problem 52

The problem is equivalent to betting $1$ increments and having $A$ start with $ki$ dollars, while $B$ starts with $k (N - i)$ dollars.

Thus, $p < \frac{1}{2}$ ,

p_{i} = \frac{1 - {(\frac{q}{p})}^{ki}}{1 - {(\frac{q}{p})}^{kN}} .

Note that,

lim k \to \infty \frac{1 - {(\frac{q}{p})}^{ki}}{1 - {(\frac{q}{p})}^{kN}} = lim k \to \infty \frac{- ki {(\frac{q}{p})}^{ki - 1}}{- kN {(\frac{q}{p})}^{kN - 1}} = \frac{i}{N} lim k \to \infty \frac{1}{{(\frac{q}{p})}^{k (N - i)}} = 0 .

This result makes sense, since $p < \frac{1}{2}$ implies that $A$ should lose a game with high degree of certainty over the long run.