Infinite Monkey Theorem

Thought Experiment

The Infinite Monkey Theorem is a fascinating concept in probability theory. It posits that a monkey hitting keys at random on a typewriter for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare¹.

Here’s a bit more detail:

• Origin: The idea dates back to Émile Borel, a French mathematician, in 1913¹.
• Concept: The theorem uses the monkey as a metaphor for an abstract device that produces an endless random sequence of letters and symbols. Given infinite time, the monkey would almost surely type every possible finite text an infinite number of times¹.
• Mathematical Basis: The term “almost surely” is a mathematical term meaning the event happens with probability 1¹.

It’s a thought experiment often used to illustrate the concepts of randomness and probability. Despite its theoretical nature, it’s a great way to understand how random processes can lead to seemingly impossible outcomes given enough time.

 

Example of the Infinite Monkey Theorem

 

Let’s imagine a simplified example to illustrate the Infinite Monkey Theorem:

Suppose we have a monkey randomly pressing keys on a typewriter that only has the letters A, B, and C. The monkey types one letter per second. We want to see if the monkey can type the sequence “ABC”.

1.     Probability of Typing “A”: The chance of the monkey typing “A” at any given second is (\frac{1}{3}).

2.     Probability of Typing “AB”: The chance of typing “A” followed by “B” is (\frac{1}{3} \times \frac{1}{3} = \frac{1}{9}).

3.     Probability of Typing “ABC”: The chance of typing “A”, then “B”, then “C” is (\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}).

While (\frac{1}{27}) might seem small, if the monkey types for an infinite amount of time, it will eventually type “ABC” an infinite number of times. This is because, given infinite time, every possible sequence of letters will occur.

This example simplifies the concept, but it captures the essence of the theorem: given enough time, random processes can produce any specific outcome, no matter how improbable it seems.

 

** The expression (\frac{1}{3}) represents a fraction where:

• The numerator (the top number) is 1.
• The denominator (the bottom number) is 3.

This fraction means that you have one part out of three equal parts. In decimal form, (\frac{1}{3}) is approximately 0.3333, repeating indefinitely.

For example, if you have a pizza and you divide it into three equal slices, each slice represents (\frac{1}{3}) of the whole pizza.