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# How Bet Size Affects the Chance of Reaching your Win Goal

3 May 2006

In "fair" gambles, nobody has an edge and payoffs mirror the odds against winning. Play such a game repetitively, rather than as a single bet, and the same sort of relationship governs the shot at hitting a specified revenue target before depleting a bankroll.

For instance, beginning with \$100, the odds you must overcome to earn \$900 before busting out a 9-to-1 profit are 9-to-1. In terms of probabilities, the chance of starting at one point and reaching a higher level before going broke equals the stake divided by the final amount. So, the probability of going from \$100 to \$1,000 is 100/1,000 or 10 percent; from \$50 to \$200 it's 50/200 or 25 percent; from \$250 to \$500 it's 250/500 or 50 percent. And so on. This, regardless of how much you bet or the volatility the characteristic ups and downs of the game.

These estimates provide upper bounds on likelihood of success. In a real casino, the house will have an edge which lowers the probabilities somewhat.

When edge enters the picture, however, bet size and volatility also become factors. You can't reliably predict the impact of one independently of the others. The links among these elements are complex. But knowing the trends, if not the details, may help you evaluate your options. Of the three parameters, once you've chosen a game, bet size is the most readily controlled.

The effect is strongest in low-volatility games those in which bets pay close to even money. Blackjack is the most common example. Basic Strategy at a casino with moderately liberal rules yields an edge of roughly half a percent. The standard deviation, a measure of the volatility envision it as the average number of bet units you win or lose on each hand is 1.13.

Make believe you start with \$100 and are willing to risk it all to double your money, to finish with \$200. Absent the edge, the chance of success would be 100/200 or 50 percent. Betting \$5 per round, your chance is 46 percent; at \$10 per round, it's 48 percent; and at \$25 per round, it's up to 49 percent.

A more ambitious goal might be to go from \$100 to \$500. In a fair game, your prospects would be 100/500 or 20 percent. Here, \$5 per hand gives you just 14 percent chance; at \$10 it's improved to 17 percent; and at \$25 it's increased to 19 percent.

What if, instead, you start with \$200 and will do or die to quit with \$1,000? The ideal probability you'll triumph is 200/1000 or 20 percent. The real-world figures are only 10 percent betting \$5, 14 percent at \$10, and 18 percent at \$25.

Note the trends: 1) the higher you bet, the closer you get to the ideal probability of success; 2) as earnings aspirations rise, the penalty for small bets also increases. With \$5 bets, you drop from 50 to 46 percent hoping to win \$100. With a \$400 win goal, you fall to 14 percent. The \$800 goal brings you down to 10 percent. Similarly for the higher wagers.

Increasing the edge for the same volatility makes the effect of bet size more pronounced. Give the house 1 percent edge at blackjack with the objective of turning \$200 into \$1,000. The ideal chance of your dream coming true is still 20 percent. In contrast to having a 10 percent probability of success with \$5 bets as with half as much edge, though, your chance is 4 percent.

Does this suggest it's wise to bet as high as you can? It might, if maximizing the probability of reaching some returns plateau were your sole criterion. And perhaps it is. But solid citizens usually weigh desired profit versus playing time. And, the higher your bet relative to your stake, the shorter your session is apt to be. Either because you bite the dust or reach your target quickly. But, in the latter case, would you really quit when you amassed that \$500 or \$1,000 you had in mind or tell yourself you're on a roll and should ride it out? Precisely the point thus portrayed by the perceptive punter's poet, Sumner A Ingmark:

Gamblers who think their luck runs in streaks,
Tend to be those who quit past their peaks.

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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.