Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 16 / a / iv

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- Solution iv

Solution

By the inclusion-exclusion principle for probabilities,

P (B_{1} \cup A_{2}) = P (B_{1}) + P (A_{2}) - P (B_{1} \cap A_{2}) .

After an initial banana, five of the nine remaining pieces are apples, so the intersection probability is

(1/2) (5/9) = 5/18

. Consequently

P (B_{1} \cup A_{2}) = \frac{1}{2} + \frac{1}{2} - \frac{5}{18} = \frac{13}{18} .

As a check, the complementary event is an apple followed by a banana, which also has probability

5/18

.

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