### Ending the Monkey Typewriter Problem

In this post I will show the last results concerning this problem while also showing that there is a way to stop any Monte Carlo simulation from doing useless runs.

Coding things

In this post I will show the last results concerning this problem while also showing that there is a way to stop any Monte Carlo simulation from doing useless runs.

I began this blog with two articles on the Monkey Typewriter Theorem and, in the last one, I’ve said that there should be a possibility to stop the Monte Carlo simulation at exactly the needed point (when the required precision has been attained).

In the previous article I presented a strange problem, dubbed the MT problem (the Monkey typewriter). This time we will deal with the same problem but from another point of view: the results obtained by Monte Carlo simulation.I will post here the two source files needed for simulating both cases and then I will give some results.

I bet all of you know the now famous theorem of the infinite monkeys. Starting from this idea someone devised the following problem:

Suppose there is a monkey sitting in front of a terminal consisting ofÂ n keys. The monkey will press any of those keys at random and, when all of them will be pressed, she will receive a banana as a payment for her efforts. We are mainly interested in how long will it take for her to obtain that banana in the following two cases:

- when a key is pressed it remains pressed
- when a key is pressed for the second time it will behave as though it was not pressed at all.
Practically, we are interested to see what is the average number of keystrokes for the specific number n of keys.

This is what we’ll be doing here. For the first part of our instalment we will use Numerical Methods to deal with the exact mathematical equation for the average number of keystrokes needed.

The second instalment of this problem will tackle with a Monte-Carlo approach to this problem and (maybe) with reducing the computational cost of Monte-Carlo methods, the main reason behind choosing this problem.