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Why you can win when the casino has an edge, or lose when it doesn't

23 April 2012

Most casino players ultimately blame any losses they suffer on the edge the house has on the bets it offers. Edge certainly is a consideration. And solid citizens would be wise to select games, wagers, and strategies to minimize this factor consistent with their other priorities and preferences. However, edge is significant mainly when applied to the astronomical numbers of decisions experienced by multiple players over extended time periods. Its primary impact on most individual bettors isn’t to wipe them out, but to moderate the ecstasy or exacerbate the agony attributable to other effects during specific sessions or visits.

Were casinos to offer “fair” bets, with no edge, the law of averages predicts that patrons as a whole would finish about even – wins balancing losses. But within this framework, particular players will move ahead and quit in the money or fall behind and run out of the time, inclination, or resources to try to recover. The chances of individuals being up or down by various amounts in the short run are determined by the odds of winning one or a series of bets, edge being absent.

As an example, suppose you elect to make even-money wagers. Edge would be zero were the probability of winning any coup precisely 50 percent. It’s a modest 1.42 percent for a flat bet on Pass or Come at craps, where the chance of success is 49.29 percent. And edge would be a whopping 10 percent if the prospects of victory are only 45 percent.

Assume you bet on three different even-money propositions – having edge of zero, 1.42, and 10 percent. You go 10 times for $10 each. The accompanying table gives the likelihood you’ll finish at the various possible levels of profit and loss. The data show that edge, indeed, exerts a negative influence on your fortunes. The greater the edge, the more your chance of losing and the less of winning any specified amount. But the data show that edge isn’t the principal determinant.

Chances of various gains or losses with 10 even-money bets having edges of zero, 1.42% and 10%

No of   Probability,   Probability,   Probability,   Net gain
wins         no edge    1.42% edge        10% edge    or loss

0            0.0977%       0.1124%       0.2533%     $100 loss
1            0.9766%       1.0930%       2.0724%      $80 loss
2            4.3945%       4.7806%       7.6303%      $60 loss
3           11.7188%      12.3912%      16.6478%      $40 loss
4           20.5078%      21.0774%      23.8367%      $20 loss
5           24.6094%      24.5846%      23.4033%    break even
6           20.5078%      19.9135%      15.9568%      $20 gain
7           11.7188%      11.0605%       7.4603%      $40 gain
8            4.3945%       4.0315%       2.2890%      $60 gain
9            0.9766%       0.8708%       0.4162%      $80 gain
10           0.0977%       0.0846%       0.0341%     $100 gain

Pretend that instead of playing for even-money, you’re extremely conservative and choose a wager with five ways to win for every one to lose. A bet that the roll of a single die would land on anything except a six would meet this condition. The presumption of no edge would mean that a winning $5 bet would pay $1. Given something like 600,000 rolls by many players, the law of averages would anticipate 100,000 sixes and 500,000 other results. Bettors would accordingly have dropped close to $5 x 100,000 = $500,000 and picked up approximately $1 x 500,000 = $500,000 – together coming out roughly even despite having had far more wins than losses.

Of course, you’re more apt to bet on six than 600,000 rolls, yourself. The law of averages then says you ought to win $1 five times and lose $5 once, breaking even. The most likely result, in fact, is to break even; the probability of doing so is 40.2 percent. To emerge triumphant earning $6, you have to win all six throws – a 33.5 percent probability. That leaves 26.3 percent chance of losing, which is less than that of winning but could leave you licking as much as a $30 wound.

At the long end of the odds spectrum, make believe you toss $1 on a proposition with 1 percent – one out of 100 – chance of joy. At a $99 payout, edge – and therefore the long-term average – would be zero. That is, after 100 million tries, results should approach break-even with 1 million barnburners at $99 for a total of $99,000,000 up and 99 million duds at $1 for a total of $99,000,000 down. In 100 rounds, the highest probability – nearly 37.0 percent – would be for you to break even, as well, with one win and 99 losses. However, the chance of a $100 loss – no wins in 100 tries – isn’t too much less, at 36.6 percent. The likelihood of any profit at all is 26.4 percent, less than that of losing, but the possible $100 setback is offset by potential profits from $100 to $9,900. What if you placed this bet only 10 times? Your chances would be 90.4 percent of losing all 10 and being down $10, 9.1 percent of winning once and earning a net of $90, and 0.5 percent of getting more hits and winning greater amounts. All this, with no edge.

Does this imply you can ignore the edge? No. But it suggests the need to strongly weight odds in making your choices as well. As the beloved bard, Sumner A Ingmark, advised:

In choosing bets, maintain perspective,
Know when and how to be selective.

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.