Lecture 1. Basic concepts of probability theory
Probability theory and mathematical statistics are branches of science that deal with patterns that arise as a result of mass random phenomena. Historically, the first beginnings of probability theory appeared in the theory of gambling, when it was necessary to predict how a player should perform in a given situation.
Later, probability theory was widely used in the theory of insurance companies.In the modern world, probability theory and mathematical statistics are essential in almost all fields of knowledge.
The goal of probability theory is to formulate laws for mass phenomena and to make a forecast.
The main term of probability theory and mathematical statistics is the term of experiment that involves a certain reproduction of the set of conditions in which a particular outcome is recorded.
First, we are going to consider random events.
Before considering a random event, we need to define the term "an elementary outcome". The space of elementary outcomes is the set that contains all possible results of the experiment. We are going to use the term “elementary outcomes” for these results.
For example, the simplest experiment is to flip the correct coin.
When we say correct, we mean that it is a symmetrical coin and it is equal to the probability of falling on both sides.
In other words, there will be heads and tails in about half of the cases.
The elementary outcome in this case will be some side of the coin.
We assume that the coin will definitely fall, and it will not fall on the edge.
Then the set of elementary outcomes consists of two, in this case, equally probable outcomes, since the coin is symmetric.
If we consider a non-symmetrical coin, these outcomes will no longer be equally probable.
We'll need this later.
Example 2 can also be considered.
Next, we will consider some concepts using this example – tossing a dice.
When we toss the dice, we get from 1 to 6 points.
Thus, the set of elementary outcomes consists of six different outcomes.
On the slide, numbers from 1 to 6 indicate them.
Let us consider a more complex experiment. The box contains white, blue and red balls. The experiment consists of taking two balls out of this box. Then you must describe how the experimentis carried out.
Case 1.
You get a set of two balls at the same time. In this case, the set of elementary outcomes is just an unordered pair.
There are only three options: (white, blue), (white, red), and (blue, red).
If we remove the balls in turn, while putting them away, without returning them back to the box,then the order of these balls will be important to us.
Then the set of elementary outcomes will consist of the six elements listed on the slide.
This selection is a non-return selection.
If after we view the colour of the ball and put the ball back, the set of elementary outcomes will increase to 9.
We’ll have additional elementary outcomes, when we remove the white ball both times, the blue ball or the red one both times.
Many elementary outcomes are also shown on the slide.
Let's define the term "an event".
If an omega is a set of elementary outcomes of some experiment, then an event is a subset of the omega set.
In fact, if the omega set is more than countable, then not every subset of the omega set will be an event.
We'll talk about this later.
In the experiment with the toss of a dice, the Euler-Venn diagrams show the set of all elementary outcomes as a white rectangle.
The red oval shows the set A, consisting of elementary outcomes 2, 4 and 6.
We can describe this event as follows: an even number falls out. The set B (a green oval) shows that a prime number falls out. This is a subset of the omega set consisting of elementary outcomes 2, 3, and 5.
The blue colour shows that1falls out. The event Cis also represented; it is both an event and an elementary outcome.
Types of events. If we have an event consisting of all elementary outcomes, then such an event is called reliable. If the event is an empty subset, i.e. it does not contain any elementary outcome, then this event is called impossible and is denoted by the empty set symbol.
The remaining events are called random, i.e. they are a non-trivial subset of the set of elementary outcomes.
In the dice toss experiment, event A = "an even number of points" and event B = "a prime number of points" are random, event D = "a number less than 7 falls out" isreliable, and event E =" 0 falls out " is impossible, since 0 is not among the elementary outcomes considered earlier.
Let us give the concept of the relationship between events. Events are incompatible if they cannot occur simultaneously.
In other words, these two subsets do not intersect.
If a subset of these events has a common elementary outcome, then the events arecompatible.
In the figure, the red oval and the blue oval do not intersect, so event A="an even number falls" and event B= "1 falls out" are incompatible. The events "a prime number falls out" and "an even number falls out" are compatible, since we can have 2 on the dice, which is both an even and a prime number.
Another relationship that can be between events. «Entailing».
Event A entails event B if the occurrence of event A always guaranteesthe occurrence of event B.
That is, the set of elementary outcomes favorable to event A is a subset of the set of elementary outcomes favorable to event B.
In the figure, event A= "2 falls out" entails event B = "a prime number falls out". In the figure, they are indicated in blue and green colors, respectively.
Operations on events correspond to operations on sets.The main operations are sum, product, and negation.
The sum of two events A and B, or a combination, in other words, is event C that occurs if and only if at least one of the events A or B, or both together, occur.
The product of events A and B is event D that occurs if and only if both events A and B occur.
In the figure on the Euler-Venn diagram, the yellow color indicates the sum and product of two events, respectively.
Let me give some examples.
If we consider event A= "an even number falls out", and event B= "a prime number falls out", indicated in the figure, then event A + B consists ofelementary outcomes 2, 3, 4, 5, and 6, and event A*B consists of the elementary outcome 2.
D = "2 falls out»
There is another operation on events. This is a unary operation-the negation of event A or, in other words, the opposite event, denoted by A with a dash.
This addition of the subset of elementary outcomes favorable to event A to the set of all elementary outcomes is indicated in yellow in the figure.
As an example, we can consider the case of the dice toss again.
If event A = "an even number of points falls out", then event non-A = "an odd number of points falls out".
It will consist of elementary outcomes 1, 3, and 5.