In real life we never know for sure that coin tosses are independent and unbiased. If we flip a coin 50 times and get 50 heads, we are not actually surprised at the level of an event with 1 in 2 to the −50 probability (about 1 in 10 to the −15). We are instead surprised at the level of our subjective probability that the coin is grossly biased (for example, it might have a head on both sides), which is likely much greater than that.
But in any case, it is not rare for rare events to occur, for the simple reason that the total probability of a set of mutually-exclusive rare events need not be low. That is the case with 50 coin tosses that we do assume are unbiased and independent. Any given result is very rare, but of course the total probability for all possible results is one. There’s nothing puzzling about this.
Trying to avoid rare events by choosing a restrictive sigma algebra is not a viable approach. In the sigma algebra for 50 coin tosses, we would surely want to include events for “1st toss is a head”, “2nd toss is a head”, …, “50th toss is a head”, which are all not rare, and are the sort of event one might want to refer to in practice. But sigma algebras are closed under complement and intersection, so if these events are in the sigma algebra, then so are all the events like “1st toss is a head, 2nd toss is a tail, 3rd toss is a head, …, 50th toss is a tail”, which all have probability 1 in 20 to the −50.
Yes, this is the right view.
In real life we never know for sure that coin tosses are independent and unbiased. If we flip a coin 50 times and get 50 heads, we are not actually surprised at the level of an event with 1 in 2 to the −50 probability (about 1 in 10 to the −15). We are instead surprised at the level of our subjective probability that the coin is grossly biased (for example, it might have a head on both sides), which is likely much greater than that.
But in any case, it is not rare for rare events to occur, for the simple reason that the total probability of a set of mutually-exclusive rare events need not be low. That is the case with 50 coin tosses that we do assume are unbiased and independent. Any given result is very rare, but of course the total probability for all possible results is one. There’s nothing puzzling about this.
Trying to avoid rare events by choosing a restrictive sigma algebra is not a viable approach. In the sigma algebra for 50 coin tosses, we would surely want to include events for “1st toss is a head”, “2nd toss is a head”, …, “50th toss is a head”, which are all not rare, and are the sort of event one might want to refer to in practice. But sigma algebras are closed under complement and intersection, so if these events are in the sigma algebra, then so are all the events like “1st toss is a head, 2nd toss is a tail, 3rd toss is a head, …, 50th toss is a tail”, which all have probability 1 in 20 to the −50.