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If 100 heads come in a row, one might consider the coin to be biased. However, if, say, alternate Heads and Tails happen for 100 coin flips, then the coin is likely fair. However, both '100 heads in a row' and 'alternate HT 100 times' have the same probability

If you predetermine the order of cards after a shuffle, and the order comes out to be the same, people might think something is fishy. But again, the probability of that particular order coming out was exactly the same as any other order.

If one does get 20 heads in a row, then from one point of view, one might consider the coin to be biased. However, there's another point of view. Suppose there are billions of people in the world doing coin flips. The probability of 20 heads happening to at least one of these people was pretty damn likely. From this point of view, the person might consider that, with billions of people flipping, 20 heads in a row was almost certainly gonna happen to at least one person anyway, and that person just happened to be him. So, from this point of view, there's nothing fishy in 20 heads in a row.

Even without billions of people flipping, there are two point of views. Suppose a single person plans to make a zillion coin flips. If, at near the start of his session, he gets 20 heads in a row. He concludes that 20 heads in a row was gonna happen at least once in his 'zillion flips session' anyway, so it's no surprise that this happened. However, if a person who never planned to flip a zillion times suddenly gets 20 heads in a row, he might consider the coin to be unfair. Are both of these point of views correct?

This question arises in the probability of life as well. If the universal constants were even a bit different, then life wouldn't have been possible. The probability of life is really low. So one point of view is that God finely tuned these constants to support life (biased universe). The other point of view is that, say, with a zillion random attempts for the universe to form, at least one of was gonna have finely tuned constants anyway. We are in the universe in which things are finely tuned.

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  • $\begingroup$ This is more of a philosophical than mathematical question. I recommend the introduction of William Feller’s Introduction to Probability theory and its applications Volume I, as he addresses and compares the axiomatic, statistical, philosophical, and physical interpretations of probability and the application of probability theory. $\endgroup$ – Nap D. Lover Aug 15 at 19:37
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These are good questions. That said, they are really philosophical, not mathematical. Mathematicians manage to avoid problems with terms like "fair" by working with models (not the real world of coins) where it's a precisely stated assumption.

Statisticians have a harder job. Their models are supposed to help us deal with the real world.

If I saw a run if $100$ flips that perfectly alternated heads and tails I would suspect a very talented magician doing the flips.

I don't know how (or whether) working physicists think about the anthropic principle - the fine tuning that creates a universe that welcomes us.

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