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e next venture is called determinate ambe. Here the speculator names the order in which two selected numbers will appear. Instead of 20 chances at any drawing of five numbers, he has only one chance-one chance in 8,010. He ought then to receive 8,010 times his stake, if he wins. As a matter of fact he receives only 5,100 times his stake. From this it follows that he is defrauded of 2,910 out of 8,010 parts of his stake, or very nearly three-eighths of the stake's value.
But more speculative ventures remain. The speculator can name three numbers. Now there are 704,880 possible selections of three numbers out of 90. (There are 8,010 possible selections of two numbers, as already shown, and with each of these any one of the remaining 88 numbers can be taken to make the third number; thus we have 88 times 8,010, or 704,880 sets of three numbers in all.) These can appear among the five drawn numbers in 60 different ways (5 times 4 times 8). Thus the speculator has 60 chances out of 704,880, or one chance in 11,748. He ought then to receive 11,748 times his stake, if he wins; but in reality he
receives only 5,500 times his stake in this event. Thus the lottery-keeper robs him of more than half of his just winnings, if successful, and of more than half the mathematical value of his stake at the outset. The venture in this case is called simple terse. Determinate terne is not allowed. If it were, the prize of a successful guess should be 704,880 times the stake.
Quaterne involves the selection of four numbers. With 90 numbers, 61,334,560 (704,880 times 87) different selections of four numbers can be made. Among the five drawn numbers there can only be found 120 arrangements of four numbers. Thus the speculator has only 120 chances out of 61,334,560, or one chance out of 511,038. He ought therefore, if he wins, to receive 511,038 times his stake. The prize is only 75,000 times the stake. The lottery-keeper deducts, in fact, six-sevenths of the value of the stake at each venture. Determinate quaterne is, of course, not admitted. Simple quatern is, at present, the most speculative venture adopted. Formerly quine was allowed, the speculator having five numbers, and, if all five were drawn, receiving a million times the value of his stake. He should have received 43,949,265 times its value; so that, in effect, he was deprived of more than 42 forty-thirds of the true value of his venture.
The following table shows the amount by which the terms of the Geneva system reduce the value of the stake in these different cases, the stake being set at 1/. for convenience:
Actual Worth Robbery per
of 1l.. Stake 1l. Stake
d. s. d.
Simple drawing · 16 8 4
Determinate drawing 15 6 3/4 4 5 1/4
Simple ambe . 13 6 6 6
Determinate ambe 12 9 7 3
Terne . 9 4 1/2 10 7 1/2
Quaterne 2 111/4 17 0 3/4
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