♣ K-Dub ♣

An urn contains p balls labeled +1 and p balls labeled -1. The balls will be drawn out uniformly at random, without replacement, until the urn is empty. Before each ball is drawn, the player decides whether or not to place a bet on the next drawn ball.The payoff is the value of the ball drawn if the player bets. Otherwise, the ball is drawn, but nothing else happens. An optimal strategy is to place a bet if there are more +1 balls in the urn than -1 balls left in the urn. If there are an equal number of +1 and -1 balls, either option (pass/bet) is optimal, and if there are more -1 balls than +1 balls, the player will not bet. Here, assume the player will opt to pass on any "neutral" urn, e.g. the player won't bet on the first ball.

The player has a "bank" B to work with. If at any time during the game the player's bank is emptied, the player is ruined. It is my suspicion that if B is a fixed quantity, the probablility of ruin goes to 1 as p goes to infinity. How can I prove this? TIA