on the previous question
A and B both pick a random integer separately. What is the probability that A and B end up with the same integer?
Since the number of integers is infinite, the probability that A and B end up with the same integer is infinitely small. lim 1/x, x--> infinity = 0, so the probability is 0.
But it is possible that A and B end up with the same integer. As such, should we conclude that "probability = 0" and "cannot taking place" are not mutually exclusive events?
...
A few thoughts (some inputs are courtesy of my philosophy tutor):
1. Yes, 'probability = 0' and 'event cannot take place' are not mutually exclusive events. 'Probability = 0' merely implies that the event will almost surely not take place/the probability of the event is infinitely small. If we repeat the scenario a finite number of times (assuming that we can) , we would not expect the event to take place at all. Nevertheless, it is possible that the event takes place.
2. It is not possible to pick a random integer. Since the size of the set of integers is infinite, we are practically excluded from picking integers that are too big to consider. Even if we were to spend our whole lives reciting the digits of the number or defining a number (e.g. 1234^12345 + 1), we could still conceive of (infinitely many) numbers greater than that number.
3. The probability is different from the limit. The probability is smaller than any positive number that we can give but since there cannot exist a number x such that 1/x=0, the probability is not 0. it is unfortunate that mathematics does not have a symbol to mean 'an infinitely small positive number'. As such, the best that we can say about this probability is that using the idea of mathematical limits, it is 0.
Due to (2), any selection of integers by A or B would not be random because it does not consider several arbitrarily large numbers. Here, we can ask if it is possible to construct a probability distribution that assigns an equal probability to each integer. If such a probability distribution exists, then it would have to assign the probability 0 to each integer else the sum of probabilities will be infinite. Therefore, it appears that we cannot construct such a probability distribution, at least given our current understanding of numbers and probability.
(3) might suggest that we need some richer conception of number that can capture the idea of an infinitely small but non-zero number. In standard analysis, there cannot be such a number. However, my tutor points out that in non-standard analysis, such a number might be permissible and I would be better placed to study the possibility of alternative axiomatic systems in 2 years time.
We could then rephrase the question more accurately as: if the probability of an event, as given by the concept of a mathematical limit, is 0, is that event possible? My answer would then be yes. Otherwise, we will get paradoxes in the above and other related scenarios (such as the probability of a point on a plane being selected).
I hope this helps to clarify certain issues. Comments are welcome.
Since the number of integers is infinite, the probability that A and B end up with the same integer is infinitely small. lim 1/x, x--> infinity = 0, so the probability is 0.
But it is possible that A and B end up with the same integer. As such, should we conclude that "probability = 0" and "cannot taking place" are not mutually exclusive events?
...
A few thoughts (some inputs are courtesy of my philosophy tutor):
1. Yes, 'probability = 0' and 'event cannot take place' are not mutually exclusive events. 'Probability = 0' merely implies that the event will almost surely not take place/the probability of the event is infinitely small. If we repeat the scenario a finite number of times (assuming that we can) , we would not expect the event to take place at all. Nevertheless, it is possible that the event takes place.
2. It is not possible to pick a random integer. Since the size of the set of integers is infinite, we are practically excluded from picking integers that are too big to consider. Even if we were to spend our whole lives reciting the digits of the number or defining a number (e.g. 1234^12345 + 1), we could still conceive of (infinitely many) numbers greater than that number.
3. The probability is different from the limit. The probability is smaller than any positive number that we can give but since there cannot exist a number x such that 1/x=0, the probability is not 0. it is unfortunate that mathematics does not have a symbol to mean 'an infinitely small positive number'. As such, the best that we can say about this probability is that using the idea of mathematical limits, it is 0.
Due to (2), any selection of integers by A or B would not be random because it does not consider several arbitrarily large numbers. Here, we can ask if it is possible to construct a probability distribution that assigns an equal probability to each integer. If such a probability distribution exists, then it would have to assign the probability 0 to each integer else the sum of probabilities will be infinite. Therefore, it appears that we cannot construct such a probability distribution, at least given our current understanding of numbers and probability.
(3) might suggest that we need some richer conception of number that can capture the idea of an infinitely small but non-zero number. In standard analysis, there cannot be such a number. However, my tutor points out that in non-standard analysis, such a number might be permissible and I would be better placed to study the possibility of alternative axiomatic systems in 2 years time.
We could then rephrase the question more accurately as: if the probability of an event, as given by the concept of a mathematical limit, is 0, is that event possible? My answer would then be yes. Otherwise, we will get paradoxes in the above and other related scenarios (such as the probability of a point on a plane being selected).
I hope this helps to clarify certain issues. Comments are welcome.