In a fair game starting at k with target N: probability of reaching N first is k/N, and the expected duration is k(N - k).
That duration is much larger than intuition suggests - starting at 50 targeting 100 takes 2,500 rounds on average, because the walk drifts nowhere.
With a bias, the probability becomes (1 - r^k)/(1 - r^N) where r = (1-p)/p - exponential in the number of units rather than linear, so small edges compound into near-certainty.
The casino result. Against an infinitely rich opponent, ruin is certain even at fair odds. Bankroll matters more than edge.