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If you had a set of 7 purely theoretical objects, and 5 were blue and 2 were green, the probability of a randomly chosen object being red would be 0 and it would never happen

However, if you were to pick a random point on a sphere, the probability of getting any specific point would be 0, but obviously you would pick a point on the sphere.

Likewise, if you picked a random point on the number line, the chances of picking a transcendental number is 100% but it could also not happen

Why are these two actions different? Is it because picking a red object contradicts the definition of the set? In that case, is probability distinct from whether or not an event will occur entirely?

Or am I wrong entirely about the nature of the probability of an picking an item from infinite set?

Logos King
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  • It's just because you have a different probability distribution on each set. You choose the distribution, so you choose which events have probability 0. But which events are impossible are decided before you even introduce probabilities (event $A$ is impossible means $A = \emptyset$). – Mason Jun 14 '22 at 00:33
  • @Mason Yeah that makes a lot of sense. Does the set of contradictory events not exist then? – Logos King Jun 14 '22 at 04:23
  • I should point out that the probability of picking a transcendental number from the real number line is actually undefined. This is because $\aleph_1 / \aleph_1$ is undefined. – MathGeek Jun 15 '22 at 18:46

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