I've started to learn some probabilty and it made think about this question: let us assume we randomize virtually any number between 0 and 1. What is the probability for this number to be irrational?
Probabilty of picking an irrational number
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probability
1 Answers
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Modern probability uses measure theory. In particular the Lebesgue measure.
This means that any countable set has probability zero to be chosen from. In particular this means that the probability for choosing an irrational number is $1$.
In fact not just irrational, but also transcendental and normal, and any other property which occurs outside a countable set.
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0When the random number generator of your PC outputs consecutive binary digits of a "random" real between $0$ and $1$ then it will surely print the binary expansion of a rational, because in the long run the output will be periodic. – 2012-10-22