0

I can pick a number in the interval from all real numbers from 0 to 1. Say this number is 0.42. Now the probability for drawing this number most be zero since we have infinity many numbers in the interval 0 to 1. Is this true and is it not a paradox ?

Ole Petersen
  • 215
  • 9
    This is true, and no, it is not a paradox. It's only a paradox if you think that "with probability $0$" and "impossible" mean the same thing. They don't. – 5xum Nov 04 '16 at 13:36
  • 5
    Related : http://math.stackexchange.com/questions/180283/why-is-the-probability-that-a-continuous-random-variable-takes-a-specific-value/180301#180301 – Ethan Bolker Nov 04 '16 at 13:38
  • I've heard of this referred to as the paradox of the dartboard (video) – Ben Grossmann Nov 04 '16 at 13:41
  • In the standard sense,it is zero. If you want to consider things in a "standard" and "uniform" way, you can consider the Lebesgue measure, which is a uniform measure over the Borel σ-algebra with the usual meaning of length, area or volume, depending on dimension.For example, for R the Lebesgue measure of the interval (a,b) is (b−a). Now the rationals have Lebesgue measure zero, hence the fact. –  Nov 04 '16 at 13:41
  • A probability is always something depending of the $\sigma$-algebra of events so it can change according this last definition. – Piquito Nov 04 '16 at 15:27

1 Answers1

0

If we could have a real way to pick a random number between 0 and 1, which would be extremely difficult, there would be a infinitely small (infinitesimal) chance of picking each number.

PearlSek
  • 171
  • 1
  • 2
  • 11