What Are the Probabilities of Different Types of Numbers in an Infinite Book?

[starfish99]

Suppose I had an infinitely large book that had every real number between 0 and 10. I open this book at random and randomly select a number.
What is the probability that the number is rational?
What is the probability that the number is irrational, but not transcendental?
What is the probability that the number is transcendental?

Would the sum of these probabilities be 100%

 

[JonF]

Even an infinitely large book couldn't hold the reals, since books are essentially list, ordered by page number and character.

 

[disregardthat]

Ignoring the fact that the reals are uncountable, and assuming the lebesgue measure as a measure of probability, the answers are successively 0,0 and 1, and yes the sum is 1.