That's not how math works. Of those 8 checks, the result will only be changed twice.
I mean, heck, if the fighter isn't even running out of Second Winds, then it's not adjusting that many results is it?
That is how it works. What you are missing is on most checks there is no difference between a 0 and a +4. Without second wind if a 0 and a +4 PC make 5 rolls, the most likely result is that they are different on only one of those five rolls.
On 80% of checks these two would be identical without considering Tactical Mind.
Rolling a D20 generates a uniform distribution between 1 and 20. On a DC 15 check, the only time someone with a 0 differs from someone with a +4 is when the number rolled on the D20 is between 11 and 14. On all other numbers the results are the same.
Which means that the MAXIMUM this could impact is 5 rolls.
We are talking about a bias being applied which affects the chance of succeeding on a check, that bias exists on every roll as long as a use of 2nd Wind is available. 5 checks attempted in one day is where the results most favor the PC with
2 uses of 2nd Wind (they are actually closer if they only attempt 1 or 2 checks than it is if they attempt 5).
If we compare a PC with a +4 and a PC with a +0 and
2 uses of Tactical Mind rolling against a DC15 check hereare the averages:
1 check in a day: +4 averages 0.5 successes, +0 with Tactical Mind average 0.57 successes and uses on average 0.28 uses of Tactical Mind
2 checks in a day: +4 averages 1 success, +0 with Tactical Mind average 1.15 successes and uses 0.55 uses of Tactical Mind
3 checks in a day: +4 averages 1.5 successes, +0 with Tactical Mind average 1.70 successes and uses 0.8 uses of Tactical Mind
4 checks in a day: +4 averages 2 successes,+0 with Tactical Mind average 2.23 successes and uses 1.02 uses of Tactical Mind
5 checks in a day: +4 averages 2.5 successes, +0 with Tactical Mind average 2.72 successes and uses 1.22 uses of Tactical Mind
6 checks in a day: +4 averages 3 successes, +0 with Tactical Mind average 3.18 successes and uses 1.38 uses of Tactical Mind
7 checks in a day: +4 averages 3.5 successes, +0 with Tactical Mind average 3.61 successes and uses 1.51 uses of Tactical Mind
8 checks in a day: +4 averages 4 successes, +0 with Tactical Mind average 4.02 successes and uses 1.61 uses of Tactical Mind
9 checks in a day: +4 averages 4.5 successes, +0 with Tactical Mind average 4.4 successes and uses 1.70 uses of Tactical Mind
You need to make 9 checks on average in a day before the person with the +4 is better at passing DC15 checks than a person with a +0 and
2 uses of Tactical Mind available.
4-5 checks is actually where the difference most favors the person with
2 uses of Tactical Mind.
And, as you say, since you're not going to run out of Second WInds, that means you're only influencing 2-3 results per long rest.
Right and changing 2 results out of 8 means most of the time you will have outperformed someone with a much better ability.
To put it another way that might make it easier - It only changes the outcome 2-3 times per Long Rest, but it only needs to change the outcome 1-2 times per Long rest for the Fighter to be better in play than the PC with the higher ability.
IMO, the problem with Tactical Mind is that it's random. If it was simply, "Ok, burn a Second Wind and turn a failed check into a success",
All skill checks are Random unless your bonus is enough to ensure success and you do not burn a use of Second Wind unless you turn a failed check into success. The use does not get used up if it is still a failure. That is why it remains a bias to the mean for so long.
For example I roll a 7 on a DC15 check, with my 0 it is a 7, with +4 it is an 11. Both of these are failures. I try Tactical Mind and I roll a 6 it is also still a failure but I did not burn any 2nd Wind uses then we both roll an 8 and fail again I can try TM again and then we both roll a 16 and it doesn't matter and then we both roll a 12 you succeed and I can try to succeed ....
No matter how you slice this, it's only going to change 2 maybe 3 results per long rest.
It only needs to change it twice for the lower score Fighter to be better in play.
1. How often? How often is Tactical Mind making a difference? Because something like Guidance makes a difference all the time.
Ok this is where I think you are getting lost. You are applying different standards for evaluating these two and are not grasping the statistics involved. In one case you are talking about discreet uses and in another you are talking about a bias to the mean roll. The bias does not always affect a specific discreet check, in fact it rarely does but the bias alters the mean and the expected result as long as it is in place. To put it another way, having Guidance does not always make a difference and having a high ability score does not always make a difference, in fact they rarely do.
For example, Guidance results in a 1-4 uniform distribution summed with a uniform 1-20 distribution. Adding these together results in a trapezoidal distribution with a mean of 13. Guidance affects 12.5% of checks, not "all the time". 87.5% of the time if you use Guidance and then try a check the result will be the same as if you did not use Guidance. On 87.5% of rolls guidance is meaningless.
Similarly having +4 applies a constant bias to the mean of the uniform 1-20 Distribution resulting in a mean of 14.5. The +4 only affects the outcome of an individual check 20% of the time. 80% of the time the result is the same as if you had a 0 and your +4 does not matter at all.
These things apply a bias to the mean 100% of the time (or 100% of the times it is used for guidance), and they change the expected result, but they actually affect very few discreet rolls.
Tactical Mind is the same - as long as it is not used up (i.e. you still have Second Wind uses available and intend to try it on a fail) it applies a 1-10 uniform distribution to some results on the normal 1-20 distribution, resulting in a non-standard distribution with a mean that varies based on the DC, but it also affects few individual checks (generally 27.5%) while it is in play. Because a use only expires when it affects a check, this means the bias remains in effect for many checks before it loses potency.
None of these things (Guidance, skill score, Tactical Mind) have an effect on the outcome every time you make a check. All of them affect individual checks infrequently, but all of them have a statistical power which affects the mean and expected result as long as they are in place.
SO, how can it matter 8 times if you only have 5 Second Winds per day?
Because it affects the mean result until it is used up. With a 0 vs a +4 it will make the 0 statistically more powerful through 8 checks with 2 uses.
I want to emphasize this - 8 checks is for
2 uses a day, not 5.
With 5 uses it is going to result in a better chance of success for
23 checks as compared to someone with a +4.
Central to understanding this is realizing that a +4 and a +0 would get the same result on 80% of checks, it is only a narrow window (11-14 on a DC15) that there is a difference to the to PCs to start with, without even considering 2nd wind. Any time the D20 is not an 11-14 the +4 can't possibly do better than the +0.
So if we don't consider Tactical Mind, and we roll 20 DC15 checks the most likely result is that we get different results only 4 times out of 20 (+4 passes 4 times when +0 fails), 6 times they both succeed, 10 times they both fail. Tactical Mind can turn some these where there are differences into successes AND in addition it can turn some of those where BOTH PC failed into successes up to 5 times total.
So if both PCs roll 20 DC15 checks and the +0 has 5 uses of tactical mind available the most likely outcome is:
PC with +4
Mean outcome with 20 attempted checks - Passes 10 checks and fails 10 checks
PC with +0
- passes 6 checks outright
- fails 4 checks outright
- has 10 checks where TM might be used, statistically on average he ends up using tactical mind a mean 4.44 times out of this 10 for 4.44 more successes (10.44 successes total).
Mean outcome with 20 attempted checks - Passes 10.44 checks and fails 9.56 checks
Keep in mind we are comparing a PC with an 18 to a Fighter with a 10! Are you starting to understand why this ability is so powerful?