House Profits in Negative Edge Games

This is a simulation of gambling profits with plots of the profits or losses of individual gamblers along with a density plot of all the gambling together. The gambling consists of successive independent plays to win one unit ($1 for instance) with probability and lose one unit with probability , where ranges from 0.5 down to 0.3, reflecting the negative edge of casino games. The plot also includes a heavy line for the expectation stake of a player after n plays, simply , where is the player's initial bankroll. The gambler is busted and play ends when his stake is exhausted.

Charting shows the perspective of the house where player losses are colored green (as house profits). Also the simulation shows the efficacy of the house odds with in the range of 0.4 to just under 0.5. In this range the gamblers mostly lose money (so the house profits consistently) while most gamblers stay in the game for dozens of plays, with not too many bust. Likewise many common casino games have win probabilities in this range.