	39	45	51	56	62	67	73	78	84	89	95	101	106	112
7	0.3415	3	6	9	12	15	18	21	24	27	30	33	36	39	41	44	47	50	53	56	59
8	0.6004	2	4	5	7	9	10	12	14	15	17	19	20	22	24	25	27	29	30	32	34
9	0.975	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
10	1.4746	1	2	3	3	4	5	5	6	7	7	8	9	9	10	11	11	12	13	13	14
11	2.0992	1	1	2	2	3	3	4	4	5	5	6	6	7	7	8	8	9	9	10	10
12	2.8413	1	1	2	2	2	3	3	3	4	4	4	5	5	5	6	6	6	7	7	8
13	3.6806	1	1	1	2	2	2	2	3	3	3	3	4	4	4	5	5	5	5	6	6
14	4.5893	1	1	1	1	2	2	2	2	2	3	3	3	3	4	4	4	4	4	5	5
15	5.5719	1	1	1	1	1	2	2	2	2	2	2	3	3	3	3	3	4	4	4	4
16	6.6007	1	1	1	1	1	1	2	2	2	2	2	2	2	3	3	3	3	3	3	4
17	7.5833	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	3	3	3	3	3
18	8.5659	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	3	3	3
I've also made this table that gives the probability of at least one Critical Failure among a given number of rolls, depending on skill:

Code:
Skill	Chance of CritFail\Rolls	1	2	3	4	5	6	7	8	9	10	15	20	25	30	45	50	75	100	150	200
3	0.2589	0.26	0.45	0.59	0.7	0.78	0.83	0.88	0.91	0.93	0.95	0.99	1	1	1	1	1	1	1	1	1
4	0.1617	0.16	0.3	0.41	0.51	0.59	0.65	0.71	0.76	0.8	0.83	0.93	0.97	0.99	0.99	1	1	1	1	1	1
5	0.0923	0.09	0.18	0.25	0.32	0.38	0.44	0.49	0.54	0.58	0.62	0.77	0.86	0.91	0.95	0.99	0.99	1	1	1	1
6	0.0461	0.05	0.09	0.13	0.17	0.21	0.25	0.28	0.31	0.35	0.38	0.51	0.61	0.69	0.76	0.88	0.91	0.97	0.99	1	1
7	0.0184	0.02	0.04	0.05	0.07	0.09	0.11	0.12	0.14	0.15	0.17	0.24	0.31	0.37	0.43	0.57	0.6	0.75	0.84	0.94	0.98
8	0.0184	0.02	0.04	0.05	0.07	0.09	0.11	0.12	0.14	0.15	0.17	0.24	0.31	0.37	0.43	0.57	0.6	0.75	0.84	0.94	0.98
9	0.0184	0.02	0.04	0.05	0.07	0.09	0.11	0.12	0.14	0.15	0.17	0.24	0.31	0.37	0.43	0.57	0.6	0.75	0.84	0.94	0.98
10	0.0184	0.02	0.04	0.05	0.07	0.09	0.11	0.12	0.14	0.15	0.17	0.24	0.31	0.37	0.43	0.57	0.6	0.75	0.84	0.94	0.98
11	0.0184	0.02	0.04	0.05	0.07	0.09	0.11	0.12	0.14	0.15	0.17	0.24	0.31	0.37	0.43	0.57	0.6	0.75	0.84	0.94	0.98
12	0.0184	0.02	0.04	0.05	0.07	0.09	0.11	0.12	0.14	0.15	0.17	0.24	0.31	0.37	0.43	0.57	0.6	0.75	0.84	0.94	0.98
13	0.0184	0.02	0.04	0.05	0.07	0.09	0.11	0.12	0.14	0.15	0.17	0.24	0.31	0.37	0.43	0.57	0.6	0.75	0.84	0.94	0.98
14	0.0184	0.02	0.04	0.05	0.07	0.09	0.11	0.12	0.14	0.15	0.17	0.24	0.31	0.37	0.43	0.57	0.6	0.75	0.84	0.94	0.98
15	0.0184	0.02	0.04	0.05	0.07	0.09	0.11	0.12	0.14	0.15	0.17	0.24	0.31	0.37	0.43	0.57	0.6	0.75	0.84	0.94	0.98
16	0.0046	0	0.01	0.01	0.02	0.02	0.03	0.03	0.04	0.04	0.05	0.07	0.09	0.11	0.13	0.19	0.21	0.29	0.37	0.5	0.6
17	0.0046	0	0.01	0.01	0.02	0.02	0.03	0.03	0.04	0.04	0.05	0.07	0.09	0.11	0.13	0.19	0.21	0.29	0.37	0.5	0.6
18	0.0046	0	0.01	0.01	0.02	0.02	0.03	0.03	0.04	0.04	0.05	0.07	0.09	0.11	0.13	0.19	0.21	0.29	0.37	0.5	0.6

Last edited by Eukie; 01-21-2016 at 08:18 PM.
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Old 01-20-2016, 10:56 PM   #3
trooper6
 
Join Date: Sep 2004
Location: Medford, MA
Default Re: Gathering Successes Thresholds (Call for Math Wizards)

That's amazing. Could you explain how you did the math so I could learn something about this?
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Old 01-21-2016, 01:50 AM   #4
Curmudgeon
 
Join Date: Sep 2011
Default Re: Gathering Successes Thresholds (Call for Math Wizards)

Quote:
Originally Posted by trooper6
So on average, given x skill level, how many turns will it take to achieve y success margins?

How would one construct a chart or table that could be made with the x axis being skill level, y axis success margins and the then data in the table the average number of turns it would take?

If I had that I could decide on difficulty levels.
A bit different from Eukie's tables since I started from slightly different assumptions:

[php]effective skill Successes/Roll Rolls/Success
3 -0.250
4 -0.120
5 -0.005
6 0.134 7.448
7 0.324 3.086
8 0.583 1.714
9 0.958 1.043
10 1.458 0.686
11 2.083 0.480
12 2.605 0.384
13 3.662 0.273
14 4.569 0.219
15 5.551 0.180
16 6.579 0.152
17 7.560 0.132
18 8.556 0.117[/php]

As to the math, this would apply to both Eukie's tables and mine, allowing for our different assumptions. The probability for threshold margins of success for any effective skill level is the sum of probabilities for each possible die roll and the probability for each die roll is (margin of success +1 [or +2, if a critical success] x probability of that die roll).

For example, for skill level 18, a roll of 6 has a threshold probability of 12 [margin of success]+2 [for being a critical success] = 14 x 10/216 (probability of rolling a 6 on 3d6) = 0.648 to 3 decimal places.
For skill level 18, a roll of 7 has a threshold probability of 11 [margin of success] +1 [for being an ordinary success] = 12 x 15/216 (probability of rolling a 7 on 3d6) = 0.833 to three decimal places.

The probability for skill level 18 as a whole is the sum: 0.079 [probability of a 3] + 0.222 [probability of a 4] + 0.417 [5]+ 0.648 [6] + 0.833 [7] + 1.069 [8] + 1.157 [9] + 1.125 [10] + 1 [11] + 0.810 [12] + 0.583 [13] + 0.347 [14] + 0.185 [15] + 0.083 [16] +0 [17] -0.005 [18] = 8.556 successes gained on any given roll.

The number of rolls/success is the reciprocal of the number of successes/roll or 1/(successes/roll).

Where Eukie and I differ is in calculating the effects of a critical failure. Eukie treated them as ordinary failures, giving a probability of zero. I treated them as giving a value of -1, effectively undoing one success. While this isn't strictly accurate as a critical failure removes all the successes, it's useful in determining which skill levels are likely to accumulate more critical failures than successes (whether critical or ordinary).As can be seen in my table, any time the effective skill is reduced to 5 or less, there is very little hope of getting even a single success.

My tables don't provide a lookup for the number of turns to garner y successes but the rolls/success for skill levels 6-9 and the successes/roll for skills 10+ should provide a useful guide for judging that.

Last edited by Curmudgeon; 01-21-2016 at 02:46 AM. Reason: hit submit by accident
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Old 01-21-2016, 04:19 AM   #5
Eukie
 
Join Date: Dec 2013
Default Re: Gathering Successes Thresholds (Call for Math Wizards)

Quote:
Originally Posted by Curmudgeon View Post
A bit different from Eukie's tables since I started from slightly different assumptions:

As to the math, this would apply to both Eukie's tables and mine, allowing for our different assumptions. The probability for threshold margins of success for any effective skill level is the sum of probabilities for each possible die roll and the probability for each die roll is (margin of success +1 [or +2, if a critical success] x probability of that die roll).

For example, for skill level 18, a roll of 6 has a threshold probability of 12 [margin of success]+2 [for being a critical success] = 14 x 10/216 (probability of rolling a 6 on 3d6) = 0.648 to 3 decimal places.
For skill level 18, a roll of 7 has a threshold probability of 11 [margin of success] +1 [for being an ordinary success] = 12 x 15/216 (probability of rolling a 7 on 3d6) = 0.833 to three decimal places.
I believe all rolls that succeed have a MOS of at least 1; on a roll of 10 against effective skill 10, you get MOS 1, on a roll of 9 against effective skill 10, you get MOS 2, etc. At least that's the assumption I worked under.
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Old 01-21-2016, 05:06 AM   #6
Curmudgeon
 
Join Date: Sep 2011
Default Re: Gathering Successes Thresholds (Call for Math Wizards)

Quote:
Originally Posted by Eukie
I believe all rolls that succeed have a MOS of at least 1; on a roll of 10 against effective skill 10, you get MOS 1, on a roll of 9 against effective skill 10, you get MOS 2, etc. At least that's the assumption I worked under.
Not according to Degree of Success or Failure [B347].

Quote:
Example: If you have effective skill 18 and roll a 12, you succeed; your margin of success is 6.
From that, (18-12=6), it seems obvious that rolling your skill exactly would yield (18-18=0), 0 margin of success.

I'd interpret that as succeeding by the skin of your teeth, so you're not better positioned than simple success would indicate while someone with a margin of success of 1 succeeded and has managed gain one or more minor advantages into the bargain, so that he's competitively better off.
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Old 01-21-2016, 11:01 AM   #7
Brandy
 
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Default Re: Gathering Successes Thresholds (Call for Math Wizards)

Quote:
Originally Posted by Curmudgeon View Post
I'd interpret that as succeeding by the skin of your teeth, so you're not better positioned than simple success would indicate while someone with a margin of success of 1 succeeded and has managed gain one or more minor advantages into the bargain, so that he's competitively better off.
The OP indicates that they are awarding 1+MOS to the total success pool for a made roll.
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Old 01-21-2016, 11:33 AM   #8
Maz
 
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Quote:
Originally Posted by trooper6 View Post
There is this thing that I do in my GURPS campaigns for long skill tasks while in combat time that makes everything very exciting. It is sort of inspired by other game systems but also inspired by energy gathering magic from Thaumatology.
I have used it for great effect in D&D 4e. I hadn't really thought much of using it in GURPS but i's a really good idea. Gives lot more interesting thins to do for none-combat characters. Good idea!
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Old 01-21-2016, 12:21 PM   #9
Curmudgeon
 
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Default Re: Gathering Successes Thresholds (Call for Math Wizards)

Quote:
Originally Posted by Bookman
The OP indicates that they are awarding 1+MOS to the total success pool for a made roll.
Actually, he's only awarding 1+MOS on an ordinary success, it's 2+MOS on a critical success; which is how it was calculated for gathering successes in my table in post #4. However, it doesn't change how margin of success itself is calculated which was what was being addressed in the post you quoted.
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Old 01-21-2016, 12:32 PM   #10
trooper6
 
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Default Re: Gathering Successes Thresholds (Call for Math Wizards)

I'm thinking about writing this up as a quick little article for Pyramid...but I do have one thing I go back and forth on...and that is the result of failure, and I'd love people's thoughts.

So far I have ruled that a failure results in no progress, crit fail wipes out all progress and/or no longer allows for any more attempts (you jam the lock, your terminal is disconnected from the mainframe, etc). This is somewhat generous to the players because there is no negative consequences for a failure beyond taking more time.

Here is my question: do you think there should be some sort of penalty for failure? -1 to accumulated MoS? Should it mirror the positive side and give -1 and the Margin of Failure on a failed roll? Do you think it is more fair to mirror the the positive and the negative completely? What you gain is what you can lose? This would allow for contests of skills to work in this system like a tug of war.

Thoughts?
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