What's the average skill gain from successes in Call of Cthulhu?
As you use skills in Call of Cthulhu 7e, any successes get marked on your character sheet, and can be used to make skill advancement checks when requested by the Keeper. Any extreme successes are also marked.
To succeed a skill increase check you must roll higher than your skill number. If it was an Extreme success you roll the increase check with a penalty die.
If that result is above the skill number, you add 1d10 to that skill.
e.g. if you roll against a Pilot(boat) of 40, and you get a 5, that's an extreme success. If asked by the keeper, you can roll 2d00 and keep the highest (the penalty die) and see if it's over your skill (in this case 40). Say you pass by rolling a 30 and a 50, you then add 1d10 to Pilot(Boat)
That got me wondering, are you more likely to gain skill increases with skill values that are already high (as you can pass with an extreme success and get the penalty die on your advancement check), or one that's low, as it's easier to roll over it even if that's less frequent due to the initial check being lower?
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Ignoring the extreme success case for now, and supposing the skill gives a probability p of success, then the probabilities of succeeding on the skill roll and then failing the learning roll (that is, getting the increase) is p(1-p). This is a parabel with top at 1/2 and zeros at 0 and 1. It is symmetric as all parabels are. (The function is a second degree polynomial with a negative highest order term.)
However, if you get to test the skill several times, say n times, then you have n chances to succeed, while you still only get a single test to confirm that you fail. I can calculate the probabilities if desired, but the main point is that lower skills are indeed favoured - any of the rolls succeeding gives the advancement chance, while there is only a single try at "failing" that one.
The extreme success case happens rarely and essentially gives two attemps at the test to fail, so it will not affect the big picture, but it will somewhat improve the chances of success, especially where the skill level is high. A reasonable relative change, but still a small absolute one.
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The peak is about 55
It looks like, based on this Anydice calculation that peak skill increase is about 55.
I haven't done an analytical calculation of the probabilities, but my guess is that the initial pass (under) and then the skill advancement check (over) make the distribution symmetric.
That is a skill level of 5 (where the probability of a success then a subsequent skill advancement is 0.05×0.95=0.0475) is equivalent to when the skill level is 95 (i.e. 0.95×0.05=0.0475), but for a skill level of 50 (i.e. 0.5×0.5=0.25).
But the probability contribution from both the penalty die, and needing to initially roll under ⅕ the make higher rolls very slightly 'better' but not to completely break the underlying symmetry and stop the much higher rolls being worse.
Here's a visual representation of that distribution, but you need to look at the numbers to see subtle shift to larger values:
Although, it looks like the balance shifts downwards for repeated rolls at the same skill level: https://anydice.com/program/32abe/summary/table
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