Frequently Asked Questions about the Gambler's Fallacy
Last update: November 3, 2005
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Gambler's Fallacy FAQ
Please understand this: There is no such thing as an event being "due." An event is not more likely just because it has not happened for a long time. For example, many people mistakenly believe that if one color in roulette has won several times in a row then the other color is overdue and they should bet on it. While the ratio of reds to blacks will always approach 50/50 in the long term, it can not be concluded that this will happen in the short term. It does not matter what the history of past spins is; every trial in games of luck like roulette are independent, and each color is equally likely to come up every time. If you don't believe me, try guessing the toss of 1,000 coin flips using any method of prediction you like. I guarantee that 99% of the time you will get between 459 and 541 correct.
The idea that events can be "due" is known as the Gambler's Fallacy. Below are common questions I get about it.
I disagree with you when you said that if a roulette ball has landed on red 20 times in a row it is equally likely to land in red as black on the next spin.
This kind of comment never comes with any kind of mathematical reasoning or evidence behind it. Little metallic balls do not have a memory and can not defy the laws of physics by jumping in one color or the other. If you don't believe me please read any introductory book on probability. I don't have time to tutor the entire world on this one person at a time.
The probability of red winning 20 times in a row in roulette is (18/38)20 = 1 in 3,091,874. Thus, if red has already hit the last 19 spins the odds are that the next spin will almost certainly be black because of the long shot against 20 reds.
First, the odds of 19 reds in a row and then a black are the same as 20 reds in a row. The past does not matter. The probability of red after 19 reds is 18/38.
What is the probability of winning an even money bet in roulette a n times in a row?
(18/38)n. For example, the probability of 20 reds in a row is (18/38)20 = 1 in 3,091,874.
Doesn't the above contradict what you said before? If the probability of 20 reds in a row is 1 in 3,091,874 how can the probability of red after 19 previous reds be only 18/38?
No! When dealing with games like roulette, past events do not affect future events. The probability of getting 20 reds in a row is always the same from the first spin only. After each red you get, the probability of finishing all 20 increases because there are fewer more reds you need. Before the first spin the probability is 1 in 3,091,874 but after you already get 19 reds the probability of finishing drops to 18/38 .
I still say you're wrong. As you said the probability of 20 reds in a row is 1 in 3,091,874. That is a fact that can not be changed. The laws of probability do not change just because the last 19 spins were red. Thus the 20th spin would have a 1 in 3,091,874 probability of being red and 3,091,873 in 3,091,874 of being black.
Oy. The laws of probability never change. If 19 reds already occurred, most of the work has already been done to get to 20. The probability of getting that far to begin with is (18/38)19 and the probability of finishing is 18/38.
I understand that for every 38 spins on the roulette wheel about 12 numbers will never hit and about 26 will hit twice or more. So the odds of a number hitting are about 26/38. Wouldn't it be profitable to track 37 numbers and on the 38th spin bet on a number that hasn't hit yet? Wouldn't the odds of it hitting be 26/38?
No! No! No! First about 14 will never hit, 15 will hit once, and 9 will hit twice or more, on average. For every spin that a number doesn't hit the odds of it never hitting within the 38 spins increases. By the 38th spin there will still be about 14 numbers that have never hit, surely the probability of each of them hitting can't be 26/38. Still the past has no relevance on the future in games of luck like roulette. On the 38th spin every number is equally likely to hit. Tracking numbers is a waste of time unless one is testing for a biased wheel.
I still say you're wrong.
I don't really care.
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