So, one thing I encountered is called “Infinite monkey theorem”.
It says that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.
In fact the monkey would almost surely type every possible finite text an infinite number of times. However, the probability of a universe full of monkeys typing a complete work such as Shakespeare’s Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero).
At first it seemed complete utter BS, but it has a proof and when something’s proved, you have to digest it because proofs are convincing at times.
Suppose the typewriter has 50 keys, and the word to be typed is banana. If the keys are pressed randomly and independently, it means that each key has an equal chance of being pressed. Then, the chance that the first letter typed is ‘b’ is 1/50, and the chance that the second letter typed is a is also 1/50, and so on. Therefore, the chance of the first six letters spelling banana is
(1/50) × (1/50) × (1/50) × (1/50) × (1/50) × (1/50) = (1/50)^6
= 1/15 625 000 000 , less than one in 15 billion, but not zero, hence a possible outcome.
From the above, the chance of not typing banana in a given block of 6 letters is 1 − (1/50)^6. Because each block is typed independently, the chance Xn of not typing banana in any of the first n blocks of 6 letters is
Xn = (1–(1/50)^6)^n
As n grows, Xn gets smaller. For an n of a million, Xn is roughly 0.9999, but for an n of 10 billion Xn is roughly 0.53 and for an n of 100 billion it is roughly 0.0017. As n approaches infinity, the probability Xn approaches zero; that is, by making n large enough, Xn can be made as small as is desired and the chance of typing banana approaches 100%.