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 / v

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- Solution v

Solution

Let

I_{j}

be the indicator random variable of an apple in lunch

j

. The number remaining is

R = 5 - I_{1} - I_{2} - I_{3}

. Each expected value

E [I_{j}] = P (A_{j}) = 1/2

. The linearity of expectation therefore gives

E [R] = 5 - 3 (\frac{1}{2}) = \frac{7}{2} apples .

Although the indicator random variables are dependent under sampling without replacement, the linearity of expectation needs no independence assumption.

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