1. ## A question

I just thought about a problem. Say you have a circle (there is an infinity of radius of the circle), and you want to pick up one radius. If there is a finite number of elements (here radius), the probability to pick up one on n elements is $\frac{1}{n}$ if the probability to pick up one is equal to the probability to pick any other element. I believe that in my example, I can make n tends to $+\infty$. As I will pick up one radius, the sum of all the probabilities to pick a particular radius must be 1. So I can rewrite $1=\sum lim \frac{1}{n}$ as $n$ tends to $+\infty$. Seems very strange for me. Can someone tell me if I did something wrong?

2. Oh... As n tends to $\infty$, the sum I made is in fact lim $n*\frac{1}{n}$ as tends to $+\infty$. So of course it's always equal to $1$. Nice problem! You can try to ask to someone that knows a bit of probability : "Explain me how this could be : if there are an infinity of radius in a circle and I want to chose exactly one. There is a probability of 0 (well $\frac{1}{+\infty}$) to pick up one in particular. Repeating this with "all" the radius, none of them have a probability to get chosen greater than 0. So how can you explain that I will chose one of them?"

3. Originally Posted by arbolis
Oh... As n tends to $\infty$, the sum I made is in fact lim $n*\frac{1}{n}$ as tends to $+\infty$. So of course it's always equal to $1$. Nice problem! You can try to ask to someone that knows a bit of probability : "Explain me how this could be : if there are an infinity of radius in a circle and I want to chose exactly one. There is a probability of 0 (well $\frac{1}{+\infty}$) to pick up one in particular. Repeating this with "all" the radius, none of them have a probability to get chosen greater than 0. So how can you explain that I will chose one of them?"
A probability of 0 does not mean that the event cannot happen.

RonL

4. A probability of 0 does not mean that the event cannot happen.
For me this is totally counter-intuitive, but true. In my example, you have an infinity of zero-probable events, but while summing the probabilities it makes 1, so it will always happens. Unless I'm misunderstanding you.
Another counter-intuitiveness for me, is like when you see for the first time lim when x tends to $+\infty$ of $(1+\frac{1}{x})^x$. At first your intuition tells you that $\frac{1}{x}$ tends to $0$, so the limit would be simply 1, while it is false.
Edit : I think I just realized. When you chose a radius, it has a probability of 0 to have been chosen! This is crazy, you're right! An event with a probability of 0 to happens can happens.

5. The counter intuitiveness stems from the unreliable habit of predicting things from examples.

As a proof for Captain's statement, what is the probability of picking 0.5 from the interval [0,1]?

Since there are infinite reals in the interval, The chance is 0. So does that mean I can never pick 0.5? Of course not.. "I pick point 0.5", there... I picked it