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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1865 Excerpt: ...trial, 4 for head at the second, and so on; the sum which the player ought to give is 2o 1 + 2 (1--) + 2 (1-o) +... + 2-1 (1-w) -1, which we will call fl. D'Alembert suggests, if I understand him rightly, that if we know nothing about the value of a we may take as a solution of the problem, for the sum which the player ought to give I Slda. J 0 But this involves all the difficulty of the ordinary solution, for the result is infinite when n is. D'Alembert is however very obscure here; see his pages 45, 46. He seems to say that I lda will be greater than, equal to, or 0 less than n, according as n is greater than, equal to, or less than 5. But this result is false; and the argument unintelligible or inconclusive. We may easily see by calculation that I fldm = n when J o n = 1; and that for any value of n from 2 to 6 inclusive i Idea is less than n; and that when n is 7 or any greater number 531. D'Alembert then proposes a method of solving the Petersburg Problem which shall avoid the infinite result; this method is perfectly arbitrary. He says, if tail has arrived at the first throw, let the chance that head arrives at the next be t, and not, where a is some small quantity; if tail has arrived at the first throw, and at the second, let the chance that head arrives at the next throw be---, and not; if tail has arrived at the first throw, at the second, and at the third, let the chance that head arrives at the next throw be--a + c an(j n0 . so on The quantities a, b, c, ... are supposed small positive quantities, and subjected to the limitation that their sum is less than unity, so that every chance may be less than unity. On this supposition if the game be as it is described in Art. 389, it may be shewn that A ought to give half of th...

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