Answers to questions posed

Chapter 2 : With fair dice having eight or ten sides, the chance of an even number is 1/2, and the chance of a multiple of six is 1/8 or 1/10 respectively. The respective chances of a multiple of three are 1/4 (two in eight), which gives independence, and 3/10, which does not.

Chapter 6 : In the Colour of Money puzzle, you should aim to bank ￡12,000 at your first choice. If it works, you hope to bank ￡3,000 next time; if it fails, you must try for ￡15,000 next time. Your overall winning chance is (6/12)*(10/11)+(6/12)*(4/11)=84/132=7/11, which is higher than any alternative tactic.

In 46 similar poker games, the nine times a Spade arises, you make a net profit of 50 chips; the other 37 times, you lose the extra stake. If you can stay in for 12 chips or fewer, you’ll be ahead on average, but if you have to pay 13 chips or more, you’ll expect to lose more than you gain.

Chapter 7 : The chance of a miscarriage is 1 in　m , the chance of a Down’s baby is 1 in　n , both　m and　n large and m＞n. Take the test if y＞x+n/m.

Suppose Anne’s sister Celia has　n sons, all healthy. If Betty is not a carrier, all Celia’s sons are sure to be healthy; if Betty is a carrier, the chance Celia will not be a carrier (and hence all her sons are healthy) is 1/2, and the chance she is a carrier, but nevertheless all her sons are healthy, is (1/2) n +1 . So the posterior odds of Betty being a carrier, after the information about all Anne’s healthy nephews, are the prior odds (i.e. those computed using the information about Anne’s healthy brothers), multiplied by all these factors of the form (1/2+(1/2) n +1 ) from her sisters. Convert this figure into the probability that Betty is a carrier, and halve it to find the chance Anne is a carrier.

Chapter 8 : In the randomized response examples, suppose the proportion of smokers is　x , and all answers are truthful. Then, in the first case, the overall proportion of ‘Agree’ answers is 0.8 x +0.2(1- x ); equate this to 1/3, solve to find　x =2/9. In the second case, the proportion of ‘Yes’ answers is 0.8 x +0.2/2; setting this equal to 1/3 gives　x =7/24.

Chapter 9 : Let　x be the mean number of throws to get HH. To get the first H takes two throws, on average. After this, we need at least one extra throw; and even then, half the time we must begin again. So　x = 2 +1+( x /2), hence　x = 6.

In Penney-ante, if your opponent selects HHH, you should choose THH, and your winning chance is 7/8; if she picks HHT, again select THH, and win 3/4 of the time; to her HTH, respond HHT, against THH, use TTH – in both these cases, expect to win 2/3 of the time. Symmetry leads to the best choices against TTT, TTH, THT, and HTT.