Infinite Monkeys and Literature
Imagine there're infinitely many monkeys typing on infinitely many typewriters.
If monkeys type at , at which time point is the word almost guaranteed to be typed on infinitely many different scripts?
Give your answer in seconds ( ).
The answer is 1.
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1 solution
The word L I T E R A T U R E contains 10 characters, which can be typed by a monkey in 1 second.
Assuming a typical typewriter has 50 keys, when there is only one monkey, the possibility for it to type L I T E R A T U R E in 1 second is
( 5 0 1 ) 1 0 = 9 7 6 5 6 2 5 0 0 0 0 0 0 0 0 0 0 1 .
The probability of the monkey not typing L I T E R A T U R E in 1 second is
1 − 9 7 6 5 6 2 5 0 0 0 0 0 0 0 0 0 0 1 .
We can conclude that, in this case, The chance of not typing L I T E R A T U R E for n monkeys in 1 second is
As n grows, X n gets smaller.As n approaches infinity, the probability X n approaches zero.
Therefore, when there's infinitely many monkeys, it is almost surely that an infinite amount of L I T E R A T U R E will be typed in 1 second , so are the other possible combinations of 10-character words.