How bet size and other factors affect your chances of extending your bankroll

22 October 2012

By Alan Krigman

Casino aficionados who are in the hole but not broke often don’t quit to cut their losses but stay in action, hoping to climb back out. The question therefore arises of gambling in a way that minimizes risk of going bust. The answer involves choosing what to play and – when there’s a choice – propositions on which to bet and/or strategies to follow, as well as the size bets to make.

Notwithstanding premonitions commonly claimed post facto by winners of big jackpots, nobody can predict how they’ll fare in any particular instance. The probabilities that a bankroll will suffice for any desired session duration, however, can be calculated using risk of ruin analyses.

Many solid citizens visit casinos knowing what and how they’ll play. Someone may be a craps devotee who always bets Pass with triple Odds and place two numbers. Or a blackjack buff who follows strict Basic Strategy. Or a video poker fan who goes for deuces wild games with certain payoff sets. In such cases, the house’s percent edge over them, the representative bankroll jumps up or down per round – the volatility as characterized by what statisticians call standard deviation, and the average number of rounds per hour are pre-established.

The element remaining for folks like these to determine is bet size. Large bets are tempting because, when successful, they can yield big profits or rapid recovery from earlier setbacks. But, when they bomb, fanny packs can run quickly dry. In a practical sense, bet size isn’t absolute, but some fraction of an individual’s bankroll. Gurus sometimes value bankroll in terms of a lifetime allocation of gambling money. For the majority of recreational casino visitors, including “regulars,” session-by-session stakes serve as more realistic starting points.

Imagine you go for Player at minibaccarat and, even if you fall behind, want to stay at the table for at least four hours without depleting an intended gambling budget. This is about 400 rounds. The house’s edge will be 1.2 percent and the standard deviation will be roughly equal to your wager. Betting 2 percent of your initial bankroll (e.g., $5 for a $250 stake), the probability of still being in contention after four hours is 97.8 percent. Double the bet to 4 percent of your bankroll (e.g., $10 for a $250 stake) and your prospects for surviving this long drop to 72.0 percent. Betting 10 percent of your bankroll (e.g., $25 for a $250 stake) reduces your outlook to 31.1 percent. And if you let avarice get in the way of reason with bets of 20 percent of your bankroll (e.g., $50 for a $250 stake), the likelihood you’ll withstand the downswings is only 15.2 percent.

Say you have a friend who prefers the leisurely pace of roulette. Pretend this person normally bets on Red at a double-zero roulette table where both Black and Green lose. The house has 5.3 percent edge, over fourfold that on Player at baccarat. Red, like Player, pays even money so the volatility is roughly the same on both propositions – a standard deviation about equal to the bet. For 400 rounds, five or six hours in this game, chances fall to about 88.8 percent of survival with bets at 2 percent of the stake, 43.0 percent of survival at 4 percent of the stake, 11.8 percent of survival at 10 percent of the stake, and 4.8 percent of survival at 20 percent of the stake.

Your friend might get bored with even-money bets, and opt to stick with roulette but wager on four-spot corners. These are less apt to win but pay 8-to-1. They still give the house 5.3 percent edge, but have volatilities with standard deviations about 2.75 times the wager. Now for 400 rounds, the probabilities are 50.2 percent of survival with bets at 2 percent of the bankroll, 24.3 percent of survival at 4 percent of the bankroll, 9.1 percent of survival at 10 percent of the bankroll, and 4.4 percent of survival at 20 percent of the bankroll.

Decreasing edge, standard deviation, fraction of the bankroll bet per round, and number of rounds all enhance the probability of surviving sessions – and conversely. As a very rough rule of thumb (for the mathematically inclined, based on a multiple regression and having an 87 percent correlation with the results of risk of ruin analyses), changes in the probability of survival are:

• 2 percent rise (drop) for every 1.0 percent drop (rise) in edge,
• 5.5 percent rise (drop) for every one wager-equivalent drop (rise) in standard deviation,
• 3.0 percent rise (drop) for every 1 percent of starting bankroll drop (rise) in amount bet,
• 1.5 percent rise (drop) for every 100-round drop (rise) in number of rounds.

Uninitiated know-it-alls tend to believe that casino gambling is pretty much a matter of making your bets and taking your chances. Sophisticated players are aware that alternate games and strategies have unique characteristics, affording opportunities to pick and choose to suit their personal preferences. Accepting smaller profits for lower prospects of biting the dust is one of many such criteria. As the punters’ poet, Sumner A Ingmark, proclaimed:
Those in the know, not amateurs,
Consider the parameters.