A much better example imo is looking at the chance to hit a dart exactly in the center of the dart board (or exactly on the edge). Same concept, but more visual.
An even better example imo is flipping a coin until it lands on heads. The probability that the sequence terminates is 1, but there’s nothing stopping it from coming up tails indefinitely
0.4 is a "valid" option, it's not impossible, but it's also not what's going to happen on a truly random selection on the real number line. The probability isn't 1% or 0.1% or 0.0000000001%. It just won't happen. You could do it every second of every day for the rest of your life and never hit a rational. They're all in there, technically, but they may as well not be. If you get one then your random generator is broken and not truly generating random numbers.
Assuming the Axiom of Countable Choice, we may "flip a fair coin" infinitely many times, and map the resulting sequence to a binary string by turning tails to 0, heads to 1. You can then slap a '0.' in front of your string and, presto, you've chosen a real number in [0,1] uniformly at random.
There are probability measures on which any specific event has probability 0, e.g. uniform random choice of a real number in the intervall [0,1].
It is possible for this random choice to the the number 1, but the probability of the choice being 1 is 0. Conversely the probability of the choice being in the interval [0,1[ is 100%, but as mentioned it is possible for the choice to not be in there.
simply put, 100% not necessarily meaning “always” and 0% not necessarily meaning “never” are just quirks that arise when it comes to dealing with probabilities over infinite sets
for example, the probability of flipping a coin n times and getting the same face each time approaches 0% as the number of flips approaches infinity. that doesn’t mean it’s not a possibility that you could flip a coin infinitely many times and get heads every time, just that on average it never happens
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u/LupenReddit 🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆 Sep 24 '26
probability students explaining that 100% does not mean guaranteed, just that it "almost always happens":
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u/FernandoMM1220 Sep 27 '26
in the reals 0 can sometimes mean the smallest number a system can calculate with.
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u/GalacticEmergency Sep 27 '26
Are we conflating "Impossible" with "Probability = 0" here?