Diversity and Mechanisms in Swarm Intelligence

2. Infinite Monkey Theorem And Swarm Intelligence

There is a well-known thought experiment, called the infinite monkey theorem, which states that the probability of producing any given text will almost surely be one if an infinite number of monkeys randomly type for an infinitely long time (Marsaglia, 1993; Gut, 2005). In other words, the infinite monkeys can be expected to reproduce the whole works of Shakespeare. For example, to reproduce the text “algorithm” (9 characters), for a random sequence of n characters on a 101-key computer keyboard, the probability of a consecutive 9-character random string to be “algorithm” is p=(1/101)⁹ ≈ 8.4 × 10^-19, which is extremely small. However, the importance here is this probability is not zero. Therefore, for an infinitely long sequence n→∞, the probability of reproducing the collected works of Shakespeare is one, though the formal rigorous mathematical analysis requires Borel-Cantelli’s lemma (Marsaglia & Zaman, 1993; Prokhorov, 2002; Gut, 2005).