If two points are chosen at random on the circumference of the circle, find the probability that the selected points form the diameter of the circle.

Question

Question:
If two points are chosen at random on the circumference of the circle, find the probability that the selected points form the diameter of the circle.

My thoughts:
For the $2$ points to be the diameter of the circle, they must be diametrically opposite of each other. If the first point says $A$ is selected, then the number of possible choices for the second point let say $B$ will be only $1$.

When one point is chosen from the uncountable infinite points on a circle there are only one of those infinite points that will form a diameter with the first point.

So the probability is given by, $$P(E) = \dfrac{1}{\infty} \equiv 0$$

So the probability must be $0$.

How is it possible that the probability of the event is $0$ even if the event is possible? The answer given in my textbook is also $0$ but I'm not quite sure about this. This question is confusing. Can anyone help me out?

This is a general property of continuous distributions....the probability of any particular outcome is $0$. No need for such a complicated example...just consider the uniform distribution on $[0,1]$. The probability of getting $\frac 12$ (or any other specified value) is $0$. — lulu, May 07 '22 at 18:43
In the same way, a probability of 1 does not mean a guaranteed event. Instead, we talk about almost sure or almost impossible events. — Vercassivelaunos, May 07 '22 at 20:37
Would this answer your question canonically? Note that your attempt is correct. — Sarvesh Ravichandran Iyer, May 30 '22 at 11:26
This is because of the same reason why the probability of hitting the bullseye with the dart is zero even if that's possible. — Nandeesh Bhatrai, May 31 '22 at 09:25
Dude. Don't delete your account. Just take a break from this site. :( — tryst with freedom, Jun 23 '22 at 20:32

MathGeek · Accepted Answer · 2022-05-31T19:08:33.980

The points on a circle are an absolutely continuous probability distribution, which means that we are picking two points out of an uncountably infinite amount of points. The answer to this question is indeed $0$. Even though this may seem unintuitive, just because the probability of an event is $0$ does not mean that it is an impossible event. This is especially true for continuous distributions. Also, even if we have a countably infinite amount of possibilites, the probability will still be $0$. This means that the probability of picking a certain random natural number out of all the natural numbers is $0$, even though it is not impossible.

If two points are chosen at random on the circumference of the circle, find the probability that the selected points form the diameter of the circle.

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