A course of mathematics prepares a lot of surprises for students, one of which is a task in probability theory. With the solution of such tasks, 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. We offer all this to learn.
Science and its application
Since we offer an accelerated course "probability theory for dummies", it is first necessary to introduce basic concepts and letter abbreviations. To begin with, we will determine the very concept of "theory of probability." What kind of science is this and why is it needed? Probability theory is one of the branches of mathematics that studies random phenomena and quantities. She also considers the laws, properties and operations performed with these random variables. What is it for? Science has become widespread in the study of natural phenomena. Any natural and physical processes can not do without the presence of chance. Even if during the experiment the results were recorded as accurately as possible, when repeating the same test, the result is most likely not to be the same.
We will certainly consider examples of problems in probability theory ; you yourself can see this. The outcome depends on many different factors that are almost impossible to take into account or register, but nevertheless they have a huge impact on the outcome of the experience. Vivid examples are the tasks of determining the trajectory of planetary motion or determining the weather forecast, the likelihood of meeting a familiar person on the way to work, and determining the athlete’s jump height. Likewise, probability theory is of great help to brokers on stock exchanges. The task of probability theory, 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 consider a little later, only one type is studied - random ones. Nevertheless, it is necessary to know that events can be of three types:
- Impossible.
- Reliable.
- Random.
We offer a little stipulation of each of them. An impossible event will never happen, under any circumstances. Examples include: freezing water at positive temperature, pulling a cube from a bag of balls.
A reliable event always occurs with a 100% guarantee if all conditions are met. For example: you received a salary for the work done, received a diploma of higher professional education, if you studied in good faith, passed exams and defended your diploma, and so on.
With random events, things are a little more complicated: in the course of an experiment, it can happen or not, for example, pulling an ace out of a card deck with no more than three attempts. The result can be obtained both on the first attempt, and, in general, can not be obtained. It is the probability of the origin of the event that science is studying.
Probability
In a general sense, this is an assessment of the possibility of a successful outcome of the experience at which the event occurs. Probability is assessed at a qualitative level, especially if quantification is not possible or difficult. A task in probability theory with a solution, more precisely with an estimate of the probability of an event, implies finding the very possible fraction of a successful outcome. Probability in mathematics is the numerical characteristic of an event. It takes values from zero to one, is indicated by the letter P. If P is zero, then the event cannot occur, if it is one, then the event will happen with one hundred percent probability. The more P approaches unity, the greater the probability of a successful outcome, and vice versa, if close to zero, then the event will happen with low probability.
Abbreviations
The probability theory problem that you will soon be faced with may contain the following abbreviations:
- !;
- {};
- N;
- P and P (X);
- A, B, C, etc.
- n;
- m.
Some others are possible: additional explanations will be made as necessary. We suggest, for starters, to clarify the abbreviations presented above. The factorial is the first on our list. In order to make it clear, we give examples: 5! = 1 * 2 * 3 * 4 * 5 or 3! = 1 * 2 * 3. Further, the given sets are written in curly brackets, for example: {1; 2; 3; 4; ..; n} or {10; 140; 400; 562}. The following notation is a set of natural numbers, quite often found in tasks in probability theory. As mentioned earlier, P is the probability, and P (X) is the probability of the occurrence of event X. Events are written in capital letters of the Latin alphabet, for example: A - caught a white ball, B - blue, C - red or, respectively,,. The small letter n is the number of all possible outcomes, and m is the number of successful ones. Hence we get the rule of finding classical probability in elementary problems: P = m / n. Probability theory "for dummies" is probably limited by this knowledge. Now for fixing we pass to the solution.
Task 1. Combinatorics
The student group consists of thirty people, from which it is necessary to choose the headman, his deputy and the trade union. You need to find the number of ways to do this. A similar task can be found on the exam. The theory of probability, the solution of the problems of which we are now considering, may include problems from the course of combinatorics, finding classical probability, geometric, and problems on the basic formulas. In this example, we solve the task from the course of combinatorics. We pass 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 a position of a trade union.
All that remains for us 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. Relocation
At the conference, 6 participants speak, the order is determined by drawing lots. We need to find the number of possible draw options. In this example, we are considering a permutation of six elements, that is, we need to find 6!
In the acronym paragraph, we already mentioned what it is and how it is calculated. 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 probability theory considers. How to solve tasks 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 each. 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, then you can give it in the form of factorials.
Task 4
Three people guessed numbers from one to ten. Find the probability that someone matches the numbers. First we need to find out the number of all outcomes - in our case it is a thousand, that is, ten to a third degree. Now we find the number of options when everyone guessed different numbers, for this we multiply ten, nine and eight. Where did these numbers come from? The first one guesses the number, he has ten options, the second has already nine, and the third needs to choose from the eight remaining ones, so we get 720 possible options. As we already calculated earlier, there are 1000 variants in total, and 720 without repetitions, therefore, we are interested in the remaining 280. Now we need a formula for finding the classical probability: P =. We got the answer: 0.28.