 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/discrete-mathematics-continuous-mathematics-and-probability/8/a/iii/solution

Let

I_{c}

c

appears at least once. Then

X = I_{R} + I_{B} + I_{G}

. For a colour of single-roll probability

p_{c}

, independence of rolls gives

P (I_{c} = 0) = (1 - p_{c})^{n}

E X = c \sum (1 - (1 - p_{c})^{n}) = 3 - (\frac{1}{2})^{n} - (\frac{2}{3})^{n} - (\frac{5}{6})^{n} .

The three indicators are not independent, but linearity of expectation does not require them to be. This is the expected number of observed categories.

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