Editing Openai/6924d898-a68c-8009-beac-944d19ca0daf (section)

=== Assistant: Right, totally fair to call out that “linear” was garbage for this kind of thing. ===
Right, totally fair to call out that “linear” was garbage for this kind of thing. Let’s do it your way: treat the flip as following a normal CDF and actually solve for where P=0.5P = 0.5P=0.5.

We’ll keep it simple and explicit.

==== Let’s model Luffy’s win probability as a normal CDF over the fight index iii: ====

PL(i)=Φ ⁣(i−μσ)P_L(i) = \Phi\!\left(\frac{i - \mu}{\sigma}\right)PL(i)=Φ(σi−μ)
where:
* Φ\PhiΦ is the standard normal CDF
* μ\muμ = the “flip mean” = fight number where PL=0.5P_L = 0.5PL=0.5
* σ\sigmaσ = spread of the transition

Then Naruto’s win chance is:

PN(i)=1−PL(i)P_N(i) = 1 - P_L(i)PN(i)=1−PL(i)
We know:
* Fight 1: Luffy’s chance is effectively 0 (Naruto hard-wins). A normal curve can’t hit 0, so we treat it as very small, say PL(1)≈0.001P_L(1) \approx 0.001PL(1)≈0.001.
* Fight 100: Luffy’s chance is our agreed 0.8.

So:

PL(1)=Φ ⁣(1−μσ)≈0.001P_L(1) = \Phi\!\left(\frac{1-\mu}{\sigma}\right) \approx 0.001PL(1)=Φ(σ1−μ)≈0.001
PL(100)=Φ ⁣(100−μσ)=0.8P_L(100) = \Phi\!\left(\frac{100-\mu}{\sigma}\right) = 0.8PL(100)=Φ(σ100−μ)=0.8
The corresponding z-scores:
* Φ−1(0.001)≈−3.09\Phi^{-1}(0.001) \approx -3.09Φ−1(0.001)≈−3.09
* Φ−1(0.8)≈0.84\Phi^{-1}(0.8) \approx 0.84Φ−1(0.8)≈0.84

So we have:

1−μσ=−3.09\frac{1-\mu}{\sigma} = -3.09σ1−μ=−3.09
100−μσ=0.84\frac{100-\mu}{\sigma} = 0.84σ100−μ=0.84
Solve those two equations for μ\muμ and σ\sigmaσ.

==== Subtract the first equation from the second: ====

100−μσ−1−μσ=0.84−(−3.09)\frac{100-\mu}{\sigma} - \frac{1-\mu}{\sigma} = 0.84 - (-3.09)σ100−μ−σ1−μ=0.84−(−3.09)
Left side:

99σ\frac{99}{\sigma}σ99
Right side:

0.84+3.09=3.930.84 + 3.09 = 3.930.84+3.09=3.93
So:

99σ=3.93⇒σ≈993.93≈25.2\frac{99}{\sigma} = 3.93 \Rightarrow \sigma \approx \frac{99}{3.93} \approx 25.2σ99=3.93⇒σ≈3.9399≈25.2
Now plug back into one equation, say the second:

100−μ25.2=0.84\frac{100-\mu}{25.2} = 0.8425.2100−μ=0.84
100−μ=0.84×25.2≈21.2100 - \mu = 0.84 \times 25.2 \approx 21.2100−μ=0.84×25.2≈21.2
μ≈100−21.2≈78.8\mu \approx 100 - 21.2 \approx 78.8μ≈100−21.2≈78.8
So the mean of the flip—where the normal CDF gives Luffy a 50% chance—is around:

: 

That’s your P=0.5P = 0.5P=0.5 point.

==== Just to sanity check: ====

z50=50−μσ≈50−78.825.2≈−1.14z_{50} = \frac{50 - \mu}{\sigma} \approx \frac{50 - 78.8}{25.2} \approx -1.14z50=σ50−μ≈25.250−78.8≈−1.14
PL(50)=Φ(z50)≈0.126P_L(50) = \Phi(z_{50}) \approx 0.126PL(50)=Φ(z50)≈0.126
PN(50)≈0.874P_N(50) \approx 0.874PN(50)≈0.874
So under a normal-distribution flip with:
* Fight 1 ≈ “essentially 0” for Luffy
* Fight 100 = 0.8 for Luffy

you get:
* Fight 50: Naruto still wins ~87% of the time
* Fight ~79: the matchup hits 50/50

Which lines up with your intuition:
the curve stays flat (Naruto-dominated) for a long time and only really drops off late.

So:

>