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d fairly represent the price of a chance. And while that unpractical result (for the number of speculators would not be unlimited) is true, the practical result is easily proved, that the larger the number of venturers the greater should be the price for each chance-a relation which absolutely forbids the employment of this method of keeping lotteries.
Let us see how this ca, be shown. De Morgan has given a demonstration, but it is not one to be very readily understood by those not versed in mathematical methods of reasoning. I believe, however, that the following proof will be found easy to understand, while at the same time satisfactory and convincing.
Suppose that eight ventures only are made, and that among the eight, four, or exactly half, toss head the first time; of the remaining four, two half-toss head at the second trial; of the remaining two, one tosses head at the third trial; while the other tosses head at the fourth trial. This may be regarded as representing what might on the average be expected from eight trials, though in reality it does not; for of course, if it did, the average price per chance, inferred from eight such trials, would be the true average for eight million trials, or for eight million times eight million. Still it fairly represents all that could be hoped for from a single set of eight ventures. Now we see that the sums paid in prizes, in this case, would be four times 21. for those who tossed ' head' at the first trial; twice 4l. for those who tossed 'head' at the second trial; 81. for him who tossed 'head' at the
third trial; and 16/. for the last and most fortunate of the eight; or 40l. in all. This gives an average of 5l. for each chance.
Now suppose there are sixteen ventures, and treat this number in the same way. We get eight who receive 21. each; four who receive 41. each; two who receive 81. each; one who receives 16/.; and one who receives 82/. The total, then, is 96/., giving an average of 61. for each chance.
Next take thirty-two ventures. Sixteen receive 21. each; eight 41. each; four 81. each; two 16/. each; one 32/.; and one 64/.; a total of 224/., giving an average of 71. for each venture.
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