TITLE: Are there many fewer rational numbers than reals?
QUESTION [31 upvotes]: Today my professor asked me to figure out the probability of getting a rational number from $[0,1]$.
His answer was that the probability is $0$.
Why is this?

REPLY [2 votes]: The probability measure in your problem is just a special type of measure. Any concrete single number has probability zero because as a set, it is a set of zero measure. Measure of all rationals in $[0,1]$ is also $0$ (see this question for explanation), and so the probability of getting rational number is $0$.