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Gambler’s Fallacy

(also known as: the Monte Carlo fallacy, the doctrine of the maturity of chances)

Description: Reasoning that, in a situation that is pure random chance, the outcome can be affected by previous outcomes.

Example #1:

I have flipped heads five times in a row.  As a result, the next flip will probably be tails.

Explanation: The odds for each and every flip are calculated independently from other flips.  The chance for each flip is 50/50, no matter how many times heads came up before.

Example #2:

Eric: For my lottery numbers, I chose 6, 14, 22, 35, 38, 40.  What did you choose?

Steve: I chose 1, 2, 3, 4, 5, 6.

Eric: You idiot!  Those numbers will never come up!

Explanation: “Common sense” is contrary to logic and probability, when people think that any possible lottery number is more probable than any other.  This is because we see meaning in patterns -- but probability doesn’t.  Because of what is called the clustering illusion, we give the numbers 1, 2, 3, 4, 5, and 6 special meaning when arranged in that order, random chance is just as likely to produce a 1 as the first number as it is a 6.  Now the second number produced is only affected by the first selection in that the first number is no longer a possible choice, but still, the number 2 has the same odds of being selected as 14, and so on.

Example #3:

Maury: Please put all my chips on red 21.

Dealer: Are you sure you want to do that?  Red 21 just came up in the last spin.

Maury:  I didn’t know that!  Thank you!  Put it on black 15 instead.  I can’t believe I almost made that mistake!

Explanation: The dealer (or whatever you call the person spinning the roulette wheel) really should know better -- the fact that red 21 just came up is completely irrelevant to the chances that it will come up again for the next spin.  If it did, to us, that would seem “weird,” but it is simply the inevitable result of probability.

Exception: If you think something is random, but it really isn’t -- like a loaded die, then previous outcomes can be used as an indicator of future outcomes.

Tip: Gamble for fun, not for the money, and don’t wager more than you wouldn’t mind losing.  Remember, at least as far as casinos go, the odds are against you.


Tversky, A., & Kahneman, D. (1974). Judgment under Uncertainty: Heuristics and Biases. Science, 185(4157), 1124–1131.

Registered User Comments

former student
Monday, July 10, 2017 - 04:49:00 AM
I do love it when mathematicians “prove” something which is obviously untrue to the real world. Gambler’s Fallacy is a case in point. A mathematician will tell you that all tosses of a true coin will be random and therefore independent. So according to their calculations you can have 100 heads and no tails. In the real world this would be amazingly unlikely. So what is happening? Why does 1000,000 tosses result in almost 50/50. Is there some ‘fairness’ fairy doing this? The logical answer is no. The world and the universe do not care about the result or the past results. If something is 50/50 then it will mostly turn out to be 50/50. A small sample just reflects the big picture but can have some anomalies that are out of sync.

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Tuesday, July 02, 2019 - 03:42:13 PM
Yes the probability is that any particular set of, say, 10,000 rolls will be fairly close to 50/50, However if you do a series of 10,000 sets of 10,000 rolls each you will find some sets that come in quite lopsided in one direction and some that come in quite lopsided in the other direction. Even as one side grows disproportionately the coin still doesn't care about history; it's just as likely to add another tail to a count of 800T/200H as it is to add a head.

The total of all of those rolls is still pretty likely to come in near 50/50 though. Most sets will be centered close to that, with fewer as you move farther from that.

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Thursday, December 13, 2018 - 06:25:41 PM
The 'Gambler's Fallacy' itself, seems to be the real fallacy. As a legit coin toss is 50/50, flipping either heads or tails infinitely every time is not only unlikely but mathematically, physically and with regard to the occult, impossible. Unless you're also able to walk on water. So, if you manage to flip let's say, 10 heads in a row, the odds do in fact favor tails over the next several flips, likely until the results balance out again. Disagree, grab a coin and try for yourself.

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Bo Bennett, PhD
Thursday, December 13, 2018 - 08:43:39 PM
Given a fair coin, the odds are 50/50 (ignoring any insignificant balancing issue that favors one side) every time. The coin does not "remember" the previous flips. Every time is like starting from scratch. It only appears to "balance out" as you say because as the number of flips increases, the difference in percentage decreases. Fallacies are fallacies because they go against "common sense." This fallacy is a perfect example of that, and the reason casinos make billions of dollars and people who don't understand this fallacy contribute to those billions.

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Monday, February 13, 2017 - 04:08:23 PM
I feel like people often confuse things like this with processes that take into account all parts of the event.

In another fallacy, you had the event where a coin was flipped 99 times and landed on heads, and two people were asked the probability of it being heads or tails on the 100th flip- one man clearly gave into this fallacy, while the other gave the correct 50/50 answer. However, I think someone could read too much into it if they were given this scenario instead: starting from the first flip, what's the probability that I can get this coin to land on heads 100 times in a row?

Obviously, the answer is extremely low (something like .5^100 if I'm correct because you continue to reiterate the 50/50 chance to the 100th flip), but do people often confuse the entire event with just isolated parts? I think this is similar to the composition over division fallacy (or whatever you call it in your works).

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