= Solution
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> gives
$$
\boxed{\mathbb P(M)=\sum_{d=1}^{\infty}\alpha q^{d-1}=\frac\alpha{\alpha+\beta-\alpha\beta}}.
$$
There is no missing probability of never breaking down: survival for $d$ days has probability $q^d\to0$.
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