I would like to solve this gambler problem using interpolation or any other technique that allows you to solve it with only pen and paper (without knowing the known gambler's problem formulas). Of course, I don't want to lose 30 minutes in solving the system of 8 linear equations, that would take too long.
Smith is in jail and had 3 dollars and can get out on bail if he has 8 dollars. A guard agrees to make a series of bets with him, with Smith’s probability of winning p=0.4 and probability of losing p=0.6. Find the probability that he can leave on bail before losing all his money if he bets 1 every time versus if he bets as much as possible each round to get to 8 dollars but nothing more than that. Which is the optimal strategy?
If the probability was 50% I would arrive to the known formula $\frac{i}{N}$ by interpolation:
knowing that $p8 = 1$ and $p0 = 0 $
$\frac{p8-p0}{p3-p0} = \frac{1-0}{x-0}$
$\frac{8-0}{3-0} = \frac{1-0}{x-0}$
$\frac{8}{3} = \frac{1}{x}$
$X = \frac{3}{8}$
In this case the probability is $p=0.4$ and $q=0.6$. The formula would be $\frac{1-(\frac{q}{p})^i}{1-(\frac{q}{p})^N}$ but how can I arrive to that by interpolation?