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Is there a cap on the chance you'll make big bucks with a small bankroll?

27 February 2012

Many casino aficionados try their luck with the idea of investing stakes it wouldn’t unduly pain them to sacrifice, seeking rewards they covet. Establishing targets at both ends of the scale offers these players reasonable guidelines for their action. Guidelines, as opposed to strict criteria, because the approach is most felicitous when individuals are disciplined enough to avoid exceeding loss limits but flexible enough to accept less than their win goals.

Quantifying barriers and aspirations can go further, however. As a set, the values establish a ceiling on the chance a player will be successful. To see how this works, picture a million patrons entering the casino with $100 each. In total, they bring in $100 million. If, ideally, neither the bosses nor the bettors have an inherent advantage on the action, the million solid citizens will take roughly the same $100 million away. Of course, some folks will be ahead and others behind.

Pretend that the million players all keep going until they lose or win $100, quitting with either nothing or $200. Because there’s no edge, the net will be zero. For this to happen, half – 500,000 – must finish in each category. Given these conditions, every person’s chance of success at the outset will have been 50 percent. Instead, make believe the players all continue until they either earn $900 or lose their $100. Now, 10 percent of the million – 100,000 – will leave with $900 profit and 90 percent – 900,000 – will go home broke. Each individual will therefore have had prospects of 10 and 90 percent for ecstasy and agony, respectively.

To calculate the ideal – no-edge – chance of success with any pair of end points, divide the bottom-of-the-barrel bankroll by the total a player would have if the venture bears fruit. For the lose-or-win $100 example, attainment is a $100/$200 or 50 percent shot. For those who’ll invest $100 attempting to earn $900, the likelihood of joy is $100/$1,000 or 10 percent.

The real world is complicated by house advantage. The chance of reaching any profit level before exhausting a stake is less than the ideal. This, because the casino effectively siphons money off of every transaction, making the goal incrementally tougher to reach. Anyone who considers loss limits and win targets important, if not key, should know how to tame the influence of the edge.

One way is to minimize house advantage consistent with games you enjoy playing. Depending on the circumstances, this may involve your selection of tables or machines, propositions on which to bet within games, and strategies to follow when you can make choices during rounds.

Another method is to gamble without undergoing so many rounds that edge erodes your fortune significantly. Several approaches can be used for this purpose.

You may, for instance focus on high payoff propositions. These may be longshots or wagers having multiple possible outcomes including at least one with substantial returns. Nail a biggie and you’ll have reached or exceeded your profit objective in relatively few passes, before edge has operated on many bets. The downsides are that the chance of scoring a longshot is low, and most returns on payoff schedules are at the low end so you’re stuck making numerous bets.

An alternative is to bet on propositions having high probabilities of hitting but small payoffs, then parlay or press your wins. The result is that a run of favorable coups positions you with more moolah on the line than you could justify or afford drawing directly out of your stash. This is sometimes referred to as “playing with the house’s money.”

Of course, after a win, the payoff is yours and not the casino’s, so “the house’s money” is folderol. Still, it lets you make series of bets such as $5, $10, $20, etc when losing intermediate gains only depletes your reserve by the initial wager. The catch is that although the odds to be overcome at each stage are short, they escalate as the series lengthens and the amount grows. The chance you’ll win even money on Pass at craps is 49.29 percent. To parlay a $5 outlay into a $2,555 profit requires nine successive passes. The ninth has the same 49.29 percent probability as any other but the chance of nine in a row is 0.4929 multiplied by itself nine times – which equals 0.1717 percent, or one out of 582. Compare this figure with the ideal no-edge chance of winning $2,555 before losing $5, which is $5/$2,560 or 0.1953 percent, or one out of 512. Since you’ve only gone nine times, the edge hasn’t diminished your prospects very much. Not that they were so good at the getgo. In contrast, say you bet $5 per round. It’s possible to amass $2,555 before losing your original $5, but the chance is 0.000001390 percent – one out of 71,956,656.

So, you can trim the impact of the edge on your chance of glory, but still hit a wall with the odds tied to gains and losses. As the renowned rhymer, Sumner A Ingmark, was moved to muse:
While owing to edge, your distress is compounded,
By bankroll and win goals your chances are bounded.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.