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Paradoxes of Probability and Other Statistical Strangeness

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Re: Paradoxes of Probability and Other Statistical Strangeness

#11

Post by red75prime »

georgewsinger wrote: post #6
Sinergy2 wrote: post #4

Earlier quoted context omitted.

Please describe how it is possible to pick such a number. For example, I can readily imagine how to pick a random 32b float, but that it is an entirely problem with a nonzero probability.

Usually in math we assume the axiom of choice :) https://en.wikipedia.org/wiki/Axiom_of_choice I'm assuming this could somehow lead to such a "random" pick in the technical sense. In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer. Perhaps a mathematician could show how to pick one on some abstract machine with infinite resources, and not constrained b…

Run a random number generator (0-9) for each decimal position after the dot in parallel. This should do a trick.

Re: Paradoxes of Probability and Other Statistical Strangeness

#13

Post by philipov »

georgewsinger wrote: post #3

Another "paradox": even though it's possible to randomly pick a rational number from the reals, the probability of this happening is 0.

I think this only sounds like a paradox if it is phrased poorly. The accurate way to state it is "The probability of randomly picking a specific number is 0" and that sounds reasonable. The probability of successfully picking any number is 1.

Re: Paradoxes of Probability and Other Statistical Strangeness

#14

Post by Mattasher »

btilly wrote: post #9
georgewsinger wrote: post #3

Another "paradox": even though it's possible to randomly pick a rational number from the reals, the probability of this happening is 0.

There is no uniform probability distribution on the reals. Perhaps you meant the interval from 0 to 1?

There are many ways to pick randomly from the rationals with a non-uniform distribution.

Re: Paradoxes of Probability and Other Statistical Strangeness

#15

Post by cdavid »

philipov wrote: post #13
georgewsinger wrote: post #3

Another "paradox": even though it's possible to randomly pick a rational number from the reals, the probability of this happening is 0.

I think this only sounds like a paradox if it is phrased poorly. The accurate way to state it is "The probability of randomly picking a specific number is 0" and that sounds reasonable. The probability of successfully picking any number is 1.

That's a different statement: OP is alluding to the fact the measure of Q is 0 when using the "standard" sigma algebra on the real line, while you are saying that the measure of a number of 0.

[edit] strictly speaking, you would restrict yourself to a bounded interval, e.g. if you pick a random number from a uniform distribution on [0, 1], the probability that this number is rational is 0.

Re: Paradoxes of Probability and Other Statistical Strangeness

#17

Post by philipov »

cdavid wrote: post #15
philipov wrote: post #13

Earlier quoted context omitted.

I think this only sounds like a paradox if it is phrased poorly. The accurate way to state it is "The probability of randomly picking a specific number is 0" and that sounds reasonable. The probability of successfully picking any number is 1.

That's a different statement: OP is alluding to the fact the measure of Q is 0 when using the "standard" sigma algebra on the real line, while you are saying that the measure of a number of 0. [edit] strictly speaking, you would restrict yourself to a bounded interval, e.g. if you pick a random number from a uniform distribution on [0, 1], the probability that this number is rational is 0.

oh, yeah, but that's because although Q is dense, it is not a dense subset of R and locally that's equivalent to saying a single point is not dense in R

Re: Paradoxes of Probability and Other Statistical Strangeness

#18

Post by cdavid »

georgewsinger wrote: post #3

Another "paradox": even though it's possible to randomly pick a rational number from the reals, the probability of this happening is 0.

I find that result fairly intuitive, when you understand how measure theory came up to be.

A much more surprising result is that most irrational are normal numbers, but we know almost no normal number (morally speaking, a normal number is an irrational number where each digit is equiprobable in any base).

Re: Paradoxes of Probability and Other Statistical Strangeness

#19

Post by Houshalter »

By far the most unintuitive paradox for me personally is the one presented here: https://youtu.be/go3xtDdsNQM?t=3m27s

"Mr. Jones has 2 children. What is the probability he has a girl if he has a boy born on Tuesday?" Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre.

Re: Paradoxes of Probability and Other Statistical Strangeness

#20

Post by Houshalter »

philipov wrote: post #13
georgewsinger wrote: post #3

Another "paradox": even though it's possible to randomly pick a rational number from the reals, the probability of this happening is 0.

I think this only sounds like a paradox if it is phrased poorly. The accurate way to state it is "The probability of randomly picking a specific number is 0" and that sounds reasonable. The probability of successfully picking any number is 1.

The paradox is that, after picking a random number, you have just done a thing which has probability zero. Doing a thing that has zero probability shouldn't be possible. Ever.
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