Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 8 / b / ii / Solution

Compute all three probabilities. From the preceding part, $\mathbb{P}(A)=e^{-6}$. Integrating the joint probability density function over $u>v$ givesOn $A\cap B$ one has $v>2$ and $u>v$, soTherefore $A$ and $B$ are independent events. The minimum waiting time and the identity of the first clock are independent for competing exponential clocks; unequal rates change the winner probabilities but do not destroy that independence.