$365(x)^{364}-365(x)^{365}+x^{365}=0.9$

Question

I can't seem to solve this, I tried using multiple software but it says it doesn't support this kind of equation: $$365(x)^{364}-365(x)^{365}+x^{365}=0.9$$

Context: Hey! I was having fun with tricked coin flip probability and I came up with that equation at a certain point. If you replace $365$ for $n$ (the exponent $364$ would be $n-1$), and consider it as the number of coin flips, and $p$ as the probability that the favored side of the coin comes up, then $0.9$ (or whatever number, say $y$) is the probability that $A$ happens strictly more than $B$, that the favored side happens strictly more than the unfavored. So: Given that the favored side is greater than the unfavored $90\%$ of the time after $365$ throws, what is the probability of the favored side happening every throw?

The equation I came up with would be formally written: $np^{n-1}-np^{n}+p^{n}=y$

Answer 1

As

$$ (365-364x)x^{364}=0.9 $$

defining

$$ f(x) = \ln(365-364x)+364\ln x-\ln 0.9 $$

and calling $\phi(x) = x -\frac{f(x)}{f'(x)}$ the iterative process

$$ x_{k+1} = \phi(x_k) $$

converges quickly to the solution, giving an initial guess $x_0 \in(0,1)$