Probabilistic World

Calculating Compound Event Probabilities

Posted on Written by 5 Comments

Venn diagram of three eventsYou can think of probabilities as measures of uncertainty in the occurrence of an event, the truth of a hypothesis, and so on.

These measures are numbers between 0 and 1. Zero means the event is impossible to occur and 1 means the event is certain to occur.

If you have many events of interest, you can measure their probabilities separately, but you can also measure probabilities of different combinations of these events.

Say you’re following the national soccer championships of England, Spain, and Italy. You want to calculate the probabilities of Arsenal, Barcelona, and Juventus becoming national champions next season. These probabilities are:

  • P(Arsenal)
  • P(Barcelona)
  • P(Juventus)

But what if you want to calculate the probability of both Arsenal and Barcelona becoming champions? Or the probability that at least one of the three teams does?

In this post, I’m going to show how probabilities of such combinations of events are calculated. I’m going to give the general formulas, as well as the intuition behind them. To do that, I’m first going to introduce a few relevant concepts from probability theory.

[Read more…]

Probability: What Is It, Really?

Posted on Written by 12 Comments

A ruler, a pen and a calculator on a notebook.Throughout history, we have come up with better and more accurate ways to measure physical quantities like time, length, mass, and temperature. This has been crucial for our scientific and technological development.

Each of these quantities has a precise definition and is informative about some aspect of the current state of the physical world. For example, the mass of an object can tell you how much work is necessary to lift it at a certain height. The outside air temperature determines the kind of clothes you would wear when you go out. And so on.

Probabilities are also quantities that measure something — they have a very precise and unambiguous mathematical definition. But still, they don’t relate to things in the physical world as straightforwardly and as intuitively as measures like mass and length.

[Read more…]

Bayes’ Theorem: An Informal Derivation

Posted on Written by 12 Comments

A man, a dog, a cat, and a hamster staring outside from a high building's window with Bayes' theorem formula in the foreground

If you’re reading this post, I’ll assume you are familiar with Bayes’ theorem. If not, take a look at my introductory post on the topic.

Here I’m going to explore the intuitive origins of the theorem. I’m sure that after reading this post you’ll have a good feeling for where the theorem comes from. I’m also sure you will find the simplicity of its mathematical derivation impressive. For that, some familiarity with sample spaces (which I discussed in this post) would come in handy.

So, what does Bayes’ theorem state again?

[Read more…]

What Is a Sample Space?

Posted on Written by Leave a Comment

A photograph of two white and three brown large standard dice.The concept of a sample space is fundamental to probability theory. It is the set of all possibilities (or possible outcomes) of some uncertain process.

For example, the sample space of the process of flipping a coin is a set with 2 elements. Each represents one of the two possible outcomes: “heads” and “tails”. The sample space of rolling a die is a set with 6 elements and each represents one of the six sides of the die. And so on. Other terms you may come across are event space and possibility space.

Before getting to the details of sample spaces, I first want to properly define the concept of probabilities. [Read more…]