Does a subset of $R$ contain equal number of rational and irrational numbers? How to prove?
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You have asked two different questions.
A number chosen uniformly at random between 1 and 10 has probability zero of being rational.
An interval of reals contains a countable infinity of rationals, an uncountable (i.e., much larger) infinity of irrationals. The rationals have measure (i.e., length) zero; all the measure is in the irrationals.