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?
Best Answer
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.