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alue of each trial. Buffon and each of three correspondents of De Morgan's made 2,048 trials--an experiment which even the most enthusiastic student of chances will not greatly care to repeat. Buffon's results, the only set we shall separately quote, were as follows. In 1.061 trials, ' head'
showed at the first tossing; in 494, at the second; in 232, at the third; in 187, at the fourth; in 56, at the 1llth; in 29, at the sixth; in 25, at the seventh; in 8, at the eighth; in 6, at the ninth. The 2,048 trials, estimated according to the Petersburg system, would have given 20,1141. in all, or nearly 101. per game. According to our method, since 2,048 is the eleventh power oŁ 21., the average value of each chance would be 18/.; 1 and Buffon's result is quite as near as could be expected in a single experiment on 2,048 trials.
But when we take the four experiments collectively, getting in this way the results of 8,192 trials (of which De Morgan, strangely enough, does not seem to have thought), we and the average value of each chance
I note that De Morgan obtains the value 11/. instead of 13l. But he strangely omits one of the last pair of trials altogether. Thus, he says, ' in the long run, and on 2,048 trials, we might expect two sets in which "heads" should not appear till the tenth throw,' which is right, ' and one in which no such thing should take place till the eleventh,' which is also right. But it is because there will probably be four trials of which two only will probably give heads,' that we may expect two to give ' tails' yet once more. The two which gave ' heads ' are the two first mentioned by De Morgan, in which ' heads' appear at the tenth throw. Of the two remaining we expect one to give ' head,' the other ' tail.' The former is the one' next mentioned by De Morgan, in which ' head' appears at the eleventh throw. The other in which ' tail' may be expected to appear is the most valuable of all. Even if 'head ' appears at the next or twelfth tossing, this trial brings a prize worth twice as many pounds as the total number of trials--and therefore adds 21. to the average value of each trial. It is quite true that Buffon's experiment chances to give a result even less than De Morgan's value, and still further therefore from mine. But as will be seen, the other experiment gave an average result above his estimate, and even above mine. It cannot possibly be correct to omit all consideration of the most profitable trial of all.
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