The Monte Carlo Fallacy

Posted On 30 Apr 2014
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The Monte Carlo Fallacy

The thing that struck me most about the casino in Monte Carlo wasn’t the array of Ferrari’s and Lamborghini’s (mostly in red) parked outside, but the beauty and quiet ambience within. Not so on August 18, 1913 when a stunning and now famous event took place at their roulette table. On that date black came up a record twenty-six times in a row and there was a near-panicky to bet on red, beginning around the time the ball landed on black fifteen consecutive times. Players were in a red-betting frenzy, doubling and tripling their stakes after black came up the twentieth time. They all believed there wasn’t a chance in a million of another black landing. And so it went for 26 spins with the casino winning millions of francs. Gamblers lost millions of francs, in fact, due to the mistaken belief that this unusual streak somehow caused an imbalance in the randomness of the wheel. Be assured that before tossing the dice or flipping a coin there is a fixed probability that is not biased by recent but past events.

Video Poker Fallacy: The Monte Carlo Fallacy, also called the gambler’s fallacy, has eked its way into all forms of gaming. For example, people playing slots will continue feeding a machine because they are convinced that it’s “due” for a win or that somebody else will come along and win “their” money. Whether the slot machine won or lost in the recent past has zero to do with predicting future spins. Video poker is just like tossing the dice or flipping a coin – in that; previous but recent results can’t be used to predict future outcomes. Before each coin toss the odds of landing on heads or tails are equal or 50/50 regardless of the previous flip. As Arthur Conan Doyle said (“The Complete Sherlock Holmes”), “It is easy to be wise after the event.”

Understanding the Data: To clarify how hand frequency data can be used to indicate the odds of specific winning hands I’ll refer to Table 1.1. Here you’ll see how often each hand occurs in 9/6 Double Double Bonus (9/6DDB), one of the most popular video poker games on the casino floor.


Table 1.1: 9/6DDB Hand Frequency

Hand: 9/6DDB Occurs Once In:
Royal 40,799
Straight Flush 9,123
4 Aces with 2, 3 or 4 16,236
Four 2’s, 3’s, 4’s w/ A-4 6,983
4 Aces 5,761
Four 2’s, 3’s, or 4’s 2,601
4 Fives-Kings 613
Full House 92
Flush 88
Straight 78
3 of a Kind 13
2 Pair 8
Jacks or Better 5


Note that I rounded occurrences to the nearest whole number.

Notice that 4 Aces with a kicker occurs once every 16,799 hands. Even those with poor math skills understand that it’s a rare event and that’s precisely why the casino can offer such a big return on this hand. Before the game starts the odds are the same, even if you just had the Aces with a kicker. So you can get back-to-back Royals or Aces with the kicker. It’s possible but improbable, therefore it makes a good story. On the other hand there are players at the wrong end of the bell-shaped curve who may go several royal cycles without a royal. (A royal cycle is the theoretical number of hands before a royal, in this game it’s 40,799.) The important thing to understand by studying this table is that before each game begins your odds are constant, assuming correct play, regardless of the end results in previous hands.

Using the Results: You should always play according to Zamzow (wrote the WinPoker software for 100% accurate play) regardless of previous results. Just like the Monte Carlo roulette players who lost their shirts in 1913, you will make costly errors in the long run anytime you play video poker using hunches or past results. Just because you have held three Aces several times instead of a dealt full house in 9/6DDB but failed to get the fourth Ace does not mean you should change from the correct playing strategy. In 9/6DDB, you should always hold just the three Aces going for the fourth Ace instead of the full house. (You will find free strategy cards in the back of my book “The Video Poker Edge” for the core games available in casinos. Each game requires a separate strategy based on winning hand payouts.)

Final Thoughts: If anybody in 1913 Monte Carlo roulette game were alive today they’d probably still be lamenting their gambler’s remorse for their strategy fallacy. Smart video poker players should always use mathematically correct strategy and never believe past hands on a device will influence future results.


Linda Boyd ,a long-time table game player before turning to video poker, also writes for “Southern Gaming” and “Midwest Gaming and Travel” her book, “The Video Poker Edge”, includes free removable pay schedules and free strategy cards for the most popular games. Look for her story in the latest editions of “American Casino Guide” (ACG) and her YouTube videos, also with ACG. The 2010 Second Edition of her book is available in both paperback and kindle. Available at 877-900-BOOK,, bookstores or



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