The author expects you to have a profound knowledge of linear algebra, analytic geometry, and calculus.

Forewarned is forearmed.

The purpose of probability theory is to build up a mathematical framework for various random experiments. Yeah, there will be a bunch of formulas to construct all that mathematical grounds.

**A random experiment** is a procedure whose collection of every possible outcome can be described before its performance and can be repeated under the same conditions.

**A sample** is an outcome of a random experiment. **A sample space** is a collection of samples:

**An event** is a subset of a sample space. **An event space** is a collection of events, which complies with the sigma-algebra axioms:

I. an event space contains an entire sample space:

II. an event space is closed under complementation:

III. an event space is closed under countable unions:

**Mutually exclusive events **are those which can’t happen at the same time:

**Collectively exhaustive events **are those the union of which is the sample space:

**A partition** of an event space is such events which are collectively exhaustive and mutually exclusive simultaneously:

**A probability measure** is a function from an event space into the real numbers, which complies with Kolmogorov’s axioms below:

I. the probability measure of an entire sample space is equal to 1:

II. the probability measure of an event is greater than or equal to 0:

III. the probability measure of a union of mutually exclusive events is equal to the sum of probabilities of events:

There are some properties you need to keep in mind as a result of the axioms above:

I. the sum of the probability of an event and the probability of its complement is equal to 1:

II. the probability of a null event is equal to 0:

III. the probability of an event is greater than or equal to the probability of its subevent and less than or equal to the probability of its superevent:

IV. the probability of an event is greater than or equal to 0 and less than or equal to 1:

V. the probability of a union of two events is equal to the sum of the probability of the first event and the probability of the second, subtracting the probability of their intersection:

VI. the probability of a difference between two events is equal to the difference between the probability of the first event and the probability of their intersection:

**A probability space** is a triplet, consisting of a sample space, an event space, and a probability measure:

There are two types of probabilities:

I. **A classical probability** of an event (a discrete case) is the number of samples in that event divided into the total number of samples:

II. **A geometric probability** of an event (a continuous case) is the area of samples in that event divided into the total area of samples:

Let’s utilize that knowledge and apply that framework to some examples from real life.

**A discrete example.** We cast dice. The question is: what are the probabilities of a) both equal and even; b) both equal and odd; c) their union; d) their intersection?

In this case, casting dice is a random experiment, which means we should define a probability space:

so that we can calculate the probabilities:

**A continuous example.** We toss darts. The question is: what are the probabilities of the dart landing in the a) red circle (0 < r ≤ 2); b) blue circle (2 < r ≤ 4)?

In this case, tossing darts is a random experiment, which means we should define a probability space:

so that we can calculate the probabilities: