Instruction

1

Carefully read the problem statement. Find the number of favorable outcomes and the total number. Suppose you want to solve the following problem: in the box are 10 bananas, 3 of them are immature. You need to determine what is the probability that randomly removed the banana will be ripe. In this case, we apply the classical definition of probability. Calculate the probability using the formula: p= M/N, where:

- M - number of favorable outcomes

- N is the total number of all outcomes.

- M - number of favorable outcomes

- N is the total number of all outcomes.

2

Calculate the number of favorable outcomes. In this case, 7 bananas (10 - 3). The total number of all outcomes in this case is equal to the total amount of bananas, that is 10. Calculate the probability of substituting values in the formula: 7/10= 0,7. Therefore, the probability that randomly removed the banana to be ripe, will be equal to 0.7.

3

Using the theorem of addition of probabilities, solve the problem, if by its terms the events in it are incompatible. For example, in the box for needlework are spools of thread of different colours: 3 of them with white thread, 1 - green, 2 - blue and 3 - black. You need to determine what is the probability that the coil will be removed with colored thread (not white). To solve this task according to the theorem of addition of probabilities, use the formula: p=P1+P2+P3....

4

Determine how many coils sitting in the box: 3+1+2+3=9 the coils (this is the total number of all outcomes). Calculate the probability to remove the coil: with green yarn - P1 = 1/9 = 0,11, with blue threads - P2 = 2/9 = 0,22, with black thread - P3 = 3/9 = 0,33. Add the resulting numbers: p = 0,11+0,22+0,33 = 0,66 - the likelihood that the coil will be removed with a colored thread. So, using the definition of probability to solve simple problems on probability.

Note

To solve more complex problems on the probability to apply the theorem of multiplication of probabilities, formulae of Laplace, Bayes and Bernoulli, depending on the compatibility of events and the number of their outcomes in terms of these tasks.

# Advice 2 : How to take the theory of probability

Theory

**of probability**is called a branch of mathematics which is devoted to the study of regularities of random phenomena. This item, if not separately, in math class, pass almost all students, even if they are studying Humanities specialties. And not for everyone to take the exam on this subject is an easy task.Instruction

1

Read lectures. It would be great if you wrote your lectures and progressively all examples and problems, but if you have no opportunity to use his lectures, ask someone else. If you ever were in trouble with math, it's likely you will be able to understand what studying the subject and how to solve it tasks. Compared with mathematical analysis, theory

**of probability**easier.2

Write Cribs. If you've never been good with math and understand the subject just by reading the lectures, I can't write Cribs. At least that's

**the theory**, which is not to be solved, you just need to write, it is possible to transfer from one paper to another. What should I do with the examples? The simplest examples can still be solved with existing theories or with similar tasks from the lectures. Simply substitute the sample data in the solution of similar problems and calculate the result. If you absolutely can't decide practical tasks, ask friends that know anything about the theory**of probability**to help you to solve your assignments.3

Abstracted from happening. This advice is suitable for any exam, not only for the theory

**of probability**. When you go into a stupor and can't solve the problem or to recall**a theory**, is distracted from what is happening on the exam. Look around, not think about anything. You will cease to worry and to seek a solution in a very narrow plane. If you do it right and will do anything not to think and just will internally keep track of its state, the solution, or at least the hint of it will come to you in the head itself. If this act fails, the old fashioned way or ask for help from classmates, or write off**the theory**or task.# Advice 3 : How to calculate the probability

In order to calculate

**the probability**of the event that you need to apply basic concepts of probability theory, to count the number of all possible events, to get the most accurate result.You will need

- a sheet of paper, pen

Instruction

1

The probability of the event means, in essence, share the belief that a certain result will occur or not. Let you have some event And, for example, roll of the dice is an equilateral cube. We need to calculate

**the probability**that it will drop 2 points. In order to calculate**the probability**P of an event A is found by dividing the number of favorable events n cases of loss 2 points to the total number of events m.2

Count the number of cases of loss of 2 points on the dice. This is possible only in one case – when the dice will be 2 points, in any other case, the amount will be more. Thus, the number of favorable events n = 1.

3

Count the number of cases of loss of any of the numbers on the dice. 1 dice the possible options of hair loss points:

1, 2, 3, 4, 5, 6. Thus, the number of all favourable cases m = 6.

1, 2, 3, 4, 5, 6. Thus, the number of all favourable cases m = 6.

4

Calculate

**the probability**of loss of 2 points on the dice: P = n/m= 1/6. Thus, only**the likelihood**th 1/6 of the dice will be 2 points, the chances are slim.5

If there are several auspicious events – for example, it is necessary that bones fell to (less than or equal to 4 points, then you need to add the total number of favorable events n = n1 + n2 + ...+ nx and divide it by the total number of cases. In this case, the cube will be up to 4 points if you get the following points: 1, 2, 3, 4 – all 4 options. Thus, the number of favorable events n = 4. Now

P = n/m= 4/6 = 2/3 – already more than half, the risk to lose is the third (if you roll 5 or 6).

**the probability**of loss up to 4 points on the dice:P = n/m= 4/6 = 2/3 – already more than half, the risk to lose is the third (if you roll 5 or 6).

6

In order to correctly calculate

**the probability**, don't forget to count absolutely all possible results, which will be the denominator, and remember that if something is not taken into account, the result will show the big share of probability, which can be a mistake. Upon the occurrence of simultaneous outcomes of several events is sometimes important priority of getting a result, then the total number of events is further increased.# Advice 4 : How to calculate the probability

In mathematical statistics the most basic and most important concept is considered to be

**the probability**of an event. The probability describes the degree of possibility of occurrence of an event. How to calculate**the probability**?Instruction

1

The probability of an event is the ratio of the number of all favorable outcomes to the number of all possible outcomes. A favorable outcome is the outcome, which invariably leads to the realization of the event. To understand this better, you need to disassemble a simple example with dice. The probability of rolling a three when throwing dice, is calculated as follows. Just when throwing dice, there are six possible events. They are defined by the number of its faces. But in this case there is only one favorable outcome - a loss of three. Then

**the probability**to throw all three in a single throw of the dice is equal to 1/6. It should be noted that the probability of any event is in the interval from zero to one.2

If the desired event can be easily decomposed into several events that are incompatible with each other, then

**the probability**of this desired event is equal to the sum of the probabilities of each of the incompatible events. In mathematical statistics, this statement is called the theorem of addition of probabilities. It can also be considered when throwing dice. Now you need to determine**the probability**of loss of the odd numbers. Such numbers on the three dice, namely 1, 3 and 5. The probability of each of them is 1/6. By theorem find**the probability**of loss of the odd numbers. It is equal to the sum of the probabilities of each of these events: 3/6 = 1/2.3

Sometimes, you need to determine

**the probability**of occurrence of two independent from each other events. Events Can be considered as independent if their probability of non-occurrence or occurrence does not depend from each other. In this case,**the probability of**find as the product of the probabilities of occurrence of both events. To understand it better, you need to try to find**the probability**of hair loss at the same time two sixes on two dice. These events are independent from each other. The probability that on the dice roll of a six is 1/6. So**the probability**of occurrence of two sixes is 1/36.# Advice 5 : How to learn probability theory

The theory

**of probability**is one of the most important areas of mathematics that studies the regularity of random phenomena: random variables, random events, their properties and operations that can make. To master this difficult science, you will have to put a lot of effort.You will need

- - list of questions for the exam;
- tutorials Wentzel E. S. or B. E. Gmurman.

Instruction

1

If you during the semester passed the words of the teacher by the ears, begin the study of the theory

**of probability**with the absorption of essential definitions. Remember that this random variable, what are examples of random variables (dropped points on the tiles), what classes they share. Remember, what are the events, and that such a probability space. If a student floats in the ticket, most likely, the teacher will begin to ask the basic things, so knowing the definitions of these terms will be useful.2

One of the most frequent moves of the teacher – to check knowledge of basic formulas. Write down on a separate sheet of important formulas, mark, which means you do not understand every symbol, and several times a day zadubrovie them. You now have the basis for the exam and for further study of the theory

**of probability**.3

Take a sheet with a list of questions for the exam and read it. Mark those questions which you know, then those tasks for which you will be able to give incomplete and vague answer, and proceed to the study of the third category of questions, the answers to which are unknown to you. After you cope with this task, read again the material on the points, knowing which you are very confident.

4

If you know that the ticket will be given a task, take a few days for the solution of typical examples of the theory

**of probability**. Most likely the above teacher will appreciate the student who correctly coped with the practical task, though, and couldn't give a clear answer to a theoretical question than one who understands the theory with no practical skills. Write down some examples of solutions of problems on a separate sheet of paper and regularly re-read it.5

If you study

**the theory****of probability**on their own and for their own pleasure, the most important thing for you is to find a good tutorial, written in plain language. Please note on the books authorship E. S. Wentzel, B. E. Gmurman.# Advice 6 : How to calculate the probability of the event

Under

**the probabilityu**is generally understood numerically expressed measure of the ability of the onset of an**event**. In practical application, this measure acts as a ratio of the number of observations in which a particular event occurred, to the total number of observations in a random experiment.You will need

- paper;
- pencil;
- calculator.

Instruction

1

For example, the calculation of the probability let us consider the simplest situation in which you want to define the amount of confidence that a standard set of cards containing 36 elements, you randomly pull any ACE. In this case, the probability P(a) will equal a fraction, the numerator of which is the number of favorable outcomes X and the denominator is the total number possible in the experiment of event Y.

2

Determine the number of favorable outcomes. In this example it will be 4 since in a standard deck of cards has exactly this number of aces of different suits.

3

Count the total number of possible events. Each card in the set has its own unique advantage, so for a standard deck of 36 possible options of a single choice. Of course, prior experience should take the condition under which all the cards are in the deck and are not repeated.

4

Select the likelihood that you removed from the deck one card is any ACE. To do this, use the formula: P(a) = X/Y = 4/36 = 1/9. In other words, the probability that the taking of a set of one card, you will receive the ACE, a relatively small and equal to about 0,11.

5

Change the conditions of the experiment. Let's say that you are going to calculate the probability of occurrence of

**events**, when taken at random card from the same set will be the ACE of spades. The number of favorable outcomes corresponding to the condition of the experiment, changed to become equal to 1, since the set only one card of the specified advantages.6

Put new data in the above formula P(a). So, P(a) = 1/36. In other words, the probability of a positive outcome of the second experiment decreased four times and made up approximately 0,027.

7

When calculating the probability of occurrence of

**events**in the experiment consider that you need to count all possible outcomes, reflected in the denominator. Otherwise, the result will present a distorted picture of the probability.