What are the odds of 6 monkeys on typewriters typing out the opening sentence of The Hobbit?

Question

Suppose you have 6 monkeys typing random letters on keyboards. Each monkey can make 10¹⁷ attempts. These keyboards have 48 keys (no arrow keys, caps lock, tab, shift key, or other special keys.) You want to see if the monkeys will be able to randomly type the opening sentence form The Hobbit, which is 45 key strokes. What are their chances of success?

Answer 1

essentially zero

so small it's approaching 1/infinity = 0

similar to Creationists' statistical arguments that evolution by natural selection and mutation is virtually impossible, and super unlikely to have ever happened. But response is the Sherlock Holmes' argument, if you eliminate every other alternative, then what's left is true, no matter how super unlikely.

Answer 2

To determine the monkeys' chances of typing the opening sentence of The Hobbit randomly, let’s break the problem into smaller steps.

Step 1: Total possible combinations

The opening sentence is 45 characters long, and each character can be one of 48 possible keys. The total number of possible combinations of 45 keystrokes is:

48⁴⁵. This equals approximately 5.27 × 10⁷⁵.

Step 2: Probability of success in one attempt

The probability of typing the correct sequence in one attempt is the reciprocal of the total number of combinations:

1/48⁴⁵, or approximately 1.9 × 10^-76.

Step 3: Total attempts by the monkeys

Each monkey can make 10¹⁷ attempts, and there are 6 monkeys. Therefore, the total number of attempts is:

6 × 10¹⁷

Step 4: Probability of failure in one attempt

The probability of failing to type the correct sequence in one attempt is:

1 - (1.9 × 10^-76), which is approximately 1 since the success probability is so tiny.

Step 5: Probability of failing all attempts

If the probability of failure in one attempt is very close to 1, the probability of failure across all attempts is calculated as:

(Probability of failing in one attempt) raised to the power of the total number of attempts.

This equals (1 - 1.9 × 10^-76) raised to the power of 6 × 10¹⁷.

For practical purposes, this is still approximately 1 because even after 6 × 10¹⁷ attempts, the probability of success remains extremely small.

Step 6: Probability of at least one success

The probability of at least one success is:

1 - (Probability of failing all attempts).

Since the probability of failing all attempts is effectively 1, the probability of success is effectively 0.

Conclusion

Despite the 6 monkeys making 10¹⁷ attempts each, the chances of them randomly typing the opening sentence of The Hobbit are astronomically low—essentially zero. The vast number of possible combinations, 5.27 × 10⁷⁵, far exceeds the number of attempts they can make.