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Math Help - A question

  1. #1
    MHF Contributor arbolis's Avatar
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    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?
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  2. #2
    MHF Contributor arbolis's Avatar
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    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?"
    Last edited by arbolis; June 5th 2008 at 10:57 AM.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by arbolis View Post
    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
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  4. #4
    MHF Contributor arbolis's Avatar
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    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.
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  5. #5
    Lord of certain Rings
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    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
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