無限猴子定理

（重定向自无限猴子定理

证明

直接证明

(1/50)×(1/50)×(1/50)×(1/50)×(1/50)×(1/50) =(1/50)6

${\displaystyle X_{n}=\left(1-{\frac {1}{50^{6}}}\right)^{n}\,}$

无限长的字符串

• 给定一个无限长的字符串，其中的每一个字符都是随机产生的，那么任意有限的字符串都会作为一个子字符串出现在其中（事实上要出现无限多次）。
• 给定一个序列，其中有无限多个无限长的字符串，其中每一个字符串中的每一个字符都是随机产生的，那么任意有限的字符串都会出现在其中某些字符串的开头（事实上是无限多个字符串的开头）。

${\displaystyle \sum _{i=1}^{\infty }P(E_{k})=\sum _{i=1}^{\infty }p=\infty ,}$

注釋

1. ^ 这表明在给定的六个字母中键入“香蕉”的可能性趋于1。
2. ^ The first theorem is proven by a similar if more indirect route in Gut, Allan. Probability: A Graduate Course. Springer. 2005: 97–100. ISBN 0387228330.
3. ^ Using the Hamlet text from gutenberg, there are 132680 alphabetical letters and 199749 characters overall.
4. ^ For any required string of 130,000 letters from the set a-z, the average number of letters that needs to be typed until the string appears is (rounded) 3.4 × 10183,946, except in the case that all letters of the required string are equal, in which case the value is about 4% more, 3.6 × 10183,946. In that case failure to have the correct string starting from a particular position reduces with about 4% the probability of a correct string starting from the next position（i.e., for overlapping positions the events of having the correct string are not independent; in this case there is a positive correlation between the two successes, so the chance of success after a failure is smaller than the chance of success in general）. The figure 3.4 × 10183,946 is derived from n = 26130000 by taking the logarithm of both sides: log10(n) = 1300000×log10(26) = 183946.5352, therefore n = 100.5352 × 10183946 = 3.429 × 10183946.
5. ^ 26 letters ×2 for capitalisation, 12 for punctuation characters = 64, 199749×log10(64) = 4.4 × 10360,783.

参考文献

1. ^ Émile Borel. Mécanique Statistique et Irréversibilité. J. Phys. (Paris). Series 5. 1913, 3: 189–196.
2. ^ Isaac, Richard E. The Pleasures of Probability. Springer. 1995: 48–50. ISBN 038794415X. Isaac generalizes this argument immediately to variable text and alphabet size; the common main conclusion is on p. 50.
3. ^ Kittel, Charles; Kroemer, Herbert. Thermal Physics 2nd. W. H. Freeman Company. 1980: 53. ISBN 0-7167-1088-9.
4. ^ Notes Towards the Complete Works of Shakespeare (PDF). [2019-04-02]. 原始内容存档于2013-01-20.
5. ^ No words to describe monkeys' play. [2017-08-01]. （原始内容存档于2019-01-21）.
6. ^ A Few More Million Amazonian Monkeys | Jesse Anderson. [2020-02-16]. （原始内容存档于2021-04-02） （美国英语）.