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I take a coin and with the assumption that it is a fair coin. I throw it 10 times and I get a sequence of 10 consecutive heads. I feel something is unusual and strange, may be the coin is not fair. But the outcome I got is as likely as any other sequences of heads and tails. However there is something really strange here, the probability of getting no tails in 10 throws of a fair coin is really small. So indeed there is something strange here. And common sense says the coin is probably not fair.

But isn't "containing no tails" just an arbitrary property of my outcome? May be my outcome is not unusual considering lots of other properties? Or may be you can come by an unusual property for any sequence of heads and tails?

My question is why seeing no tails in 10 throws is a good reason to doubt fairness of the coin?

P.S.: To abide with the laws of stackexchange my formal question is what I stated above. But my real question is something more general and vague: when we see something strange on what grounds we can say what we saw is just a low probability result of the way I think the world works or I should change my view about how the world works? Are there certain things that I should check before changing my view? I would be grateful if you can help me with my real question too.

Saeed Zargar
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    If you have a set of like $20$ outcomes you consider unique beforehand, and one of those appears then that is unusual. But if I give you a unique string beforehand and it turns up then it will also be unusual, like for example $HTHHTTTHTHT$ – Asinomás Apr 20 '21 at 13:46
  • If what you described is a one time event, to me it would not be enough reason to doubt the fairness of the coin. However, if this happens (maybe less: tails comes up more often than heads) again and again after thwoing the coin a huge amount of times then it would be a good enough reason to to doubt the fairness of the coin. There is some subjectivity involved. – Daniel Apr 20 '21 at 13:47
  • $2^{10}=1024$. This means that when a thousand mathematicians read your question, and each does the experiment on his own (with a standard coin), you may expect that one of them gets $10$ heads. – Christian Blatter Apr 20 '21 at 13:55
  • In terms of your assumption of a fair coin, a possible response is to quote Oliver Cromwell – Henry Apr 20 '21 at 13:57
  • @JohnDouma: Thanks but I don't think it answers my question, the first answer to that question says it is not surprising, but it is. If I play a game with somebody and get 100 heads in a row of course it is surprising, if the other guy says you are losing just because you are having a bad day I wouldn't believe them. I guess there can be some subjectivity or background knowledge here but I want to pin it down, where is it exactly? – Saeed Zargar Apr 20 '21 at 13:58
  • Is this question closed? If so, it is kind of sad, it has got 5 upvotes and two answers and several comments. I admit that my question is not exactly mathy (prove this,find x) and it is somehow philosophical but considering the replies in this question & the question associated, a good number of people familiar with the subject had found it relevant enough, may be if the question got more time I could have more answers to think about, probably more people could find better explanations about the question. Anyway it is the way it is, may be I should ask it in another forum. – Saeed Zargar Apr 20 '21 at 14:35
  • I greatly appreciate all the comments and answers and will think about them. – Saeed Zargar Apr 20 '21 at 14:35

2 Answers2

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This is an excellent question that does not have a good mathematical answer. It's essentially philosophical. Making sense of probabilities in the real world (as opposed to the mathematical world) is hard.

Several observations about your particular question.

If $1000$ people flip a coin $10$ times then (on average) one will see all heads and one all tails. (That's a good approximation because $2^{10} \approx 10^3$.) Those two people will think their coins are strange, but no one looking at the whole ensemble would be surprised.

If I flipped a coin and saw $10$ heads in a row I might begin to be a little suspicious. Then I would continue the experiment. Each new flip that turned up heads would increase my subjective estimate of the probability that the coin was unfair. There are mathematical models for updating that probability - but they are just models. The subjectivity is in deciding what mathematics best matches the actual observations.

Ethan Bolker
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Theoretically, the probability of getting no tails (or all heads) in $10$ independent tosses of a fair coin is $(\frac{1}{2})^{10}\approx 0.0009$ which is very low (not zero though). The assumption demands the independent nature of all tosses which I suspect is acheived in your case; unless the tossing fingers, tossing speeds, humidity, dampness in air, etc. are different in all cases.

Nitin Uniyal
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