<p>The 2nd equation optimizerdad posted is probably the right setup:
1 - (1-p)^3 = 1.2 (p^3)</p>
<p>The 1’s cancel so you can factor out a p and solve a simple quadratic equation.</p>
<p>(actually the reason you can tell 1 -(1-p)^3 = 0.2 + p^3 is wrong is because it has imaginary roots–for example if you changed 0.2 to 0.8. But if you use (1+fail%)p^3 instead, your quadratic equation will always have a positive real solution.)</p>