Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 35 / 4 / b / i / Solution

Condition on the bulb being alive at age $\tau$. The next hour's survival likelihood contribution isIt does not depend on the unknown age $\tau$. This is memorylessness of the exponential distribution: the residual lifetime given survival has the same exponential distribution as a new lifetime.