Re: Probability in an infinite sample space


Robert Israel wrote:
In article <ITSnetNOTcom#virgil-FDCF25.20173719032006@xxxxxxxxxxxxxxxxxxxxxxxx>,
Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx> wrote:
In article <1142823881.738119.307930@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"mikeh106@xxxxxxxxxxx" <mikeh106@xxxxxxxxxxx> wrote:

If you choose a natural number at random, that it will be a multiple of
3 is there a 1/3, 1/2, or undefined chance?

Does it mean something to select "at random" from an infinite number of
objects?

To select something "at random" from any set means that every member of
the set has the same chance of being selected as any other.

That requires that a selection at random from any infinite set must
assign probability zero to each member's being drawn.

For countably infinite sets, this is impossible, as countably many 0's
must still add up to 0.

True. What one can say is that the set of multiples of three has
asymptotic density 1/3, i.e. if P_N is the probability of obtaining
a multiple of 3 when selecting a member of {1,...,N} with equal
probabilities, then lim_{N -> infty} P_N = 1/3.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Another way of looking at it would be to say that picking a random
natural number n would entail the following three equally likely
events.
n =_ 0 (mod 3), n =_ 1 (mod 3) or n =_ 2 (mod 3)
so the probability of getting a multiple of 3 is the probability of
getting the event 0 =_ mod (3). That is 1/3.
Muhammad
(PS. If someone has already pointed it out in the thread, I apologize.)

.

Relevant Pages

  • Re: Probability in an infinite sample space
    ... is there a 1/3, 1/2, or undefined chance? ... That requires that a selection at random from any infinite set must ... assign probability zero to each member's being drawn. ... their union has asymptotic density sum_j d_j. ...
    (sci.math)
  • Re: Probability in an infinite sample space
    ... is there a 1/3, 1/2, or undefined chance? ... Does it mean something to select "at random" from an infinite number of ... You need to specify the probability distribution. ... as we await bird flu shots and fight a real cold war. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If one of the elements is to be chosen from the set, then the probability of one being chosen is 1. ... then that 1 representing the fact that one of those will be chosen is defined to be the sum of the probabilities of each. ... If you can establish a uniform probability distribution, then you can say each possibility has the same probability. ... If we are discussing an infinite set of possiblities, and I define infinitesimal in terms of that infinity, it's not circular. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... the sum of a countable number of 0's; he simply "wants" it to be 1. ... then the probability of one being chosen is 1. ... the question is why the standard system can't accomodate the infinite case, ... The probability of each is 0 in standard analysis, ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... therefore there must exist a uniform distribution on the naturals. ... If you use a 2 element set, you still get a total probability of 1. ... the sum of a countable number of 0's; he simply "wants" it to be 1. ... the question is why the standard system can't accomodate the infinite case, ...
    (sci.math)