Today's maths PSA: Events of probability 0 are not necessarily impossible.

@ColinTheMathmo Is there an example where this applies? Or is it just a question of semantica, e.g. "X is possible in general, but in the given circumstances the probability is 0"?

@michal @ColinTheMathmo Pick a real number between 0 and 1, the probability that it is rational is 0.

(I think this is a standard example, but would be interested in knowing more “concrete” examples.)

@michal @leonardopacheco @ColinTheMathmo A probability is always a real number in the interval [0,1]. So no infinitesimals.

@michal @leonardopacheco @ColinTheMathmo This is maybe a good example of potential mismatch between the real world and one of our mathematical models. There are no actual circles in real life because real things are made of atoms. There are no real numbers in real life for similar reasons. Taking our shaky intuitions about probability (Monty Hall problem etc) and asking for the math to match them when also talking about “zero probability events” is asking for trouble!

@michal In standard probability theory, all probability values are real numbers, so there are no true infinitesimals. (I worked on non-Archimedean probability theory which has infinitesimal values precisely for these types of problems.)
@leonardopacheco Another standard example is an infinite sequence of tosses with a fair coin. It's logically possible for the coin to come up heads on every toss, but the probability of the event zero. Not sure if that's more concrete, though.


@SylviaFysica Infinite sequences of coin tosses feel concrete enough for me, thanks! And they may be a good enough approximation for events which repeat many many times. (Also first time I’ve heard of non-archimedean probability, will take a look at it later.)

@leonardopacheco I wrote an introduction chapter about the topic that covers its history, mathematics, and philosophical discussion (aimed at formal epistemologists). :OpenAccess:

@ColinTheMathmo more strongly there are circumstances where an outcome of probability 0 is inevitable. Choose a random x uniformly from zero to one. All choices have probability zero but you still have to pick something.

@11011110 @ColinTheMathmo There's also the situation that almost all random choices from the [0,1] interval are literally undescribable.

Hm, now I wonder what the situation would be if we restricted ourselves to rational numbers. That seems like a weird distribution on the entire set of natural numbers!

@11011110 @ColinTheMathmo I've never seen the Weierstrass p used for anything other than elliptic functions. Is it normal to use \(\wp(X)\) to mean the countable subsets of \(X\)?

@11011110 @ColinTheMathmo Oh, no, of course, they mean the power set, that's why it's also a subset.

Okay! But that's also weird! Using \(\wp\) for power set!

@ColinTheMathmo Is this Colin's Paradox?

What happens when a zero probability meets an infinite trial space?

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