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During a normal conversation, my friend asked me one simple question

What is the probability of choosing the center point of a sphere if we are allowed to choose any point on the surface of the sphere$?$

She said that the answer is $0$ as total number points on the surface of the sphere is $\infty$ so $\frac{1}{\infty}=0$

But what I thought that probability can't be $0$ as there is always a possibility that we choose the center point (however less it may be). So the probability should be $0^{+}$. But I am not sure whether my answer is right or not. Here $$0^{+}=\lim_{x\to\infty}\frac{1}{x}$$

Any help is greatly appreciated.

  • First, infinity is not a number so $\frac{1}{\infty}$ is not legal. Second, $0^+$ is not a number distinguishable from $0$ or $0^{-}$. There is only one zero. There are no infinitesimals which are not equal to zero in standard real analysis or in probability. – JMoravitz Jul 19 '22 at 14:18
  • @JMoravitz so is there an answer to this question$?$ –  Jul 19 '22 at 14:20
  • Third... "center point of a sphere if picking from the surface" Isn't the center... in the center of the sphere and not on its surface? Like if the sphere was the earth isn't the center the core of the earth? Or are you talking about picking somewhere on the equator? or the north pole? – JMoravitz Jul 19 '22 at 14:20
  • @JMoravitz actually I meant a hypothetical scenario where magically you can pick the center.....but in general this question can be applied to any specific point say $A$ –  Jul 19 '22 at 14:21
  • Fourth, if talking about picking just one point from a countably infinite collection of points uniformly at random, that is not legal. Such a distribution does not exist. If picking a single point from an uncountably infinite set that is possible though not probable. See here. Finally, consider asking instead the question "the probability you pick a point within a certain distance from the point" – JMoravitz Jul 19 '22 at 14:23
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    Note that zero probability does not imply impossibility of event. And possibility of event does not imply probability is positive. – Ivan Kaznacheyeu Jul 19 '22 at 15:21

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