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2004_ q1_ sols - University of New South Wales School of...

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University of New South Wales School of Actuarial Studies ACTL 1001 Actuarial Studies and Commerce Quiz 1 Solutions, 2004
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Question 1 [10 marks] Abraham de Moivre: 1. following the work of Huygens and Bernoulli, wrote on probability ( The Doctrine of Chances , 1718)—the first treatment of probability in En- glish. 2. applied the theory of probability to the problem of calculating the value of an annuity (in his 1725 paper Annuities upon Lives ). There he assumed the number alive in the life table decreased in an arithmetic progression. 3. probably inspired the work of Halley dealing with the problem of an- nuities which depend on the survival of two lives (joint life annuities). 4. first edition of Doctrine of Chances was dedicated to the great Isaac Newton 5. de Moivre was buddies with Newton and Halley (the famous Astronomer Royal) Question 2 [30 marks] We can represent our circumstances by figure 1. ( a ) The chance that no die shows the number backed = 5 6 3 . ( b ) one die = 3 · 5 6 2 · 1 6 . ( c ) two dice = 3 · 5 6 1 6 2 . ( d ) three dice = 1 6 3 . The net expectation of the player is = ( b ) + 2( c ) + 3( d ) ( a ) = 17 / 216 , and that of the banker is = ( b ) 2( c ) 3( d ) + ( a ) = 17 / 216 , 1
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Gamble at t=0 -1 1 2 3 Gains/Losses Figure 1: Net payments from the game. so the advantage lies with the banker in the long run. For a stake of a dollar it is easily seen that this advantage is just under eight cents. Here it might be contended that as the player is to receive back his own coin in addition to the prize, his expectation should be based on respective receipts of two, three or four units instead of one, two or three units, as appears in the above solution. His expectation on this basis might be argued is being 2( b ) + 3( c ) + 4( d ) ( a ) or 74/216, which would show a substantial advantage to the player.
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