The “Gambler’s Fallacy” Clarified

We’ve received several requests to clarify the “Gambler’s Fallacy” that we address in another article. The questions basically relate to, “What exactly do you mean when you say past outcomes don’t affect future outcomes, and how does that affect whether a certain number is due to hit?” Let’s see if we can clear this up.

As described in our other article, the “Gambler’s Fallacy” is the belief that a certain number has a better chance of appearing on future rolls because of the numbers that appeared on previous rolls. For example, you often see posts on other craps websites and gambling blogs bragging about their so-called craps winning system, such as, “If the shooter hasn’t rolled a 7 in five rolls, then remove your Place bets because the 7 is due to hit.” In this example, the person believes a 7 is more likely to hit on the sixth or a subsequent roll because it hasn’t hit on the previous five rolls. This thinking is pure nonsense.

Let’s look at two fundamental principles of probability and statistics:

(1) Scenarios where past outcomes do influence future outcomes, and

(2) Scenarios where past outcomes do not influence future outcomes.

A good way to understand these principles is to use an example of playing a simple game of craps with your friend. Let’s say you have a bag containing 50 red marbles and 50 blue marbles (i.e., 100 marbles total). Suppose your friend blindly pulls out one marble at a time and you bet each other whether the marble pulled out is red or blue. Assume all 100 marbles are identical except for the color (i.e., they’re exactly the same size, shape, texture, etc.) so your friend can’t feel around in the bag and distinguish which ones are red and blue. Also assume your friend doesn’t cheat by peeking inside the bag. With 50 blue marbles and 50 red marbles, the probability (i.e., odds, or chances, or whatever term you want to use) of pulling out a red or blue marble is exactly the same, which is 50%. That is, there’s a 50% chance of pulling out a red marble, and a 50% chance of pulling out a blue marble. 50 red marbles divided by a total of 100 marbles = 0.50, or 50%. 50 blue marbles divided by a total of 100 marbles = 0.50, or 50%.

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Scenario #1

In this scenario, let’s assume each time your friend pulls out a marble, he puts it on the table (i.e., the marble does not go back into the bag). Suppose on his first pull, your friend pulls out a red marble. He puts the red marble on the table and prepares to pull out another marble. Let’s pause the game momentarily to analyze the current situation. At this point, because the red marble that your friend pulled out did not go back into the bag, the bag now contains 50 blue marbles and only 49 red marbles (99 total). Because there are now more blue marbles in the bag than red marbles, the probability of pulling a certain color changes. Remember, there are only 99 marbles in the bag now. 50 blue marbles divided by 99 total marbles = 0.5050, or 50.05% (numbers are rounded). 49 red marbles divided by 99 total marbles = 0.4949, or 49.49% (numbers are rounded). As you can see, the odds of pulling a blue marble are now just a bit better than pulling a red marble. In this case, the previous outcome does, indeed, influence the odds of future outcomes. Let’s continue the game. Suppose your friend pulls another red marble out of the bag on his second pull, and places it on the table (i.e., the second marble does not go back into the bag). Let’s pause again to analyze the situation. There are still 50 blue marbles in the bag and only 48 red marbles (98 total). The probability of pulling out a blue marble on the next pull is 51.02% (50 blue marbles divided by 98 total marbles = 0.5102). The probability of pulling out a red marble on the next pull is 48.98% (48 red marbles divided by 98 total marbles = 0.4898). As you can see in this scenario, because the marbles that your friend pulls out do not go back into the bag, the outcomes of previous pulls do, indeed, affect the probabilities of future outcomes. In this scenario, you can see how the odds change from pull to pull.

Scenario #2

In this scenario, let’s assume each time your friend pulls out a marble, he puts it back into the bag before the next pull (there are always 100 marbles in the bag for every pull). Suppose on his first pull, your friend pulls out a red marble. He puts the red marble back into the bag. Let’s pause the game to analyze the current situation. At this point, because the red marble that your friend pulled goes back into the bag, the bag still contains 50 blue marbles and 50 red marbles. Because there are still 50 red and 50 blue marbles (100 total), the probability of pulling a certain color on the next pull does not change. 50 blue marbles divided by 100 total marbles = 0.50, or 50%. 50 red marbles divided by 100 total marbles = 0.50, or 50%. The odds on the next pull are still 50:50 regardless of the fact that your friend pulled a red marble on his first pull. Let’s continue the game. Suppose your friend pulls another red marble out of the bag on his second pull, and then puts it back into the bag. The odds of pulling a red or blue marble on his upcoming third pull are still 50:50. In this scenario, the odds of future outcomes never change because of previous outcomes. For example, even if your friends pulls out a red marble 10 times in a row, the bag always contains 50 red and 50 blue for a total of 100 marbles, so the odds of pulling either a red or blue marble on the 11th pull is still 50%. In this scenario, what happened in previous pulls has no effect on the odds for future pulls.

When playing craps, it’s like you’re playing in Scenario #2. The criteria for each future roll never change (i.e., there are always two six-sided dice numbered 1 through 6 which can result in 36 possible combinations). Remember, those criteria never change from roll to roll. For example, when the shooter rolls a 5, the number 5 isn’t erased from the dice or removed from the game. With two six-sided dice, there are always six ways out of 36 possible combinations to roll a 7, which means the odds of rolling a 7 are always 16.6% (i.e., 6 divided by 36 = 0.1666). So, even if the shooter hasn’t rolled a 7 in a hundred rolls, the odds of rolling a 7 on the very next roll is still 16.6% (i.e., 6 combinations out of 36 possible combinations = 6 / 36 = 0.166). Just because the number 7 hasn’t appeared in 100 rolls does not mean a 7 is “due” to hit. A 7 always has a 16.6% chance of appearing on any roll, regardless of how many times or how few times it has appeared in the past. No number is ever “due” in the game of craps. In craps, there’s no such thing as a number being “due.” The odds of any number appearing on the next roll are constant; they never change from roll to roll.

This is true for some other casino games, too, such as roulette. The roulette wheel always has 36 numbers and 0 and 00. After a spin, the dealer doesn’t erase or remove any numbers from the wheel or change their colors. There are always 18 red numbers, 18 black numbers, and the two green numbers 0 and 00. The odds of the ball landing on a red or black number are always the same after each spin. So, just because the tote board shows that a red number has hit the last eight times in a row does not mean you should run to the table and bet \$100 on black thinking “black is due.” Just as in craps, no roulette number is ever “due” to hit because of what happened on previous spins.

So, when you read on the Internet or hear some so-called craps professional talk about adjusting your betting pattern because a certain number is “due to hit,” ignore it because the guy has no clue what he’s talking about. He’s either trying to:

(1) Impress you with his craps knowledge (or complete lack thereof),

(2) Get you to bet more so you’ll lose more (such as a casino employee hoping you’ll lose more money so the casino profits more), or

(3) Con you into buying something you don’t want.

We hope this helps clarify what the “Gambler’s Fallacy” actually means. Just remember, in craps, the odds of a certain number appearing on any future roll is in no way affected by what numbers have appeared in the past. That’s what’s meant by, “Future outcomes are not affected by past outcomes.” Another way to say it is, “In craps, past outcomes never influence future outcomes.” Let’s say it one more time to ensure you understand (say it slowly and aloud so it sinks in), “In craps, no matter what numbers have appeared on previous rolls, the odds of rolling any number on the next roll never change.”