Problem on the theory of probability with a solution. Theory of probability for dummies

The course of mathematics prepares students a lot of surprises, one of which is a problem in probability theory. With the solution of similar assignments, students have a problem in almost a hundred percent of cases. To understand and understand this issue, you need to know the basic rules, axioms, definitions. To understand the text in the book, you need to know all the abbreviations. All this we offer to learn.

Science and its application

Since we offer an accelerated course "probability theory for teapots", then first we need to introduce the basic concepts and letter abbreviations. First, let's define the concept of "probability theory". What is this science and what is it for? The theory of probability is one of the branches of mathematics that studies random phenomena and quantities. She also considers the patterns, properties and operations performed with these random variables. What is it for? Science has gained wide acceptance in the study of natural phenomena. Any natural and physical processes can not do without the presence of chance. Even if the results were as accurate as possible during the experiment, if the same test is repeated, the result with a high probability will not be the same.

Examples of problems in the theory of probability, we will necessarily consider, you can see for yourself. The outcome depends on many different factors that are almost impossible to take into account or register, but nevertheless they exert a tremendous influence on the outcome of the experiment. Strong examples are the tasks of determining the trajectory of the motion of the planets or determining the weather forecast, the probability of meeting a familiar person during the journey to work and determining the height of the jump athlete. Similarly, probability theory is of great help to brokers on stock exchanges. The problem of probability theory, which has faced many problems before, will become a trivial matter for you after three or four of the examples below.

Developments

As mentioned earlier, science is studying events. The theory of probability, examples of solving problems, we will consider a little later, only one species is studied - random ones. But nevertheless it is necessary to know that events can be of three kinds:

• Impossible.
• Credible.
• Random.

We suggest a little to stipulate each of them. An impossible event will never happen, under no circumstances. Examples are: freezing of water at plus temperature, drawing a cube from a bag with balls.

A reliable event always happens with an absolute guarantee, if all the conditions are met. For example: you received a salary for the work done, received a diploma of higher professional education, if you have honesty studied, passed exams and defended a diploma and so on.

With random events, everything is a little more complicated: during the experiment it can happen or not, for example, pull an ace from the card deck, making no more than three attempts. The result can be obtained both from the first attempt, and, in general, not to receive. It is the probability of the origin of the event that science is studying.

Probability

This is in general terms an assessment of the possibility of a successful outcome of an experience in which an event occurs. Probability is assessed on a qualitative level, especially if quantification is impossible or difficult. The problem of probability theory with a solution, more accurately with an estimate of the probability of an event, implies finding the most possible part of a successful outcome. Probability in mathematics is the numerical characteristics of an event. It takes values from zero to one and is denoted by the letter P. If P is equal to zero, then the event can not occur, if to one, then the event will happen with a 100% probability. The more P approaches unity, the stronger the probability of a successful outcome, and vice versa, if close to zero, then the event will occur with a low probability.

Abbreviations

The problem of probability theory, the solution of which you will soon encounter, may contain the following abbreviations:

• !;
• {};
• N;
• P and P (X);
• A, B, C, etc.;
• N;
• M.

Possible and some others: additional explanations will be added as necessary. We suggest, to begin with, to clarify the abbreviations presented above. The first on our list is the factorial. In order to be clear, let us give some examples: 5! = 1 * 2 * 3 * 4 * 5 or 3! = 1 * 2 * 3. Further, in the curly braces write the given sets, for example: {1; 2; 3; 4; ..; n} or {10; 140; 400; 562}. The next notation is a set of natural numbers, quite often found in assignments in probability theory. As we said earlier, P is probability, and P (X) is the probability of occurrence of event X. Large letters of the Latin alphabet denote events, for example: A - a white ball has fallen, B - blue, C - red or, accordingly,. The small letter n is the number of all possible outcomes, and m is the number of successful ones. Hence we obtain the rule for finding the classical probability in elementary problems: P = m / n. Probability theory "for teapots" is probably limited to this knowledge. Now for fixing, we turn to the solution.

Problem 1. Combinatorics

The student group consists of thirty people, of which it is necessary to choose the elder, his deputy and the trade union. It is necessary to find the number of ways to do this action. A similar task can meet at the USE. The probability theory, whose solution of problems we are considering now, can include problems from the combinatorics course, finding the classical probability, the geometric probability, and the problem of the basic formulas. In this example, we solve a task from a combinatorial course. We now turn to the solution. This task is the simplest:

1. N1 = 30 - possible headmistress of the student group;
2. N2 = 29 - those who can take the post of deputy;
3. N3 = 28 people apply for the position of trade union.

All we have to do is find a possible number of options, that is, multiply all the indicators. As a result, we get: 30 * 29 * 28 = 24360.

This will be the answer to the question posed.

Problem 2. Permutation

At the conference there are 6 participants, the order is determined by the drawing of lots. We need to find the number of possible options for the draw. In this example, we are considering a permutation of six elements, that is, we need to find 6!

In the abbreviation, we have already mentioned what it is and how it is calculated. In total, it turns out that there are 720 options for the draw. At first glance, a difficult task has a very short and simple solution. These are the tasks that are considered by probability theory. How to solve problems of a higher level, we will consider in the following examples.

Task 3

A group of twenty-five students must be divided into three subgroups of six, nine and ten people. We have: n = 25, k = 3, n1 = 6, n2 = 9, n3 = 10. It remains to substitute the values in the desired formula, we get: N25 (6,9,10). After simple calculations, we get the answer - 16 360 143 800. If the task does not say that you need to get a numerical solution, you can give it in the form of factorials.

Task 4

Three people wanted numbers from one to ten. Find the probability that someone will have the same number. First we need to know the number of all outcomes - in our case this is a thousand, that is, ten in the third degree. Now we find the number of options, when all the guesses of different numbers, for this we multiply ten, nine and eight. Where did these numbers come from? The first guesses the number, it has ten options, the second has nine, and the third one has to choose from the eight remaining ones, so we get 720 possible variants. As we have already calculated, all the variants of 1000, and without repetitions of 720, therefore, we are interested in the remaining 280. Now we need a formula for finding the classical probability: P =. We received the answer: 0.28.