Question

Let X1, X2, and X3 be three independent, continuous random variables with the same distribution. Given X2 is smaller than X3, what is the conditional probability that X1 is smaller than X2?

Answer 1

Consider 7 mutually exclusive cases:

The first six are X1 > X2 > X3, X1 > X3 > X2, X2 > X1 > X3, X2 > X3 > X1, X3 > X1 > X2, X3 > X2 > X1, and

The seventh case is where at least two of X1, X2, X3 are equal. Since the Xi are continuous random variables, the probability that at least two of the Xi are equal is zero.

Since X1, X2, X3, are independent and identically distributed, the first six cases are all equally likely. This means that P(X1 > X2 > X3) = 1/6, and similarly for the other orderings.

Given X2 is smaller than X3, the conditional probability that X1 is smaller than X2 is

Since it is fiven that X2<X3, therefore all the cases in which X2<X3 are X1 > X3 > X2, X3 > X1 > X2, X3 > X2 > X1

Out of all the three cases in which X2<X3, the case in which X1<X2 is X3 > X2 > X1

Therefore Given X2 is smaller than X3, the conditional probability that X1 is smaller than X2 is​

$\inline P(X1<X2 | X2<X3)=P(X3>X2>X1)/\left \{ P(X1>X3>X2)&plus;P(X3>X1>X2)&plus;P(X3>X2>X1) \right \}$

$\inline P(X1<X2 | X2<X3)= (1/6)/\left \{ (1/6)&plus;(1/6)&plus;(1/6) \right \}$

$\inline P(X1<X2 | X2<X3)= (1/6)/(1/2)$

$\inline P(X1<X2 | X2<X3)= 2/6=1/3$