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accidental errors annuity approximate assume Banker Bernoulli Bernoulli's theorem Bertrand best value Buffon's needle called centre chapter coefficients coin constant Czuber definition discrepancy dispersion ellipse event Example expectation expression favourable formula function fundamental give given greater happen Hence increases indefinitely independent variables infinite number infinitesimal integral large number law of error less limit Maclaurin series mathematical mathematical induction mean error mean value measurements method multiplied mutually exclusive mutually exclusive events normal equations number of molecules number of successes number of trials observed values oval partial derivatives Peirce's Criterion Player Player's chance premium priori probability prob probability for success problem proportional quantity random ratio region replace residual square stake statistical Stirling's formula Suppose takes the value total number true value turn weighted mean white balls zero