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

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The discussion centers on the probabilities of selecting different types of numbers from an infinitely large book containing every real number between 0 and 10. It concludes that the probability of selecting a rational number is 0, the probability of selecting an irrational but non-transcendental number is also 0, and the probability of selecting a transcendental number is 1. This aligns with the properties of real numbers and the concept of Lebesgue measure, confirming that the total probability sums to 1.

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

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