How to calculate $p_c$ (binomial probability)
You can use the binomial approximation:
$$
(1 + x)^\alpha \approx 1 + \alpha x,
$$
which should be a very good approximation. The conditions for this being accurate are that $|\alpha x| \ll 1$.

When you substitute this expression into your total probability, you now have a polynomial you can solve for the roots of.