Probability “1” Is Not Certainty

They sometimes say that if you try something very often then you will succeed. Say you play a game of dice and in every round there is a non-zero probability that you’ll win. As the number of rounds you play becomes larger and larger then you must, eventually, win – right?


Wrong. What you can say with certainty is that the probability that you will keep losing as you play many rounds becomes very small. That means that the probability that you will win at least once becomes very large. In the limit of the number of rounds going to infinity the probability of winning actually equals one. But that does not mean that winning is a certainty. It is possible, even after many rounds, that you’ll never win. It may be infinitely improbable, but it is not impossible.

Saying that something has a zero-probability of occurring is not the same as saying that it is impossible. Just think of throwing a dart at a dartboard. The probability of the dart hitting any specific point is zero (as there are infinitely many points on the board), and yet if you throw a dart at the dartboard you’ll always hit some point. Hitting a specific point at a dartboard is highly improbable, but not impossible.

What happens if you repeat a process that can have different outcomes (such as throwing a die) very often is described by Bernoulli’s Law of Large Numbers (LLN). There are actually two different laws that bear the name LNN: the strong and the weak LLN. In a future post I will explain the difference between the strong and weak versions of the LLN and also show how these laws are often misinterpreted in modern physics.

About fbenedictus

Philosopher of physics at Utrecht University, managing editor for Foundations of Physics and international paraclimbing athlete
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One Response to Probability “1” Is Not Certainty

  1. Ivan Flis says:

    There’s an interesting manifestation in this in how chronic gamblers perceive what they do. If they’re on a losing streak, there is this notion that “they have to win” after losing multiple times in a row. A basic misconstrual of probability that turns into a particular notion of luck that turns into a pervasive behavioral habit that turns into a pathology.

    Looking forward to the explanation of the weak and the strong LLN. I only know it as the law of large numbers, without qualifications.

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