11 ms·

Not quite. Pick a random real number in the range [0,1]. Because there are an infinite amount of such numbers, the probability of picking any specific real num

Not quite. Pick a random real number in the range [0,1]. Because there are an infinite amount of such numbers, the probability of picking any specific real number is precisely 0. However, you will still end up picking some number. EDIT: Fixed range

nitrogen 10y ago
Assuming you meant [0,1], I don't think it's accurate to describe infinitesimals like lim->0 as being equal to zero. Adding an infinite number of infinitesimals gives a nonzero result, but adding an infinite number of zeros does not.
mrob 10y ago
Picking a random real number in the range [0,1] is impossible. The expected length of its simplest description is infinite, and if you cant describe it you haven't actually picked it.
- soVeryTired 10y ago
  ...a uniformly distributed real number in the range [0, 1] :) Otherwise I can just pick 0 with probability 0.5 and 1 with probability 0.5. If you can't be pedantic when discussing maths, when can you be pedantic?
deleted 10y ago
[deleted]
Retric 10y ago
No it is (1 / infinity) which is not zero. Much like how X/X is not a continuous function even though the limits at 0 are 1.
- gizmo686 10y ago
  Standard probability theory does not have a notion of infinitesimals. There have been non-standard attempts at probability theory that do involve such a notion. See [0] (found on a brief search; I have only skimmed). EDIT: Admittedly, the existence of the non-standards analysis is a good indication that this is an artifact of the formalization of probability theory; not of the underlying reality that we are trying to model. Simpson's paradox is probably a better example of a "real" "paradox" of probability. [1] [0] https://lirias.kuleuven.be/bitstream/123456789/551380/1/BenciHorstenWenmackers-BJPS-InfinitesimalProbabilities.pdf https://lirias.kuleuven.be/bitstream/123456789/551380/1/Benc... [1] https://en.wikipedia.org/wiki/Simpson's_paradox https://en.wikipedia.org/wiki/Simpson's_paradox
  - abecedarius 10y ago
    https://www.amazon.com/Radically-Elementary-Probability-Theory-AM-117/dp/0691084742 https://www.amazon.com/Radically-Elementary-Probability-Theo... is apparently another take on this problem.
  - Retric 10y ago
    While I don't disagree with what your saying, I would point out if you want to use specific definitions they don't apply in the general case. Abstractly, there must be some difference between 0.2 and 2 as the first is a valid output and the second is not. Otherwise when you sum [0.2,0.3] vs [2,3] you get the same probability of 0. But, there is a trade off that can make things a little cleaner if you define the probability at 0.2 to be 0.

nitrogen 10y ago

Assuming you meant [0,1], I don't think it's accurate to describe infinitesimals like lim->0 as being equal to zero. Adding an infinite number of infinitesimals gives a nonzero result, but adding an infinite number of zeros does not.

mrob 10y ago

Picking a random real number in the range [0,1] is impossible. The expected length of its simplest description is infinite, and if you cant describe it you haven't actually picked it.

soVeryTired 10y ago
...a uniformly distributed real number in the range [0, 1] :) Otherwise I can just pick 0 with probability 0.5 and 1 with probability 0.5. If you can't be pedantic when discussing maths, when can you be pedantic?

deleted 10y ago

[deleted]

Retric 10y ago

No it is (1 / infinity) which is not zero. Much like how X/X is not a continuous function even though the limits at 0 are 1.

gizmo686 10y ago
Standard probability theory does not have a notion of infinitesimals. There have been non-standard attempts at probability theory that do involve such a notion. See [0] (found on a brief search; I have only skimmed). EDIT: Admittedly, the existence of the non-standards analysis is a good indication that this is an artifact of the formalization of probability theory; not of the underlying reality that we are trying to model. Simpson's paradox is probably a better example of a "real" "paradox" of probability. [1] [0] https://lirias.kuleuven.be/bitstream/123456789/551380/1/BenciHorstenWenmackers-BJPS-InfinitesimalProbabilities.pdf https://lirias.kuleuven.be/bitstream/123456789/551380/1/Benc... [1] https://en.wikipedia.org/wiki/Simpson's_paradox https://en.wikipedia.org/wiki/Simpson's_paradox
- abecedarius 10y ago
  https://www.amazon.com/Radically-Elementary-Probability-Theory-AM-117/dp/0691084742 https://www.amazon.com/Radically-Elementary-Probability-Theo... is apparently another take on this problem.
- Retric 10y ago
  While I don't disagree with what your saying, I would point out if you want to use specific definitions they don't apply in the general case. Abstractly, there must be some difference between 0.2 and 2 as the first is a valid output and the second is not. Otherwise when you sum [0.2,0.3] vs [2,3] you get the same probability of 0. But, there is a trade off that can make things a little cleaner if you define the probability at 0.2 to be 0.