The “Gambler’s Fallacy” Clarified

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Gamblers fallacy

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.” click here to learn to play craps.

Here we provided some example of  gambler’s fallacy, you can head over to this page here to read about the term gambler’s fallacy.

You can now head over to the table of contents to find more great content.

Author
Written by John Nelsen in partnership with the team of craps pros at Crapspit.org
  • jagxkr

    Loved your analogy, right out of my statistics books! Now do you believe that the dice can be thrown in such a way as to create enough of systematic perturbation so that the average value of the 2 dice thrown, say over 300 rolls does not follow the Central Limit Theorem? And hence we can avoid having that dreaded number I shall not mention be the mean value of the distribution?

    • crapspit

      Hi, jagxkr. Wow, that question was a mouthful.
      For the benefit of our other readers, let’s keep it simple. For lessons on the Central Limit Theorem, mean, standard deviation, and other statistics terms, we’ll leave that to our readers to research. By “systemic perturbation,” I assume you mean shake up (or more accurately, influence) the results such that they don’t follow the normal distribution. By definition of the Central Limit Theorem, a large number of random independent rolls will have a normal distribution. In a word, my answer to your question is, “No.” If a large number of independent craps rolls are random without unnatural influence, the results will tend to follow the normal distribution (i.e., Central Limit Theorem). If you follow the casino’s rules for throwing dice, the rolls will, indeed, be random and independent. I think what you’re asking is what the dice-control cons try to sell people. In other words, I think you’re asking if there’s a way to influence the dice so their outcomes are not purely random such that we avoid the number 7 appearing as often as it should according to the normal distribution. As explained in our lesson on Dice Control, we here at the Crapspit strongly believe that “dice control” is a con designed entirely to sell you something (i.e., transfer your money from your pocket to their pocket), such as seminars, paid memberships, lessons, books, DVDs, and merchandise (e.g., practice tables, software to track your rolls, etc.). As explained in our lesson on Dice Control, there’s no way to throw a legal pair of dice such that your throwing technique will influence the outcome.
      So, let’s paraphrase your question and then reiterate our answer. You asked (paraphrased), “Do you believe that the dice can be thrown in such a way as to influence their outcome to land more or less on certain numbers. For example, do you believe you can influence the outcomes such that the number 7 won’t appear as often as it should according to the normal distribution?” For the reasons explained in our lesson on Dice Control, the answer is simply, “No.” However, there are legitimate ways (i.e., dice control is not legitimate) to influence the outcome of dice rolls. For example, you could sneak illegal dice into the game, as described in one of our other lessons. But by sneaking illegal dice into the game (e.g., using loaded dice), the results are not purely random. So, in summary, your question is based on whether you use legal or illegal dice. If you use legal dice, then the rolls will be purely random (and independent), so the outcomes over a large sample size (i.e., a large number of rolls) will resemble the normal distribution. Accept it, there’s no way you can gain an advantage other the casino when playing craps with legal dice (i.e., we believe you can’t create any “systemic perturbation” to affect the outcome of a pair of legal dice such that you decrease the appearance of the number 7). If you experience someone at the table (regardless of their throwing technique) who hits 15 points in a row, the reason they hit 15 points in a row is because of distribution variance (assuming legal dice are used), not because of some bogus dice-control skill.
      Good luck and have fun at the table!

      • Jagxkr

        To be profitable just need to roll numbers, don’t need to make passes. Some of my most profitable excursions at the doc table were when I never actually rolled a point but did roll many numbers before severing out. Place bets pay same as pass line and buying 4 and 10 pay better than pass line net take for same number

        • JeffDunstan

          dude your logic makes no sense at all. read what he said to you and youll see that you cant control the dice enough to effect the house edge. why you said all that nonsense afterward is anyones guess.

  • SevenOut

    Have you lost your marbles?

    Just poking a bit of fun at your example, but… why not have a bag of 100 dice? You reach into the bag and being blindfolded, place the 50 pairs in a row. Take the blindfold off, take pencil and paper and note the dice total outcomes for each pair. You need not shoot the dice to get accurate results, which also contradicts the Dice Influencing / Dice Control promoters outcome expectations.

    This would then contradict Scenario #1 using dice immediately.

    Scenario #2 example as an exercise of 50%/50% marble outcomes are no comparison to possible random dice outcomes. Putting 100 coins in a bag with the addition of heads or tails, removing two at a time and lining them up as pairs would create more variance… and a slightly better example of randomness over the marble example. Repeat if necessary, but after over 1,000,000 trials, the results should be getting closer to 50% / 50%.

    Short trials prove that a Craps Player who plays with the idea of short term variance of outcomes IS better prepared to leave the Craps Table as a winner. More time at the table results in time result in outcomes that were expected through mathematics.

    Removing 50 pairs of dice a second time to compare with the first 50 pair results would prove the Gambler’s Fallacy for what it is… bad theory, poor mathematics, unreasonable statistics and pure nonsense.

    Successful Craps Players expose their bankrolls over short periods of time and wager for beneficial random dice outcomes that defy the mathematical predictions that depend on infinite time and outcome statistics. Otherwise there never could be a winning Craps game, a winning Craps System nor any chance to beat a regressive Casino game if short term dice outcomes mirrored long term expectations.

    • crapspit

      SevenOut, you completely miss the point of the article, and you apparently completely misunderstand the concept of the Gambler’s Fallacy since you stated that the concept is “bad theory, poor mathematics, unreasonable statistics, and pure nonsense.”

      By making that statement, it’s easy to conclude that you don’t know what you’re talking about. Since this will be a long post, I decided to simply create a new page due to the length of the reply and link it from here: http://www.crapspit.org/gamblers-fallacy-bad-theory-statistics/