Coin flip Archives - Probabilistic World

Discrete Probability Distributions: An Overview

Posted on October 30, 2019 Written by The Cthaeh 2 Comments

A discrete probability distribution, two hands holding dice, and a background referencing the movie The Matrix

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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An Intuitive Explanation Of Expected Value

Posted on November 24, 2018 Written by The Cthaeh 7 Comments

Expected value is another central concept in probability theory. It is a measure of the “long-term average” of a random variable (random process). I know this doesn’t sound too clear, but in this post I’m going to explain exactly what it means.

There are many areas in which expected value is applied and it’s difficult to give a comprehensive list. It is used in a variety of calculations by natural scientists, data scientists, statisticians, investors, economists, financial institutions, and professional gamblers, to name just a few.

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The Law Of Large Numbers: Intuitive Introduction

Posted on November 13, 2018 Written by The Cthaeh 11 Comments

The law of large numbers is one of the most important theorems in probability theory. It states that, as a probabilistic process is repeated a large number of times, the relative frequencies of its possible outcomes will get closer and closer to their respective probabilities.

For example, flipping a regular coin many times results in approximately 50% heads and 50% tails frequency, since the probabilities of those outcomes are both 0.5.

The law of large numbers demonstrates and proves the fundamental relationship between the concepts of probability and frequency. In a way, it provides the bridge between probability theory and the real world.

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Probability: What Is It, Really?

Posted on April 8, 2016 Written by The Cthaeh 11 Comments

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.

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Coin Bias Calculation Using Bayes’ Theorem

Posted on March 21, 2016 Written by The Cthaeh 19 Comments

Why do people flip coins to resolve disputes? It usually happens when neither of two sides wants to compromise with the other about a particular decision. They choose the coin to be the unbiased agent that decides whose way things are going to go. The coin is an unbiased agent because the two possible outcomes of the flip (heads and tails) are equally likely to occur.

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Bayes’ Theorem: An Informal Derivation

Posted on February 28, 2016 Written by The Cthaeh 7 Comments

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?

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