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16.1.  (a) Yes, (b) no, (c) no, (d) no, (e) yes.

16.3.  Show that, if px/16 is the probability that the total will be x, then the corresponding gain is [px(x2 + x) − 16x]/16. (a) A loss of 0.36 euros; (b) a gain of 27/64 euros; (c) a gain of 46/16 euros, provided he takes your advice and guesses “6” each time.

16.5.  P1 = α(α + βαβ)−1; P2 = α(1 − β)(α + βαβ)−1; P3 = P2.

16.7.  The relative probabilities are X : Y : Z = 50 : 36 : 8 (in units of 10−4); 25/47.

16.9.  Take Aj as the event that a family consists of j boys and nj girls, and B as the event that the boy has at least two sisters. Apply Bayes’ theorem.

16.11.  (i) For a even, the number of ways is 1 + 3 + 5 + ··· + (a − 3), and (ii) for a odd it is 2 + 4 + 6 ...

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