The basic concept of probability theory. The laws of probability theory

Many, confronted with the concept of "probability theory," are frightened, thinking that this is something beyond the capacity, very difficult. But everything is really not so tragic. Today we will consider the basic concept of probability theory, learn how to solve problems on specific examples.

The science

What is studying such a branch of mathematics as "probability theory"? She notes the patterns of random events and magnitudes. For the first time scientists interested in this issue in the eighteenth century, when they studied gambling. The basic concept of probability theory is an event. This is any fact that is stated by experience or observation. But what is experience? Another basic concept of probability theory. It means that this set of circumstances was created not accidentally, but with a specific purpose. As for observation, then the researcher himself does not participate in the experience, but simply is a witness of these events, he has no influence whatsoever.

Developments

We learned that the basic concept of probability theory is an event, but did not consider the classification. All of them fall into the following categories:

• Credible.
• Impossible.
• Random.

Regardless of what events are observed or created during the experiment, they are all subject to this classification. We suggest to get acquainted with each species separately.

Reliable event

This is such a circumstance, before which the necessary complex of measures is made. In order to better understand the essence, it is better to give a few examples. This law is subject to physics, chemistry, economics, and higher mathematics. The theory of probability includes such an important concept as a credible event. Let's give some examples:

• We work and receive remuneration in the form of wages.
• We passed well the exams, passed the competition, for this we receive a reward in the form of admission to the educational institution.
• We have invested money in the bank, if necessary, we will get them back.

Such events are reliable. If we have met all the necessary conditions, then we will definitely get the expected result.

Impossible events

Now we are considering the elements of probability theory. We propose to proceed to an explanation of the following type of event, namely, the impossible. First, let's discuss the most important rule - the probability of an impossible event is zero.

From this formulation it is impossible to retreat when solving problems. For an explanation, we give examples of such events:

• The water froze at a temperature of plus ten (this is impossible).
• The absence of electricity does not affect production in any way (it is just as impossible as in the previous example).

More examples should not be given, as described above very clearly reflect the essence of this category. An impossible event will never happen during an experiment under any circumstances.

Random events

Studying the elements of probability theory, special attention should be paid to this type of event. They are the ones that study this science. As a result of experience, something can happen or not. In addition, the test can be carried out an unlimited number of times. Strong examples are:

• A coin toss is an experience, or a test, an eagle is an event.
• Pulling the ball out of the bag blindly - a test, a red ball was caught - this is an event and so on.

Such examples can be an unlimited number, but, in general, the essence should be clear. To summarize and systematize the knowledge gained about events, a table is given. The probability theory studies only the last kind of all presented.

Laws

The theory of probability is a science that studies the possibility of the fall of an event. Like others, it has some rules. There are the following laws of probability theory:

• Convergence of sequences of random variables.
• Law of large numbers.

When calculating the possibility of a complex one, you can use a complex of simple events to achieve a result in an easier and faster way. We note that the laws of the theory of probability are easily proved by means of certain theorems. We suggest first to get acquainted with the first law.

Convergence of sequences of random variables

Note that there are several kinds of convergence:

• The sequence of random variables is convergent in probability.
• Almost impossible.
• Mean square convergence.
• Convergence in distribution.

So, on the fly, it's very difficult to get to the point. Here are the definitions that will help you understand this topic. To begin with, the first view. A sequence is said to be convergent in probability if the following condition is satisfied: n tends to infinity, the number to which the sequence tends is greater than zero and is close to unity.

We proceed to the next form, almost certainly . It is said that the sequence converges almost surely to a random variable when n tends to infinity, and P tends to a value close to unity.

The next type is the mean square convergence . With the use of CK-convergence, the study of vector random processes reduces to the study of their coordinate random processes.

The last type remains, let's take a brief look at it and go directly to solving problems. Convergence in distribution has another name - "weak", further explain why. Weak convergence is the convergence of the distribution functions at all points of continuity of the limiting distribution function.

Be sure to fulfill the promise: weak convergence differs from all the above in that a random variable is not defined on a probability space. This is possible because the condition is formed exclusively using the distribution functions.

Law of large numbers

Excellent theorists in the proof of this law will be theorems of probability theory, such as:

• Inequality of Chebyshev.
• Chebyshev's theorem.
• Generalized Chebyshev theorem.
• Markov's theorem.

If we consider all these theorems, then this question can be delayed by several dozen sheets. In our country, the main task is to apply probability theory in practice. We suggest you to do it right now. But before this we consider the axioms of probability theory, they will be the main assistants in solving problems.

Axioms

From the first, we already met when we were talking about an impossible event. Let's remember: the probability of an impossible event is zero. An example we brought very bright and memorable: snow fell at an air temperature of thirty degrees Celsius.

The second one sounds like this: a reliable event occurs with a probability equal to one. Now let us show how this can be written using the mathematical language: P (B) = 1.

Third: A random event can occur or not, but the possibility always varies from zero to one. The closer the value to unity, the greater the chance; If the value approaches zero, the probability is very small. We write this in mathematical language: 0

Let's consider the last, fourth axiom, which sounds like this: the probability of the sum of two events equals the sum of their probabilities. We write it in mathematical language: P (A + B) = P (A) + P (B).

The axioms of probability theory are the simplest rules that can be easily remembered. Let's try to solve some problems, relying on the knowledge already gained.

Lottery ticket

First, let's look at the simplest example - the lottery. Imagine that you bought one lottery ticket for luck. What is the probability that you will win at least twenty rubles? In total, a thousand tickets are involved in the circulation, one of which has a prize of five hundred rubles, ten for a hundred rubles, fifty for twenty rubles, and one hundred for five rubles. Problems in the theory of probability are based on finding a chance of success. Now together we will analyze the solution of the above task.

If we denote by the letter A the winnings of five hundred rubles, then the probability of loss of A will be 0.001. How did we get this? Just need the number of "lucky" tickets divided by the total number of them (in this case: 1/1000).

In - this is a gain of one hundred rubles, the probability will be 0.01. Now we acted on the same principle as in the past action (10/1000)

C - the winning is equal to twenty rubles. We find the probability, it is equal to 0.05.

The rest of the tickets are not of interest to us, since their prize fund is less than the one set in the condition. Apply the fourth axiom: The probability of winning at least twenty rubles is P (A) + P (B) + P (C). The letter P denotes the probability of the origin of this event, we have already found them in previous actions. It remains only to add the necessary data, in the answer we get 0,061. This number will be the answer to the question of the assignment.

Card deck

Problems in the theory of probability are more complex, for example we take the following task. Before you is a deck of thirty-six cards. Your task is to draw two cards in a row without mixing the stack, the first and second cards must be aces, the suit does not matter.

First, let's find the probability that the first card will be an ace, for this we divide four by thirty-six. They set it aside. We get the second card, it will be an ace with a probability of three thirty-fifths. The probability of the second event depends on which card we pulled first, we wonder if this was an ace or not. From this it follows that the event B depends on the event A.

The next action is the probability of simultaneous implementation, that is, we multiply A and B. Their product is as follows: the probability of one event is multiplied by the conditional probability of the other, which we calculate, assuming that the first event occurred, that is, the first card we extended the ace.

In order for everything to become clear, we will give a designation to an element such as the conditional probability of an event. It is calculated, assuming that event A has occurred. Calculated as follows: P (B / A).

We continue the solution of our problem: P (A _* B) = P (A) _* P (B / A) or P (A _* B) = P (B) _* P (A / B). The probability is (4/36) _* ((3/35) / (4/36) .Calculate, rounding up to hundredths. We have: 0,11 _* (0,09 / 0,11) = 0,11 _* 0, 82 = 0.09 The probability that we will draw two aces in a row is nine-nine, the value is very small, it follows that the probability of the origin of the event is extremely small.

Forgotten number

We offer to analyze several more variants of the tasks that probability theory studies. You have already seen examples of the solution of some of them in this article, we will try to solve the following problem: the boy forgot the last digit of his friend's phone number, but since the call was very important, he began to type everything in turn. We need to calculate the probability that he will call no more than three times. The solution of the problem is the simplest if the rules, laws, and axioms of probability theory are known.

Before you look at the solution, try to solve it yourself. We know that the last figure can be from zero to nine, that is, only ten values. The probability to type the desired one is 1/10.

Next, we need to consider the variants of the origin of the event, suppose that the boy guessed and immediately typed in the right one, the probability of such an event is 1/10. The second option: the first miss miss, and the second one on the target. Calculate the probability of such an event: 9/10 multiplied by 1/9, in the end we get also 1/10. The third option: the first and second call were not at the address, only the third boy got where he wanted. We calculate the probability of such an event: 9/10 multiplied by 8/9 and by 1/8, we get 1/10 as a result. Other variants do not interest us on the condition of the problem, so we have to add up the results, in the end we have 3/10. Answer: the probability that the boy will call no more than three times is 0.3.

Cards with numbers

Before you are nine cards, each with a number from one to nine, the numbers do not repeat. They were put in a box and thoroughly mixed. You need to calculate the probability that

• There will be an even number;
• Two-valued.

Before moving on to the solution, let's say that m is the number of successful cases, and n is the total number of options. Let us find the probability that the number will be even. It will not be difficult to calculate that even numbers are four, this will be our m, there are probably nine variants, that is, m = 9. Then the probability is 0.44 or 4/9.

Consider the second case: the number of options is nine, and there can not be any successful outcomes, that is, m equals zero. The probability that an elongated card will contain a two-digit number is also zero.