$X_1, X_2, X_3, X_4$ are independent exponential random variables with rates 1, 2, 3, and 4, respectively.

Question

What is $P(X_1<X_2<X_3<X_4|X_2<X_4)$? I assume I need to use the formula for conditional probability so $$P(X_1<X_2<X_3<X_4|X_2<X_4)=\frac{P(X_1<X_2<X_3<X_4,X_2<X_4)}{P(X_2<X_4)}$$

The bottom of this is easy to find: $P(X_2 < X_4)= \frac{2}{2+4}=\frac13$. But I'm not sure how to do the top. Suggestions?

Answer 1

For the numerator, note that if $A=X_1<X_2<X_3<X_4$ and $B=X_2<X_4$, then $$P(B|A)=1 \rightarrow P(A,B)=P(A)P(B|A)=P(A)$$ so the numerator simplifies to $P(X_1<X_2<X_3<X_4)$. You can follow the same method as you used for the denominator to work out the numerator.