Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 6 / c / ii

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- Solution ii

Solution

By the law of total probability,

P (T = 1) = \frac{1}{6} \sum_{r = 1}^{6} \frac{7 - r}{6} = \frac{7}{12} .

Bayes' theorem therefore gives the conditional probability mass function

P (R = r ∣ T = 1) = \frac{( 1/6 ) ( 7 - r ) /6}{7/12} = \frac{7 - r}{21}, r = 1, \dots, 6.

The six probabilities sum to one. A small red outcome is more likely after an immediate success because more white faces can meet its threshold.

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