 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/iii/solution

Let

D

be the number of days up to and including the first breakdown, and

M

the event of a morning first breakdown. For

d = 1, 2, \dots

P (D = d ∣ M) = \frac{α q ^{d - 1}}{α / ( 1 - q )} = (1 - q) q^{d - 1} .

Thus

D ∣ M \sim Geom (α + β - α β) on {1, 2, \dots} .

This geometric distribution counts the day on which the first breakdown occurs. Its parameter is the probability of at least one breakdown during a day, rather than the morning breakdown probability

α

alone.

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