Craps and the Gambler's Fallacy

If you’ve spent much time playing craps, it’s certain that you’ve heard people talk about certain rolls or bets being due. Maybe a six hasn’t been rolled in a long time, so they begin playing every bet on the table that will pay off if the shooter rolls a six, or a player will bet on the field bet because they haven’t seen those numbers hit in a long time. In any case, many (if not most) gamblers believe that if a result hasn’t happened in a long time, that result should come up soon, since that result is “due.”

This belief is known as the Gambler’s fallacy. More commonly, you might have heard of it referred to as the law of averages. Not only is it entirely incorrect, it can also lead to bad decision making that will end up costing you money at the craps table.

First, let’s understand why the Gamber’s fallacy is incorrect. It only applies to events that are random and independent, such as rolling dice or flipping coins. In a game where cards are removed from a shoe and the odds change each time a card is pulled, it is not a fallacy to think that cards that haven’t been pulled yet are more likely to come out in the future!

But when the events are truly independent, the belief that something is “due” to happen really is a fallacy. Craps dice do not have a memory, and have no idea how many times in a row a certain number has been rolled. Each roll has no effect on the ones to come after it, and isn’t influenced at all by the rolls that came before it. The odds do not chance from roll to roll based on what has happened recently.

This brings us back to the law of averages, sometimes also called the law of large numbers. Many people know that in the long run, we know almost exactly how often different results will come up. For instance, if we’re flipping a fair coin, we know that 50% of the time it will land on heads, and 50% of the time it will land on tails. If this is true, you might ask, doesn’t a run of heads have to later be “balanced out” by a run of tails? Otherwise, this line or reasoning goes, the results will never get back to the expected 50-50 split.

The key thing to remember is that there’s never a guarantee of an exactly even split between heads and tails – only that the percentage of heads and tails should each approach 50% in the long run. This will happen even if there is never a particularly big run of tails to balance out that initial run of heads!

For instance, let’s imagine a coin is flipped 10 times, and all 10 times it comes up heads. Right now, our coin’s flipping percentage stands at 100% heads, and 0% tails. What happens if our coin starts flipping at the expected 50/50 split the rest of the way, without any special runs of tails?

After just 10 more flips (5 heads, 5 tails), heads has now been flipped 75% against 25% for tails. After a total of 100 flips, heads will hold just a 55/45 edge. And after 1,000 flips, the percentage of heads thrown will be down to 50.5%.

This is the true meaning of the law of averages – there are no special runs required, nor does the coin need to make up for previous flips. In a large enough sample, small runs like the ten consecutive heads we started with have an insignificant effect on the overall results.

Another reason why people believe in the Gambler’s fallacy comes from the way we think about how likely certain events are to happen. This can be understood easily by looking at a particular event in craps – rolling a seven. If you play craps frequently, you should know that there’s a one in six chance of rolling a 7 at any time. Knowing this, it’s fairly simple to determine the odds of rolling a 7 five times in a row:

6 * 6 * 6 * 6 * 6 = 7776

There’s only a one in 7776 chance of rolling five consecutive sevens! Knowing this, it’s easy to understand why seeing a run like this is extremely rare. However, it’s very easy to use this knowledge to come to some very incorrect conclusions. Here’s a question: if someone has thrown four consecutive sevens, what are the odds that they’re throw a fifth seven?
Many people will answer this question by saying one in 7776, while others will make up an answer based on the fact that they know this event is improbable. However, the answer is still one in six! The extremely long odds existed before we started throwing the sevens, but remember, the dice don’t have a memory or any reason to be biased against throwing a seven now. We’re only analyzing the odds of what will happen on the very next roll, and the previous rolls do not influence that whatsoever.

The Gambler’s fallacy can cause you to make bets with high house edges based on the false idea that certain outcomes are due, when in fact they’re just as likely to happen now as they were last roll and the roll before. Many craps betting systems are based around this fallacy, believing that the odds change based on what has already happened at the craps table. Understanding the Gambler’s fallacy will not only give you a better understanding of statistics and probability, it will help you avoid scams and bad strategies that are based around taking advantage of this common misconception.