Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/14/d/solution

For independent seeds the probability of at least one success is . Requiring it to be at least gives . For , taking logarithms and dividing by the negative number reverses the inequality:
The smallest permitted integer is therefore
The ceiling function is necessary because seeds are counted discretely. If , no finite number suffices; if , one seed suffices. The target is interpreted as at least a ninety-five percent chance, since exact equality may not be attainable with an integer count.

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