**HINTS AND ANSWERS**

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

16.3. Show that, if *p _{x}*/16 is the probability that the total will be

16.5. *P*_{1} = *α*(*α* + *β* − *αβ*)^{−1}; *P*_{2} = *α*(1 − *β*)(*α* + *β* − *αβ*)^{−1}; *P*_{3} = *P*_{2}.

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

16.9. Take *A _{j}* as the event that a family consists of

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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