# Theory of Probability and Basic Probability Rules

**Episode #1 of the course Theory of probability by Polina Durneva**

Hello!

My name is Polina Durneva, and during the next 10 days, I will be teaching you about the basics of the theory of probability. This is definitely an exciting topic, and I can’t wait to start our course!

We will start our first lesson with the most basic rule of probability that you might have heard of before, and then we will delve into a more complicated topic: mutually exclusive and not mutually exclusive events. We will continue our course by learning about discrete and continuous probability distributions that are used for different cases and problems. At the end of the course, you should be able to recognize various distributions and their applications in real and theoretical problems.

**What Is Probability?**

Let’s get started by defining probability. In basic terms, probability is the frequency of a particular event divided by the total number of all events. In the formula, we can express this as, *Probability = P(Event A) = Event A / All Events*. The formula shows how to calculate the probability of event A.

For instance, let’s assume that we have a bag that contains 10 candies and 15 rocks (what a weird combination, huh?). Our friend John wants to randomly pick a candy out of the given bag. What is the probability of success? (In our case, success refers to the event when John picks a candy without looking into the bag.) Well, the total number of all possible events is 10 + 15 = 25. We have 10 candies, so the probability of picking out a candy is 10/25 = 2/5 = 0.4 = 40%, meaning that most of the time, John will unfortunately not pick a candy out of the bag.

**Probability of Mutually Exclusive Events**

Now, let’s talk about the probability of mutually exclusive events, meaning that these events cannot happen simultaneously. The formula for the probability of mutually exclusive events takes a form of *Probability(Event A or Event B) = Probability(Event A) + Probability(Event B)*.

To better understand this idea, let’s use a simple example. Assume that you have a bag containing slips of paper with different numbers from 1 to 5, and you want to know the probability of drawing an even number. Since you draw only once, you first need to find the probabilities of drawing 2 and 4 (the only even numbers between 1 and 5). The probability of drawing 2 is 1/5 and the probability of drawing 4 is also 1/5. Therefore, the probability of drawing an even number is 1/5 + 1/5 = 2/5 = 0.4 = 40%.

**Probability of Not Mutually Exclusive Events**

But what about the probability of not mutually exclusive events? These are the events that can happen at the same time! The formula for two not mutually exclusive events takes the following form: *Probability(Event A or Event B) = Probability(Event A) + Probability(Event B) – Probability(Event A and Event B)*.

Once again, let’s use an example to understand the idea. Assume you teach at a local college and you have 50 students in your new class. You know that 20 of your students are business majors and 10 of your students are French majors. In addition, you know that 5 of your students are both business and French majors. What is the probability that you will select a student who is either a business major or a French major? Well, this would be 20/50 + 10/50 – 5/50 = 25/50 = 50%. Please note that we have to subtract the joint probability (Event A and Event B) in order to account for the overlap in our counting of French and business majors.

Well, that is it for today! Tomorrow, we will discuss the probability of dependent and independent events!

See you soon,

Polina

**Recommended book**

*Probability Theory: The Logic of Science* by E. T. Jaynes

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