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Magnitude and Minuteness—A Crux in Calculation Involving the Use of Binomial Series—Arising Out of a Problem In Probability.*
Published online by Cambridge University Press: 03 November 2016
Extract
This is a clumsily long title, but it might have been much more lengthy if it had gone on to suggest how the problem in probability arose. That would involve us in many questions, so we may say at once that the problem is equivalent to the following: If from a bag containing M different balls N drawings are made, where M, N are very large numbers (the ball being replaced after each drawing), what is approximately the chance that exactly r balls should fail to appear?
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- Copyright © Mathematical Association 1929
Footnotes
*
The substance of a paper read to the North-Eastern Branch of the Mathematical Association on May 4th, 1929.
References
page 567 note † This case is one of a set for which M = 3 m , N = 4 m -1; but one to which special interest attached, both in itself and as perhaps the first to which the following approximation could be successfully applied,
page 570 note * A difficulty may be felt from the fact that the expression for the error is appropriate only when r + s is inconsiderable. The answer is that the terms rapidly become so small that they may be left out of account altogether. We only need in effect the terms from s=0 to s=s’, where s’ is quite moderate, say 160 in the case in point. So in summing for the errors we may stop at the 160th term; but there again the sum from 0 to s’ will not be appreciably different from the sum from 0 to M’ which gives the simple result above.
page 570 note † A like method applies to the less important errors expressed by subsequent terms in the expansion of {1 – (r + s)/M} μM (§ 5); only they involve higher powers of r. Thus for the next in importance we have a cubic in r, with more complicated maxima and minima, but the general effect will be very much less.