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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Newton–Pepys problem ： ウィキペディア英語版 Newton–Pepys problem The Newton–Pepys problem is a probability problem concerning the probability of throwing sixes from a certain number of dice. In 1693 Samuel Pepys and Isaac Newton corresponded over a problem posed by Pepys in relation to a wager he planned to make. The problem was: :''Which of the following three propositions has the greatest chance of success?'' ::''A. Six fair dice are tossed independently and at least one “6” appears.'' ::''B. Twelve fair dice are tossed independently and at least two “6”s appear.'' ::''C. Eighteen fair dice are tossed independently and at least three “6”s appear.''〔(Isaac Newton as a Probabilist ), Stephen Stigler, University of Chicago〕 Pepys initially thought that outcome C had the highest probability, but Newton correctly concluded that outcome A actually has the highest probability. ==Solution== The probabilities of outcomes A, B and C are:〔 :$P(A)=1-\backslash left(\backslash frac\backslash right)^\; =\; \backslash frac\; \backslash approx\; 0.6651\backslash ,\; ,$ :$P(B)=1-\backslash sum\_^1\backslash binom\backslash left(\backslash frac\backslash right)^x\backslash left(\backslash frac\backslash right)^$ = \frac \approx 0.6187\, , :$P(C)=1-\backslash sum\_^2\backslash binom\backslash left(\backslash frac\backslash right)^x\backslash left(\backslash frac\backslash right)^$ = \frac \approx 0.5973\, . These results may be obtained by applying the binomial distribution (although Newton obtained them from first principles). In general, if P(''N'') is the probability of throwing at least ''n'' sixes with 6''n'' dice, then: :$P(N)=1-\backslash sum\_^\backslash binom\backslash left(\backslash frac\backslash right)^x\backslash left(\backslash frac\backslash right)^\backslash ,\; .$ As ''n'' grows, P(''N'') decreases monotonically towards an asymptotic limit of 1/2. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Newton–Pepys problem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.