0

A perfectly smooth sphere has an infinite number of points on it. Now imagine you throw a dart such that it will contact only one of the points. as there are an infinite number of points the probability that the dart lands on any one point is 0. yet the dart does land on a point. How does that work?

Moiz khokhar
  • 1
  • 3
    Continuous probability distributions have this property. There is a difference between $0$ probability and "impossible". – lulu Jul 05 '22 at 16:34
  • 1
    The sphere is a red herring here: the same question can be asked of the unit interval $[0,1]$, which might be a simpler setting to wrap one's head around this fundamental probability question. – Greg Martin Jul 05 '22 at 16:35
  • 1
    Analogy: What's the area of a point? $0$ of course. But the surface of the sphere is made up of a bunch of points, each with area $0$. So how does the sphere end up with positive surface area? – paw88789 Jul 05 '22 at 17:09
  • 1
    Instead of a sphere think of a regular dart board. What's the probability of hitting the bullseye? It's a small number that depends on the area of the bullseye. What happens to that probability when the bullseye area is reduced? Ignoring quantum mechanics here, if the bullseye was only an atom in the center of the dart board, how often would you be able to hit it? – Sam Jul 05 '22 at 17:16

1 Answers1

0

This concept is demonstrated by the definition of almost surely. The wikipedia article has a throwing dart example, similar to your question.

https://en.wikipedia.org/wiki/Almost_surely

Imagine throwing a dart at a unit square (a square with an area of 1) so that the dart always hits an exact point in the square, in such a way that each point in the square is equally likely to be hit. Since the square has area 1, the probability that the dart will hit any particular subregion of the square is equal to the area of that subregion. For example, the probability that the dart will hit the right half of the square is 0.5, since the right half has area 0.5.

Next, consider the event that the dart hits exactly a point in the diagonals of the unit square. Since the area of the diagonals of the square is 0, the probability that the dart will land exactly on a diagonal is 0. That is, the dart will almost never land on a diagonal (equivalently, it will almost surely not land on a diagonal), even though the set of points on the diagonals is not empty, and a point on a diagonal is no less possible than any other point.

AgainsttheOdds
  • 1