Three components have independent exponential lifetimes with rates 1, 2 and 3. What is the probability that the rate-1 component is the last of the three to fail?

Three components have independent exponential lifetimes with rates 1, 2 and 3. What is the probability that the rate-1 component is the last of the three to fail?

Approach: Condition on which component fails first, then use memorylessness to restart the race between the two survivors and add the two orderings that leave the rate-1 component standing.

7/12. The exponential race for the first failure picks component 2 with probability 2/6 and component 3 with probability 3/6. Conditioning on the first failure and using the memoryless property, the survivors then race afresh. If component 2 goes first, component 3 fails before component 1 with probability 3/(1+3) = 3/4, contributing (1/3)(3/4) = 1/4. If component 3 goes first, component 2 fails before component 1 with probability 2/(1+2) = 2/3, contributing (1/2)(2/3) = 1/3. The law of total probability gives 1/4 + 1/3 = 7/12 = 0.5833. The slowest component is last far more often than its 1/6 share of the total rate, since being last compares maxima.

Follow-up: What is the expected time until all three components have failed?

Key concepts: memoryless property, exponential race, conditioning on the first failure, law of total probability.