 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/16/a/iv/solution

The probability of a positive delay is the integral of its density:

P (S > 0) = \int_{0}^{r} \frac{d s}{R} = \frac{r}{R} .

Adding the separate no-delay event gives

P (S = 0) + P (S > 0) = (1 - \frac{r}{R}) + \frac{r}{R} = 1.

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