 Solution

ID: 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/solution

Put

q = (1 - α) (1 - β)

, the probability of surviving a full day, and

p = 1 - q = α + β - α β

. 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

q^{d - 1} α

These events are disjoint, so summing the geometric series gives

P (M) = d = 1 \sum \infty α q^{d - 1} = \frac{α}{α + β - α β} .

There is no missing probability of never breaking down: survival for

d

q^{d} \to 0

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