We put a monkey in front of a typewriter. At each instant $n$, the monkey picks at random a letter from the alphabet ($26$ letters) and then writes the letter.What is the probability of the monkey writing the word BANANA at least once? What is the probability of the monkey writing the word BANANA an infinite number of times?
I have no idea how to solve this problem so I would appreciate any suggestions. Thanks in advance.
Best Answer
If we only consder the independent(!) possibilities of the wird BANANA occuring within the first six letters, or within the second six lettres (i.e.,m positions 7 to 12), or within the third six letters (13 to 18), or ..., then we note that it occurs aeady in each of these positions with a constant positive probability $p=\frac1{26^6}$. The probability that the word does not occur within the first $6n$ letters is therefore at most $(1-p)^n$. By letting $n\to \infty$, we see that the probability that BANANA does not occur at all in the infinite letter sequence is zero. By the same argument, for any fixed $m$, the probability that the word occurs at most $m$ times within the first $6n$ letters tends to $0$ as $n\to \infty$, i.e., the probability of at most $m$ occurrences in the infinite string is zero. Consequently, the probability of infinitely many occurrences is one.