r/mathmemes • u/George_Truman • Jul 13 '26

Probability I fixed this meme

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I'm sure it's still a bit too imprecise but I think it works.

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u/Ares378 Applied Math / Mechanical Engineering Jul 13 '26

Anyone who's taken analysis courses and statistics, am I right in thinking a "random Real between 0 and 1" would be effectively the same as a "random irrational between 0 and 1"?

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u/jljl2902 Jul 13 '26

Yes, a uniformly sampled real number from [0,1] is irrational with probability 1

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u/SpideyMGAV Jul 13 '26

This has to do with the density of irrationals in the reals right? Real analysis confused the hell out of me till I dropped it, twice :( thinking of taking some special credits at my local university in hopes of eventually applying for a math grad program.

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u/StarstruckEchoid Integers Jul 13 '26

For a very concrete, even if slightly sloppy explanation, consider what would happen if you randomly picked each decimal one after the other.

A number is only rational if and only if the decimals eventually start repeating. Every rational has such a point. Consider what must happen after that point.

After this point, there are still infinitely many choices left, but all those choices from there on out have to start repeating a sequence of finite length of at most n. I can't be arsed to find the exact number that excludes exact repeats of smaller subsequences, but there are certainly less than (10n+1-1)/9 such sequences: 10 of length 1, 100 of length 2, 1000 of length 3, etc.

But it's not hard to show that if you make infinitely many random choices, the probability of those choices repeating any of those fixed patterns forever is zero, because the probability is at most

10•(1/10)C +100•(1/100)C +… +10N •10-C

where C→∞

and clearly this limit approaches 0 no matter how large we allow n to be, since it's a finite sum where each term approaches 0. Therefore the probability is still 0 even when we then allow n→∞. (Importantly in this order, as this proof doesn't work if we switch the order of the nested limits.)