What is probability of selecting an irrational number from set of real number?

Answer 1

The probability of selecting an irrational number from the set of all real numbers (selected randomly) is unity (100%). Let me explain:

consider selecting the real number by selecting a series of digits. select each digit randomly and creating a number, say from zero to one by putting a decimal place on the far left of the digits. Selecting more and more of those digits means we are selecting from more and more of the real numbers. The limit as our selection approaches infinity is a real number that is picked from any real number (between 0 and 1).

A rational number is one whose digits end up repeating the same pattern thus being able to be written as a fraction. An irrational number is a real number that is not rational.

So with those two ideas one could think of picking a random real number by picking successive digits and figuring the odds of getting a repeating sequence would get more and more absurd as one picked more and more digits.

Think of it as your winning lottery number being picked not twice but an infinite amount of times (with no other picks in between). That would be the chance of your pick being rational.

That being said the ability to fairly choose a real number from all the possible choices is very difficult since one needs to choose an infinite amount of digits. If one gets bored and stops before an infinite amount of digits is chosen, the resulting number is rational (since it has an infinite amount of repeating zeros on the end of the number). So in that sense your chances of picking an actual irrational number are the same as your chances of picking an infinite digit random number.