I've been asked the following question:
if you pick any rational number r in the interval from [0, 1] that the probability of picking this rational number is 0.
I've seen three different ways to solve this problem:
One involving how the probability of picking an irrational number is 1 and the probability of picking a rational number is 0.
Another where the length from a smaller interval contained within [0,1] and also contains r is 2m but since the length from [0,1] is 1. then the probability of picking a rational number is 2m/1=2m. However, i dont get the rest of this way. The rest of this solution involves picking even smaller intervals and the probability of picking a rational number is found by 2m where m approaches 0. i'm not understanding how you can get the probability to be 0 using this method.
Lastly, there are infinite rational numbers between 0 and 1 and since you want to pick 1 number that is the same as saying the probability of picking a rational number is the same as 1/x where x approaches infinity.
Can anyone clarify these three methods? Especially the 2nd one!