I was solving the following equation involving probabilites.
$(1-q)^{(n-1)} -p^{2}q(n-1)(1-pq)^{n-2}=0$
p and q are independent probability values and n is a natural number.
My aim is to express q in terms of p and n.
I tried to analytically solve for $q$, but I did not succeed. (I tried to collect common terms ,taking logarithms etc)
I plotted the function in MATLAB and was able to verify that a solution exists. Also, at $q=0$ and $q=1$ the LHS has different signs indicating the existence of a solution.
Can anyone help me find an analytic expression for q in terms of p and n? Is this possible? If no, what are the possible approximations?
Note $: n>>1$ is a reasonable assumption for an approximate solution, if an analytic solution does not exist.
EDIT: If there are multiple solutions, the minimum value is the one I need.