Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 7 / b / i

Put $q=(1-\alpha )(1-\beta )$, the probability of surviving a full day, and $p=1-q=\alpha +\beta -\alpha \beta$. These lie in $(0,1)$. Let $M$ be the event that the first breakdown is in the morning. By independence, surviving $d-1$ complete days and then failing in the morning of day $d$ has probability $q^{d-1}\alpha$.

These events are disjoint, so summing the geometric series givesThere is no missing probability of never breaking down: survival for $d$ days has probability $q^d\to 0$.