Quote:
Originally Posted by NovaAce
because there is indeed a non-zero chance of losing an infinite amount of flips in a row
(1/2)^∞ = 1 / 2^∞
2^∞ = ∞
1 / ∞ = 0
What you've stumbled upon are the probabilistic concepts of "almost never" and "
almost surely", of which repeated coinflips are a textbook example. The set of infinite sequences involving no tails is non-empty, hence the need to say "almost", but the probability of a sequence from that set is still zero.
If one attempts to assign any real nonzero probability to each individual sequence, the total probability will exceed 1. No can do.
However, in the nonstandard approach to analysis (where infinitesimal "hyperreal" numbers are used), it may be possible to have infinitesimal probabilities. A free and good-looking paper from 2016 makes the case:
https://academic.oup.com/bjps/article/69/2/509/2669779
What might we do with that information? I don't know. To me, saying, "My chance of flipping infinite heads is infinity to 1 against, or epsilon" does not sound any more optimistic than "zero". Both convey the same thing: call it zero, call it epsilon, call it Santa Claus, either way it simply won't happen.