Suppose a number x is picked randomly from the interval 0 1

Suppose a number x is picked randomly from the interval [0, 1]. What is the probability that x = 1/10? What is the probability that x = m/10 for some m = 0, . . . , 10? What is the probability that x = m/100 for some m = 0, . . . , 100? What is the probability that x = m/n for some m n? What is the probability that x is rational?

Solution

The answer is zero for all parts of this question. The reason is that interval (0,1) has uncountably many points. In general, for any continuous random variable, P(X = x) = 0 for any x.

For last part, P(X = rational) = sum ( P(X=x), where x = rational ) = sum (0) = 0, because probability countable union of disjoint sets is same as countable summation of probability of those disjoint sets (this is third axiom of probability distribution)

Also, the rational set is countably infinte, and interval (0,1) is uncountably infinite, so P(X=rational) = 0, P(X = irrational) = 1