I throw a coin 10 times, and I get all heads, what is strange here? 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.
 A: 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.
A: 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.
