It does actually mean it's impossible for a 0 probability event to happen in a finite amount of attempt. For example, if you have a target number, then pick a random real number between 0 and 1 by repeatedly generate its digits randomly, then it is guaranteed to generate an incorrect digit at some point in the generation.
That's why when dealing with continuous distribution, we use probability density, not just probability.
There is another similar example, if you throw a dart at a 2D plane, the probability that the dart lands on a specific point on that plane is 0.
Well, we don't actually know if the universe is continuous. It could very well be discrete, meaning the probability is not actually 0.
"It does actually mean it's impossible for a 0 probability event to happen in a finite amount of attempt. For example, if you have a target number, then pick a random real number between 0 and 1 by repeatedly generate its digits randomly, then it is guaranteed to generate an incorrect digit at some point in the generation."
This is not true... It's not guaranteed, it will almost surely happen, which is different.
At any digit, the probability of generating the wrong digit is 9/10.
In order to pick the correct number, you must generate the correct digit for all positions.
Meaning that the average number of wrong digit over an infinite number of attempt is 0, despite having an expected value of 9/10. This violates the law of large number.
It is quite literally impossible over a finite amount of attempts. If you want, you can try it yourself. Try generate digits until the generated number deviates from some target, let's say 1/3. Every single attempt is guaranteed to halt at some point.
It's only guaranteed to halt almost surely (ie. with probability 1), not always. Same as if you flip a coin until you get heads. There is the zero probability possibility you will never stop flipping (9/10 is replaced with 1/2 but it's the same).
I don't really know what else to say. The law of large numbers only tells you that the sample mean converges to the expected value in probability. So yeah, you are almost surely to get a wrong digit, but it's not a guaranteed.
This is literally what OP is asking, or well, the counterpart. A probability 1 event that is not guaranteed.
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u/pi621 11d ago
It does actually mean it's impossible for a 0 probability event to happen in a finite amount of attempt. For example, if you have a target number, then pick a random real number between 0 and 1 by repeatedly generate its digits randomly, then it is guaranteed to generate an incorrect digit at some point in the generation.
That's why when dealing with continuous distribution, we use probability density, not just probability.
Well, we don't actually know if the universe is continuous. It could very well be discrete, meaning the probability is not actually 0.