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Best of Alan Krigman

Gaming Guru

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Does Equal Edge Mean Equal Chance of Profit in Different Games?

8 September 2006

Make believe the casinos offer two similar games. One, call it 10-10, pays $10 on a $10 bet with a 47.5 percent chance of winning any coup versus a complementary 52.5 percent risk of losing it. The other, call it 9-10, pays $9 for a $10 bet with 50 percent prospects each of winning and losing on any round. The games give the house the same edge, 5 percent. And play proceeds at about the same rate, 100 decisions per hour.

These games are equivalent to the casinos. Their bean counters figure that, assuming enough action, earnings will be equal. For instance, after a million bets averaging $10 each, they'll keep close to 5 percent of $10 million or $500,000 either way.

But no single player will make a million $10 bets. Were it even humanly possible, the overwhelming likelihood of a $500,000 loss would militate against such fecklessness. Individuals are more apt to play short sessions, for reasons of time and endurance and also because volatility then affords them a good shot at grabbing a profit before edge takes its toll. And volatility is where the alternate ways the house gains its edge in these games differ. The "standard deviation," picture it as average bankroll change per round, is essentially $10 in 10-10 and $9.50 in 9-10.

A major concern to each person is the chance a session of reasonable duration will be profitable. It's accordingly sensible to ask: if 1,000 players each go 50, 100, or 200 rounds of 10-10 and 9-10, how many ought to finish even or ahead?

Computer simulation yields some answers. These numbers of players and rounds are statistically small. Variability can therefore be expected in multiple simulation runs under the random conditions prevailing in one session or another. Just as in real games. Still, the simulations show some illuminating trends.

With 50 rounds at $10 each, each bettor's gross wager is $500. The theoretical average loss is 5 percent of $500 or $25. Five simulated sessions showed losses from $23 to $26, no bias toward either game. However, numbers of solid citizens finishing even or ahead diverged substantially. In 10-10, the even-money game with a greater chance of losing than winning and higher volatility, from 396 to 430 bettors ended where they started or at a profit. In 9-10, the 50-50 game with $9 payoffs on $10 bets and lower volatility, 325 to 345 players finished 50 rounds even or ahead.

With 100 rounds at $10 each, gross wagers were $1,000 so average loss due to edge should be $50. The simulations bore this out with figures from $47 to $56 and no tilt either way. Fewer bettors wrapped up even or in the money than in the 50-round games, owing to more action on which the edge had its impact. However, those trying 10-10 again outdid their 9-10 counterparts. Figures ranged from 309 to 355 in 10-10 and 291 to 327 in 9-10.

The same patterns appear for 200-round sessions. The theoretical average loss due to edge was $100, and the simulation showed a span from $93 to $104 with no fundamental difference between the two games. Numbers of players concluding even or ahead were below those in the shorter sessions but continued to favor the higher- to the lower-volatility game. Here, the range was 245 to 287 out of 1,000 for 10-10 and 201 to 232 out of 1,000 for 9-10.

None of this suggests a secret the casino bosses don't want anyone to know. Players could and did win in both games despite the stiff 5 percent edge. More finished even or ahead in 10-10, with higher volatility but lesser probability of winning, than in 9-10 with the converse characteristics. But the averages were equal. This implies more folks but lower profit levels in the game with sessions that were easier to win. Illustrating two important precepts. First, you can earn money at a casino whether or not you know what you're doing. And second, you can make choices with trade-offs that tailor your gambling to your personal preferences. Both of which were contemplated by the Coleridge of the casinos, Sumner A Ingmark, when he composed:

While lucky is who lucky does,
The best advice that ever was,
Is work your ciphers well because,
All gambling follows math's strict laws.

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.