I would like to isolate p in the following. I am not sure if it is even possible.
a = B(1; 10, p)
B(x;n,p) is the cumulative binomial function.
I would like to isolate p in the following. I am not sure if it is even possible.
a = B(1; 10, p)
B(x;n,p) is the cumulative binomial function.
The aim is to solve $a=u(p)$ with $u(p)=(1-p)^{10}+10p(1-p)^9=(1+9p)(1-p)^9$. Since $u$ is a high degree polynomial, there can exist no formula inverting it in full generality and using only usual functions. However...
Since $u'(p)\lt0$ for every $p$ in $(0,1)$, the function $u$ is decreasing on $[0,1]$ from $u(0)=1$ to $u(1)=0$. Hence, for each $a$ in $(0,1)$, there exists a unique value $p_a$ in $(0,1)$ such that $u(p_a)=a$.