3 ms·

In an infinite string of random integers, 4 appears consecutively an arbitrary number of times. There's no way to prove, of which I am aware, that a string of

by na85 9y ago

In an infinite string of random integers, 4 appears consecutively an arbitrary number of times. There's no way to prove, of which I am aware, that a string of 24 fours is not random.

PeterisP 9y ago
Randomness describes a process, not its results. You can't prove that a string of 24 fours is not random, and the fact that some process returned 24 fours once is not proof (but is some evidence) that it's not random - however, a process that always returns 4 is not random. A process that returns a single fixed number (which was chosen by a fair dice roll) once is a random process. Using that same process twice or more is not.
jameshart 9y ago
But given a pseudorandom number generator with a known amount of internal state (eg a 32 bit seed) there are hard upper limits on the lengths of sequences like that that can plausibly be produced. Of course, to get 4 out of a PRNG you probably have to ask it for a random number in a range - if you’re requesting numbers in the range 0-1000 then you would expect fewer long sequences of fours than if you request in the range 0-5. And you can get arbitrarily many 4s by requesting numbers in the range 4-4...
thanatropism 9y ago
Tangential but related: Brownian motion (i.e. continuous-time random walk) is recurrent in one and two dimensions -- it hits zero infinite times, but in three dimensions it is not. The way my professor explained: a drunkard can always find his way back to his building, but not to his apartment.
- tzs 9y ago
  > The way my professor explained: a drunkard can always find his way back to his building, but not to his apartment. That might be true for a drunk Kitty Pryde, but is it true for a drunk ordinary human who is constrained to only move between floors via the stairs or elevator?
  - thanatropism 9y ago
    Welp, a 1-to-1 correspondence can be made between Brownian motion on a 1-dimensional manifold and Brownian motion in plain \mathbb R.

by na85 9y ago

In an infinite string of random integers, 4 appears consecutively an arbitrary number of times. There's no way to prove, of which I am aware, that a string of 24 fours is not random.

PeterisP 9y ago

Randomness describes a process, not its results. You can't prove that a string of 24 fours is not random, and the fact that some process returned 24 fours once is not proof (but is some evidence) that it's not random - however, a process that always returns 4 is not random. A process that returns a single fixed number (which was chosen by a fair dice roll) once is a random process. Using that same process twice or more is not.

jameshart 9y ago

But given a pseudorandom number generator with a known amount of internal state (eg a 32 bit seed) there are hard upper limits on the lengths of sequences like that that can plausibly be produced. Of course, to get 4 out of a PRNG you probably have to ask it for a random number in a range - if you’re requesting numbers in the range 0-1000 then you would expect fewer long sequences of fours than if you request in the range 0-5. And you can get arbitrarily many 4s by requesting numbers in the range 4-4...

thanatropism 9y ago

Tangential but related: Brownian motion (i.e. continuous-time random walk) is recurrent in one and two dimensions -- it hits zero infinite times, but in three dimensions it is not. The way my professor explained: a drunkard can always find his way back to his building, but not to his apartment.

tzs 9y ago
> The way my professor explained: a drunkard can always find his way back to his building, but not to his apartment. That might be true for a drunk Kitty Pryde, but is it true for a drunk ordinary human who is constrained to only move between floors via the stairs or elevator?
- thanatropism 9y ago
  Welp, a 1-to-1 correspondence can be made between Brownian motion on a 1-dimensional manifold and Brownian motion in plain \mathbb R.