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Given a fair coin, we sure that the previous tosses do not affect the future tosses, however, we still expect to have roughly equal amounts of heads and tails after a large number of tosses. I am confused here. I had a strong sense that says if we have 5 heads and 10 tails at the end of 15 tosses, then we would bet on heads for the next toss since we expect more heads at the future. However, this obviously contradicts the fact that every toss is independent.

Ahmet Bilal
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  • It's hard to call a soft question a duplicate of another, but have you read the answers to this question? http://math.stackexchange.com/questions/1125087/is-the-law-of-large-numbers-empirically-proven/1125158 – Erick Wong Sep 24 '16 at 17:38
  • I have read it recently and I can answer his question of "what leads to fifty fifty distribution of a coin toss?". It is just due to the symmetry of heads and tails. There is no physical difference of having heads or tails except our naming to them. My question is why I feel a contradiction? Most probably I think something wrongly, and I want to find and fix it. – Ahmet Bilal Sep 24 '16 at 18:02
  • Yep. It was very long and I was busy. But the important thing is I learned the name of my fallacy so I can investigate further, thanks to AlgorithmsX. It is resolved. – Ahmet Bilal Sep 24 '16 at 21:06
  • The point of the other question is "how can the average naturally even out after an initial lead, when the coin is supposed to be unbiased?". It's really a very similar sentiment but coming from a different position. In one case you intuitively feel that the coin should correct itself, and the other poster felt that it had to correct itself in order to maintain the law of averages. – Erick Wong Sep 24 '16 at 21:27
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    Actually, I wrote a java program to check whether I can predict the next toss by simply counting the history and betting on the less occured one. It fails to do any better than just random betting, namely 50 percent. – Ahmet Bilal Sep 24 '16 at 21:37

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Part of it comes from a glitch in human psychology called The Law of Small Numbers, in which people think that a small sample size represents the whole. It is also know as a "Hasty Generalization". Gambler's Fallacy also partially stems from the fact that humans often see randomness not as it should be, but as uniformity with slight variation.

AlgorithmsX
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  • I would think that Law of Small Numbers would cause someone to see 5 heads and 10 tails and conclude that tails should continue to occur more frequently, no? But Gambler's Fallacy fits the bill quite well. – Erick Wong Sep 24 '16 at 21:30
  • If we didn’t know that coin flips have a $50%$ chance of landing on heads, yes. – AlgorithmsX Sep 24 '16 at 23:38