Mathematics course prepares a lot of studentssurprises, one of which is the problem of probability theory. With the solution of such tasks, students have a problem in almost one 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 a crash course "theoryprobabilities for dummies ”, you first need to introduce basic concepts and letter abbreviations. To begin with, we will define the very concept of “probability theory”. What is this science and what is it for? Probability theory is one of the branches of mathematics that studies random phenomena and quantities. She also examines the patterns, properties and operations performed with these random variables. What is it for? Widespread science received in the study of natural phenomena. Any natural and physical processes are not complete without the presence of chance. Even if the results were recorded as accurately as possible during the experiment, if you repeat the same test, the result will most likely not be the same.
Examples of problems on probability theory wesure to consider, you can see for yourself. The outcome depends on many different factors, which are almost impossible to take into account or register, but nevertheless they have a tremendous impact on the outcome of the experiment. Vivid examples include the task of determining the trajectory of the planets or determining the weather forecast, the probability of meeting a familiar person while on the way to work, and determining the height of a jump in an athlete. Also, probability theory is a great help to brokers on the stock exchanges. The problem on the theory of probability, with the solution of which many problems had previously arisen, will become a mere trifle for you after three or four examples below.
Developments
As mentioned earlier, science studies events.Probability theory, examples of problem solving, we will look at a little later, studying only one type - random. But nevertheless it is necessary to know that events can be of three types:
- Impossible.
- Authentic
- Random.
We propose to discuss a little of each of them.An impossible event will never happen, under any circumstances. Examples include: freezing of water at a positive temperature, pulling the cube out of the bag with balls.
Reliable event occurs always withone hundred percent guarantee if all conditions are met. For example: you received a salary for the work done, received a diploma of higher vocational education, if you studied conscientiously, passed exams and defended a diploma, and so on.
With random events, it's a bit more complicated:during an experiment, it can happen or not, for example, to pull an ace out of a 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 and studying science.
Probability
This is a general assessment of the possibility of a successfulthe outcome of the experience at which the event occurs. Probability is assessed at a qualitative level, especially if quantification is impossible or difficult. The problem of probability theory with a solution, more precisely, with an estimate of the probability of an event, implies finding the same possible share of a successful outcome. Probability in mathematics is the numerical characteristics of an event. It takes values from zero to one, denoted by the letter P. If P equals zero, then the event cannot occur, if one, then the event will occur with one hundred percent probability. The more P approaches one, the greater the likelihood of a successful outcome, and vice versa, if it is 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:
- !
- {};
- H;
- P and P (X);
- A, B, C, and so on;
- n;
- m
Some others are possible:additional explanations will be made as necessary. We suggest, to begin with, to clarify the abbreviations presented above. First on our list is factorial. In order to be clear, we give examples: 5! = 1 * 2 * 3 * 4 * 5 or 3! = 1 * 2 * 3. Further, in curly brackets they write given sets, for example: {1; 2; 3; 4; ..; n} or {10; 140; 400; 562}. The following designation is a set of natural numbers, quite often found in tasks on the theory of probability. As mentioned earlier, P is the probability, and P (X) is the probability of the origin of the event X. The capital letters of the Latin alphabet denote events, for example: A - a white ball caught, B - blue, C - red or respectively,,. The small letter n is the number of all possible outcomes, and m is the number of successful. From here we get the rule for finding the classical probability in elementary problems: P = m / n. Probability theory "for dummies" is probably limited to this knowledge. Now for fixing proceed to the solution.
Problem 1. Combinatorics
The student group has thirty peoplefrom which it is necessary to choose the headman, his deputy and proforga. You need to find the number of ways to do this action. A similar task can meet on the exam. Probability theory, the solution of the problems of which we are now considering, may include problems from the course of combinatorics, finding the classical probability, geometric, and problems on basic formulas. In this example, we solve the task from the course of combinatorics. Go to the solution. This task is the simplest:
- n1 = 30 - possible elders of the student group;
- n2 = 29 - those who can take the post of deputy;
- n3 = 28 people apply for the position of a professional.
All we have to do is find the 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.
Task 2. Rearrangement
At the conference are 6 participants, the orderdetermined by drawing lots. We need to find the number of possible draws. In this example, we are considering a permutation of six elements, that is, we need to find 6!
At the point of abbreviations, we have already mentioned thatsuch and as calculated. Total it turns out that there are 720 variants of the draw. At first glance, a difficult task has quite a short and simple solution. These are the tasks that probability theory considers. How to solve problems of a higher level, we consider in the following examples.
Task 3
A group of twenty-five studentsmust 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 it is necessary to obtain a numerical solution, then we can give it in the form of factorials.
Task 4
Three people made a number from one to ten.Find the probability that someone has the same numbers. First we have to find out the number of all the outcomes - in our case it is a thousand, that is, ten to the third degree. Now we will find the number of variants, when everyone made different numbers, for this we multiply ten, nine and eight. Where did these numbers come from? The first one makes a number, he has ten options, the second one has already nine, and the third one has to choose from the eight remaining ones, thus we get 720 possible options. As we already considered earlier, there are 1000 variants in all, and 720 without repetitions, 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.
