well, I understand your point, but I still think it does not make sense to talk about probability without having a well-defined probability space, otherwise you will have all kinds of paradoxes.
In your example of integers, 1/2 is like a limit... A similar question can be asked: what is the probability of picking a random integer that is positive? If you restrict the sample space to [-n,n] and take limit then you will get 1/2. However you can as well look at the space [-n,10n]. When n goes to infinity it is also the set of all integers. Probability is about counting, and we know that the number of even integers has the same cardinality as the set of all integers, so we can also argue that the probability is 1...
Back to the original problem. It is true that you can scale the triangle to make it locate on the unit circle. However it is hard to argue that such operation preserves the probability. Suppose you take a circle of radius R and let R goes to infinity, you will get one probability (actually the probability does not depend on R). However if you sample the three points from a square with edge length k and let k goes to infinity, you will get a probability. In both cases, they produce the whole plane in the limit... You can also take other weird shapes and you will get all kinds of different probabilities.
I think back in the history this kind of paradox made the probability theory doubtful. When probability is based too much on intuition, it can always go wrong. For a period of time mathematicians questioned the validity of probability theory, but things were later saved by measure theory...
In your example of integers, 1/2 is like a limit... A similar question can be asked: what is the probability of picking a random integer that is positive? If you restrict the sample space to [-n,n] and take limit then you will get 1/2. However you can as well look at the space [-n,10n]. When n goes to infinity it is also the set of all integers. Probability is about counting, and we know that the number of even integers has the same cardinality as the set of all integers, so we can also argue that the probability is 1...
Back to the original problem. It is true that you can scale the triangle to make it locate on the unit circle. However it is hard to argue that such operation preserves the probability. Suppose you take a circle of radius R and let R goes to infinity, you will get one probability (actually the probability does not depend on R). However if you sample the three points from a square with edge length k and let k goes to infinity, you will get a probability. In both cases, they produce the whole plane in the limit... You can also take other weird shapes and you will get all kinds of different probabilities.
I think back in the history this kind of paradox made the probability theory doubtful. When probability is based too much on intuition, it can always go wrong. For a period of time mathematicians questioned the validity of probability theory, but things were later saved by measure theory...
选择“Disable on www.wenxuecity.com”
选择“don't run on pages on this domain”
