Step-by-step explanation:
I'm thinking of a number, let's call it X, between 0 and 10 (inclusive). If I don't tell you anything else, what would you imagine is the probability that X=0? That X=4? Assuming that I don't have any preference for any particular number, you'd imagine that the probability of each of the eleven integers 0,1,2,…,10 is the same. Since all the probabilities must add up to 1, a logical conclusion is to assign a probability of 1/11 to each of the 11 options, i.e., you'd assume that the probability that X=i is 1/11 for any integer i from 0 to 10, which we write as
Pr(X=i)=111for i=0,1,2,…,10.
Implicit in this description is the assumption that the probability that X is any other number x is zero. (Here we make a distinction between the random number X and the variable x which can stand for any fixed number.) We can write this implicit assumption as
Pr(X=x)=0if x is not one of {0,1,2,…,10}.
What would change if instead I told you that I was thinking of a number X between 0 and 1 (inclusive)? You might assume that I was thinking of either the number 0 or the number 1, and you'd assign a probability 1/2 to both options. Or, you might guess that I had more than two options in mind. There was nothing in what I said that forces you to conclude that I was thinking of an integer. Maybe I was thinking of 1/2, or 1/4, or 7/8. Once you start going down that road, the possibilities are endless. I could be thinking of any fraction between 0 and 1. But who said I was limiting myself to rational numbers? I could even be thinking of irrational numbers like 1/2√ or π/5. If we allow the possibility that the number X could any real number in the interval [0,1], then there are clearly an infinite number of possibilities. (Of course, I could have been thinking of non-integers for the number betwen 0 and 10 as well, but most people would think I was referring to integers in that case.)
Since we don't want to assume that I am favoring any particular number, then we should insist that the probability is the same for each number. In other words, the probability that the random number X is any particular number x∈[0,1] (confused?) should be some constant value; let's use c to denote this probability of any single number. But, now we run into trouble due to the fact that there are an infinite number of possibilities. If each possibility has the same probability c and the probabilities must add up to 1 and there are an infinite number of possibilities, what could the individual probability c possibly be? If c were any finite number greater than zero, once we add up an infinite number of the c's, we must get to infinity, which is definitely larger than the required sum of 1. In order to prevent the sum from blowing up to infinity, we must have c be infinitesimally small, i.e., we must insist that c=0. The probability that I chose any particular number, such as the probability that X equals 1/2, must be equal to zero. We can write this as
Pr(X=x)=0for any real number x.
What went wrong here? We know all probabilities must not be zero, because we know that the total probability must add up to one. In fact, were know that, somehow, there must be something special for the probability of numbers 0≤x≤1. We know that X is somewhere in that interval with probability one, and the probability that X is outside that interval is zero.
