Compounding Probabilities

A monkey hitting keys at random on a computer for an infinite amount of time will eventually write all the Harry Potter books. In fact, this monkey will eventually write every book that has or ever will be written. This may sound absurd at first but it is mathematically correct and it becomes intuitive once we break down the odds.

What are the chances that the monkey will type the letter ‘a’? Imagine if the keyboard only has 40 keys, the chances of typing the letter ‘a’ is 1/40 or 2.5%. Not terribly bad odds for a monkey typing at random.

Now, what are the chances that the monkey will type ‘apple’? Since the monkey types at random, there is an equal chance of striking any key and the probability of typing ‘apple’ is:

1/40 * 1/40 * 1/40 * 1/40 * 1/40 = 1/405 = 1 in a 100 million odds

Odds of one to a hundred million are not great, but still possible since it’s higher than zero.

The probability that the monkey will not type ‘apple’ is:

(1 – 1/405)n

Where n is the number of times the monkey can type ‘apple’. If the monkey is typing at random for infinity, n approaches infinity and the probability of not typing ‘apple’ approaches zero. In plain English, this means that given enough time, low probability events can compound to become a certainty.

In investing, understanding how low probabilities can compound to become near certainties is important in avoiding losses. Imagine two companies – Company A and Company B. In any given year, Company A has a 10% chance of going bankrupt while Company B has a 1% chance of going bankrupt.

In any one year, there isn’t much of a difference between 1% or 10% probability of bankruptcy. Even though a 10% chance of bankruptcy is an order of magnitude larger than a 1% chance, they are both less than 50% therefore bankruptcy is equally unlikely.

Over a 20 year period, the probabilities compound. The odds of Company A going bankrupt rises to 88%. For company B, the odds of bankruptcy rise to 18%. There is a big difference between 88% and 18% bankruptcy odds.

Over a 50 year period, Company A is 99.5% likely to go bankrupt while Company B is approximately 40% likely to go bankrupt. A 99.5% chance of going bankrupt is a near certainty. This shows how low probability events compound over time and increase the risk of a loss.

Why is this useful? In investing, avoiding losers is just as important as picking winners and understanding how probabilities can compound is a good way to avoid losers.

 

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