Welcome to the first post from my new series. Here we’re going look at a famous probability question often called the birthday problem. This is actually a more general question related to the probability of at least one coincidence after a fixed number of draws from a discrete uniform distribution.
This post is part of my series Probability Questions from the Real World.
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In my previous two posts I sketched the frame of the big picture around probability distributions. In my introductory post I gave some intuition about the general concept and talked about the two major kinds: discrete and continuous distributions. And in the follow-up post I related the concepts of mean and variance to probability distributions. I showed that this connection itself goes through two fundamental concepts from probability theory: the law of large numbers and expected value.
Now I want to build on all these posts. My plan is to start introducing commonly used discrete and continuous distributions in separate posts dedicated to each one. And I want to start with the former, since they are significantly easier to understand.
The goal of the current post is to be a final warm-up before delving into the details of specific distributions.
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If you want to take your understanding of probabilities to the next level, it’s crucial to be familiar with the concept of a probability distribution.
In short, a probability distribution is an assignment of probabilities or probability densities to all possible outcomes of a random variable.
For example, take the random process of flipping a regular coin. The outcome of each flip is a random variable with a probability distribution:
Depending on the type of random variable you’re working with, there are two general types of probability distributions: discrete and continuous. In this post, I’m going to give an overview of both kinds. And in follow-up posts I’m going to individually introduce specific frequently used probability distributions from each kind.
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What does a probability of zero mean? When people use it in everyday conversations, a statement like “the probability of something is zero” usually implies that that something isn’t going to happen. Or that it is impossible to happen. Or that it will never happen.
There’s zero chance I’m passing this exam!
Is this true? Can we really say that zero probability events are impossible to occur? I’m going to show you that this is, in fact, false. You will see zero probability events are more than possible: they happen all the time.
In this post, I want to talk about conditional dependence and independence between events. This is an important concept in probability theory and a central concept for graphical models.
In my two-part post on Bayesian belief networks, I introduced an important type of graphical models. You can read Part 1 and Part 2 by following these links.
This is actually an informal continuation of the two Bayesian networks posts. Even though I initially wanted to include it at the end of Part 2, I decided it’s an important enough topic that deserves its own space.
In the first part of this post, I gave the basic intuition behind Bayesian belief networks (or just Bayesian networks) — what they are, what they’re used for, and how information is exchanged between their nodes.
In this post, I’m going to show the math underlying everything I talked about in the previous one. It’s going to be a bit more technical, but I’m going to try to give the intuition behind the relevant equations.
If you stick to the end, I promise you’ll get a much deeper understanding of Bayesian networks. To the point of actually being able to use them for real-world calculations.