Mom washed the frame

Towards the end of a long summer vacation, it's time to slowly return to higher mathematics and solemnly open an empty Verd file in order to start creating a new section - . I confess that the first lines are not easy, but the first step is half the way, so I suggest everyone to carefully study the introductory article, after which it will be 2 times easier to master the topic! I'm not exaggerating at all. ... On the eve of the next September 1, I remember the first grade and primer .... Letters form syllables, syllables into words, words into short sentences - Mom washed the frame. Mastering terver and mathematical statistics is as easy as learning to read! However, for this it is necessary to know the key terms, concepts and designations, as well as some specific rules, to which this lesson is devoted.

But first, please accept my congratulations on the beginning (continuation, completion, appropriate note) of the academic year and accept the gift. The best gift is a book, and for self-study, I recommend the following literature:

1) Gmurman V.E. Theory of Probability and Mathematical Statistics

A legendary textbook that has gone through more than ten reprints. It differs in intelligibility and the ultimate simple presentation of the material, and the first chapters are completely accessible, I think, already for students in grades 6-7.

2) Gmurman V.E. Guide to Problem Solving in Probability and Mathematical Statistics

Reshebnik of the same Vladimir Efimovich with detailed examples and tasks.

NECESSARILY download both books from the Internet or get their paper originals! A 60s-70s version will do, which is even better for dummies. Although the phrase "probability theory for dummies" sounds rather ridiculous, since almost everything is limited to elementary arithmetic operations. They slip, however, in places derivatives and integrals, but this is only in places.

I will try to achieve the same clarity of presentation, but I must warn you that my course is focused on problem solving and theoretical calculations are kept to a minimum. Thus, if you need a detailed theory, proofs of theorems (yes, theorems!), please refer to the textbook.

For those who want learn to solve problems in a matter of days, created crash course in pdf format (according to the site). Well, right now, without postponing the matter in a long folder, we are starting to study terver and matstat - follow me!

Enough to get started =)

As you read the articles, it is useful to get acquainted (at least briefly) with additional problems of the types considered. On the page Ready-made solutions for higher mathematics the corresponding pdf-ki with examples of solutions are placed. Also, significant assistance will be provided IDZ 18.1-18.2 Ryabushko(easier) and solved IDZ according to the collection of Chudesenko(more difficult).

1) sum two events and is called the event which consists in the fact that or event or event or both events at the same time. In case the events incompatible, the last option disappears, that is, it can occur or event or event .

The rule also applies to more terms, for example, an event is what will happen at least one from events , a if the events are incompatible – that one and only one event from this sum: or event , or event , or event , or event , or event .

Plenty of examples:

The event (when throwing a die does not drop 5 points) is that or 1, or 2, or 3, or 4, or 6 points.

Event (will drop no more two points) is that 1 or 2points.

Event (there will be an even number of points) is that the or 2 or 4 or 6 points.

The event is that a card of red suit (heart) will be drawn from the deck or tambourine), and the event - that the “picture” will be extracted (jack or lady or king or ace).

A little more interesting is the case with joint events:

The event is that a club will be drawn from the deck or seven or seven of clubs According to the above definition, at least something- or any club or any seven or their "crossing" - seven clubs. It is easy to calculate that this event corresponds to 12 elementary outcomes (9 club cards + 3 remaining sevens).

The event is that tomorrow at 12.00 AT LEAST ONE of the summable joint events, namely:

- or there will be only rain / only thunder / only sun;
- or only some pair of events will come (rain + thunderstorm / rain + sun / thunderstorm + sun);
– or all three events will appear at the same time.

That is, the event includes 7 possible outcomes.

The second pillar of the algebra of events:

2) work two events and call the event, which consists in the joint appearance of these events, in other words, multiplication means that under certain circumstances there will come and event , and event . A similar statement is true for a larger number of events, for example, the work implies that under certain conditions, there will be and event , and event , and event , …, and event .

Consider a trial in which two coins are tossed and the following events:

- heads will fall on the 1st coin;
- the 1st coin will land tails;
- the 2nd coin will land heads;
- the 2nd coin will come up tails.

Then:
and on the 2nd) an eagle will fall out;
- the event consists in the fact that on both coins (on the 1st and on the 2nd) tails will fall out;
– the event is that the 1st coin will land heads and on the 2nd coin tails;
- the event is that the 1st coin will come up tails and on the 2nd coin an eagle.

It is easy to see that the events incompatible (since it cannot, for example, fall out 2 heads and 2 tails at the same time) and form full group (since taken into account all possible outcomes of tossing two coins). Let's summarize these events: . How to interpret this entry? Very simple - multiplication means logical connection And, and the addition is OR. Thus, the sum is easy to read in understandable human language: “two eagles will fall or two tails or heads on the 1st coin and on the 2nd tail or heads on the 1st coin and eagle on the 2nd coin »

This was an example when in one test several objects are involved, in this case two coins. Another scheme commonly used in practice is repeated tests when, for example, the same dice is thrown 3 times in a row. As a demonstration, consider the following events:

- in the 1st throw, 4 points will fall out;
- in the 2nd roll, 5 points will fall out;
- in the 3rd throw, 6 points will fall out.

Then the event consists in the fact that in the 1st roll 4 points will fall out and in the 2nd roll will drop 5 points and in the 3rd roll, 6 points will fall. Obviously, in the case of a die, there will be significantly more combinations (outcomes) than if we were tossing a coin.

…I understand that, perhaps, not very interesting examples are analyzed, but these are things that are often encountered in problems and there is no getting away from them. In addition to a coin, a die and a deck of cards, there are urns with colorful balls, several anonymous people shooting at targets, and a tireless worker who constantly grinds out some details =)

Event Probability

Event Probability is a central concept in probability theory. ...A deadly logical thing, but you had to start somewhere =) There are several approaches to its definition:

;
Geometric definition of probability ;
Statistical definition of probability .

In this article, I will focus on the classical definition of probabilities, which is most widely used in educational tasks.

Notation. The probability of some event is denoted by a capital Latin letter , and the event itself is taken in brackets, acting as a kind of argument. For example:

Also, a small letter is widely used to represent probability. In particular, one can abandon the cumbersome designations of events and their probabilities in favor of the following style:

is the probability that the toss of a coin will result in heads;
- the probability that 5 points will fall out as a result of throwing a dice;
is the probability that a card of the club suit will be drawn from the deck.

This option is popular in solving practical problems, since it allows you to significantly reduce the solution entry. As in the first case, it is convenient to use “talking” subscripts/superscripts here.

Everyone has long guessed about the numbers that I just wrote down above, and now we will find out how they turned out:

The classical definition of probability:

The probability of an event occurring in some test is the ratio , where:

is the total number of all equally possible, elementary outcomes of this test, which form full group of events;

- amount elementary outcomes favorable event .

When a coin is tossed, either heads or tails can fall out - these events form full group, thus, the total number of outcomes ; while each of them elementary and equally possible. The event is favored by the outcome (heads). According to the classical definition of probabilities: .

Similarly, as a result of a roll of a die, elementary equally possible outcomes may appear, forming a complete group, and the event is favored by a single outcome (rolling a five). So: .THIS IS NOT ACCEPTED TO DO (although it is not forbidden to figure out the percentages in your mind).

It is customary to use fractions of a unit, and, obviously, the probability can vary within . Moreover, if , then the event is impossible, if - authentic, and if , then we are talking about random event.

! If in the course of solving any problem you get some other probability value - look for an error!

In the classical approach to the definition of probability, the extreme values ​​(zero and one) are obtained by exactly the same reasoning. Let 1 ball be drawn at random from an urn containing 10 red balls. Consider the following events:

in a single trial, an unlikely event will not occur.

That is why you will not hit the Jackpot in the lottery if the probability of this event is, say, 0.00000001. Yes, yes, it is you - with the only ticket in a particular circulation. However, more tickets and more draws will not help you much. ... When I tell others about this, I almost always hear in response: "but someone wins." Okay, then let's do the following experiment: please buy any lottery ticket today or tomorrow (don't delay!). And if you win ... well, at least more than 10 kilo rubles, be sure to unsubscribe - I will explain why this happened. For a percentage, of course =) =)

But there is no need to be sad, because there is an opposite principle: if the probability of some event is very close to unity, then in a single test it almost certain will happen. Therefore, before a parachute jump, do not be afraid, on the contrary - smile! After all, absolutely unthinkable and fantastic circumstances must arise for both parachutes to fail.

Although all this is poetry, because, depending on the content of the event, the first principle may turn out to be cheerful, and the second - sad; or even both are parallel.

Probably enough for now, in class Tasks for the classical definition of probability we will squeeze the maximum out of the formula. In the final part of this article, we consider one important theorem:

The sum of the probabilities of events that form a complete group is equal to one. Roughly speaking, if events form a complete group, then with 100% probability one of them will happen. In the simplest case, opposite events form a complete group, for example:

- as a result of a coin toss, an eagle will fall out;
- as a result of tossing a coin, tails will fall out.

According to the theorem:

It is clear that these events are equally likely and their probabilities are the same. .

Because of the equality of probabilities, equally probable events are often called equiprobable . And here is the tongue twister for determining the degree of intoxication turned out =)

Dice example: events are opposite, so .

The theorem under consideration is convenient in that it allows you to quickly find the probability of the opposite event. So, if you know the probability that a five will fall out, it is easy to calculate the probability that it will not fall out:

This is much easier than summing up the probabilities of five elementary outcomes. For elementary outcomes, by the way, this theorem is also valid:
. For example, if is the probability that the shooter will hit the target, then is the probability that he will miss.

! In probability theory, it is undesirable to use the letters and for any other purpose.

In honor of Knowledge Day, I will not give homework =), but it is very important that you can answer the following questions:

What types of events are there?
– What is chance and equal possibility of an event?
– How do you understand the terms compatibility / incompatibility of events?
– What is a complete group of events, opposite events?
What does the addition and multiplication of events mean?
– What is the essence of the classical definition of probability?
– Why is the addition theorem for the probabilities of events forming a complete group useful?

No, you don’t need to cram anything, these are just the basics of probability theory - a kind of primer that will fit in your head pretty quickly. And so that this happens as soon as possible, I suggest that you read the lessons

Many, faced with the concept of "probability theory", are frightened, thinking that this is something overwhelming, very complex. But it's really not all that tragic. Today we will consider the basic concept of probability theory, learn how to solve problems using specific examples.

The science

What does such a branch of mathematics as “probability theory” study? She notes patterns and magnitudes. For the first time, scientists became interested in this issue back in the eighteenth century, when they studied gambling. The basic concept of probability theory is an event. It is any fact that is ascertained by experience or observation. But what is experience? Another basic concept of probability theory. It means that this composition of circumstances was not created by chance, but for a specific purpose. As for observation, here the researcher himself does not participate in the experiment, but simply is a witness to these events, he does not influence what is happening in any way.

Events

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:

• Reliable.
• Impossible.
• Random.

No matter what kind of events are observed or created in the course of experience, they are all subject to this classification. We offer to get acquainted with each of the species separately.

Credible Event

This is a circumstance before which the necessary set of measures has been taken. In order to better understand the essence, it is better to give a few examples. Physics, chemistry, economics, and higher mathematics are subject to this law. Probability theory includes such an important concept as a certain event. Here are some examples:

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

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

Impossible events

We now consider elements of probability theory. We propose to move on to an explanation of the next type of event, namely, the impossible. To begin with, we will stipulate the most important rule - the probability of an impossible event is zero.

It is impossible to deviate from this formulation when solving problems. To clarify, here are examples of such events:

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

More examples should not be given, since the ones described above very clearly reflect the essence of this category. The impossible event will never happen during the experience under any circumstances.

random events

When studying the elements, special attention should be paid to this particular type of event. That is what science is studying. As a result of experience, something may or may not happen. In addition, the test can be repeated an unlimited number of times. Prominent examples are:

• Tossing a coin is an experience, or a test, heading is an event.
• Pulling the ball out of the bag blindly is a test, a red ball is caught is an event, and so on.

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

title	definition
Credible	Events that occur with a 100% guarantee, subject to certain conditions.	Admission to an educational institution with a good passing of the entrance exam.
Impossible	Events that will never happen under any circumstances.	It is snowing at an air temperature of plus thirty degrees Celsius.
Random	An event that may or may not occur during an experiment/test.	Hit or miss when throwing a basketball into the hoop.

The laws

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

• Convergence of sequences of random variables.
• The law of large numbers.

When calculating the possibility of the complex, a complex of simple events can be used to achieve the result in an easier and faster way. Note that the laws of probability theory are easily proved with the help of some theorems. Let's start with the first law.

Convergence of sequences of random variables

Note that there are several types of convergence:

• The sequence of random variables is convergent in probability.
• Almost impossible.
• RMS convergence.
• Distribution Convergence.

So, on the fly, it's very hard to get to the bottom of it. Here are some definitions to help you understand this topic. Let's start with the first look. The sequence is called convergent in probability, if the following condition is met: n tends to infinity, the number to which the sequence tends is greater than zero and close to one.

Let's move on to the next one, almost certainly. The sequence is said to converge almost certainly to a random variable with n tending to infinity, and P tending to a value close to unity.

The next type is RMS convergence. When using SC-convergence, the study of vector random processes is reduced to the study of their coordinate random processes.

The last type remains, let's briefly analyze it in order to proceed directly to solving problems. Distribution convergence has another name - “weak”, we will explain why below. Weak convergence is the convergence of distribution functions at all points of continuity of the limiting distribution function.

We will definitely fulfill the promise: weak convergence differs from all of the above in that the random variable is not defined on the probability space. This is possible because the condition is formed exclusively using distribution functions.

Law of Large Numbers

Excellent assistants in proving this law will be theorems of probability theory, such as:

• Chebyshev's inequality.
• Chebyshev's theorem.
• Generalized Chebyshev's theorem.
• Markov's theorem.

If we consider all these theorems, then this question can drag on for several tens of sheets. Our main task is to apply the theory of probability in practice. We invite you to do this right now. But before that, let's consider the axioms of probability theory, they will be the main assistants in solving problems.

Axioms

We already met the first one when we talked about the impossible event. Let's remember: the probability of an impossible event is zero. We gave a very vivid and memorable example: snow fell at an air temperature of thirty degrees Celsius.

The second is as follows: a certain event occurs with a probability equal to one. Now let's show how to write it down using the mathematical language: P(B)=1.

Third: A random event may or may not occur, but the possibility always ranges from zero to one. The closer the value is to one, the greater the chance; if the value approaches zero, the probability is very low. Let's write it in mathematical language: 0<Р(С)<1.

Consider the last, fourth axiom, which sounds like this: the probability of the sum of two events is equal to the sum of their probabilities. We write in mathematical language: P (A + B) \u003d P (A) + P (B).

The axioms of probability theory are the simplest rules that are easy to remember. Let's try to solve some problems, based on the knowledge already gained.

Lottery ticket

To begin with, consider the simplest example - the lottery. Imagine that you bought one lottery ticket for good luck. What is the probability that you will win at least twenty rubles? In total, a thousand tickets participate in the circulation, one of which has a prize of five hundred rubles, ten of one hundred rubles, fifty of twenty rubles, and one hundred of five. Problems in probability theory are based on finding the possibility of luck. Let's take a look at the solution to the above problem together.

If we denote by the letter A a win of five hundred rubles, then the probability of getting A will be 0.001. How did we get it? You just need to divide the number of "happy" tickets by their total number (in this case: 1/1000).

B is a win of one hundred rubles, the probability will be equal to 0.01. Now we acted on the same principle as in the previous action (10/1000)

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

The remaining tickets are of no interest to us, since their prize fund is less than that specified in the condition. Let's 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 occurrence of this event, we have already found them in the previous steps. 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 task.

card deck

Problems in the theory of probability are also more complex, for example, 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 pile, the first and second cards must be aces, the suit does not matter.

To begin with, we find the probability that the first card will be an ace, for this we divide four by thirty-six. They put it aside. We take out 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 drew first, we are interested in whether it was an ace or not. It follows that event B depends on event A.

The next step is to find the probability of simultaneous implementation, that is, we multiply A and B. Their product is found as follows: we multiply the probability of one event by the conditional probability of another, which we calculate, assuming that the first event happened, that is, we drew an ace with the first card.

In order to make everything clear, let's give a designation to such an element as events. It is calculated assuming that event A has occurred. Calculated as follows: P(B/A).

Let's continue the solution of our problem: P (A * B) \u003d P (A) * P (B / A) or P (A * B) \u003d P (B) * P (A / B). The probability is (4/36) * ((3/35)/(4/36). Calculate by rounding 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 hundredths.The value is very small, from this it follows that the probability of the occurrence of the event is extremely small.

Forgotten number

We propose to analyze a few more options for tasks that are studied by probability theory. You have already seen examples of solving some of them in this article, let's 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 dial 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 looking at the solution, try to solve it yourself. We know that the last digit can be from zero to nine, that is, there are ten values ​​in total. The probability of getting the right one is 1/10.

Next, we need to consider options for the origin of the event, suppose that the boy guessed right and immediately scored the right one, the probability of such an event is 1/10. The second option: the first call is a miss, and the second is on target. We calculate the probability of such an event: multiply 9/10 by 1/9, as a result we also get 1/10. The third option: the first and second calls turned out to be at the wrong address, only from the third the boy got where he wanted. We calculate the probability of such an event: we multiply 9/10 by 8/9 and by 1/8, we get 1/10 as a result. According to the condition of the problem, we are not interested in other options, so it remains for us to add up the results, as a result we have 3/10. Answer: The probability that the boy calls no more than three times is 0.3.

Cards with numbers

There are nine cards in front of you, each of which contains a number from one to nine, the numbers are not repeated. They were placed in a box and mixed thoroughly. You need to calculate the probability that

• an even number will come up;
• two-digit.

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

We consider the second case: the number of options is nine, and there can be no successful outcomes at all, that is, m equals zero. The probability that the drawn card will contain a two-digit number is also zero.

What is a probability?

Faced with this term for the first time, I would not understand what it is. So I'll try to explain in an understandable way.

Probability is the chance that the desired event will occur.

For example, you decided to visit a friend, remember the entrance and even the floor on which he lives. But I forgot the number and location of the apartment. And now you are standing on the stairwell, and in front of you are the doors to choose from.

What is the chance (probability) that if you ring the first doorbell, your friend will open it for you? Whole apartment, and a friend lives only behind one of them. With equal chance, we can choose any door.

But what is this chance?

Doors, the right door. Probability of guessing by ringing the first door: . That is, one time out of three you will guess for sure.

We want to know by calling once, how often will we guess the door? Let's look at all the options:

1. you called to 1st a door
2. you called to 2nd a door
3. you called to 3rd a door

And now consider all the options where a friend can be:

a. Behind 1st door
b. Behind 2nd door
in. Behind 3rd door

Let's compare all the options in the form of a table. A tick indicates the options when your choice matches the location of a friend, a cross - when it does not match.

How do you see everything possibly options friend's location and your choice of which door to ring.

BUT favorable outcomes of all . That is, you will guess the times from by ringing the door once, i.e. .

This is the probability - the ratio of a favorable outcome (when your choice coincided with the location of a friend) to the number of possible events.

The definition is the formula. Probability is usually denoted p, so:

It is not very convenient to write such a formula, so let's take for - the number of favorable outcomes, and for - the total number of outcomes.

The probability can be written as a percentage, for this you need to multiply the resulting result by:

Probably, the word “outcomes” caught your eye. Since mathematicians call various actions (for us, such an action is a doorbell) experiments, it is customary to call the result of such experiments an outcome.

Well, the outcomes are favorable and unfavorable.

Let's go back to our example. Let's say we rang at one of the doors, but a stranger opened it for us. We didn't guess. What is the probability that if we ring one of the remaining doors, our friend will open it for us?

If you thought that, then this is a mistake. Let's figure it out.

We have two doors left. So we have possible steps:

1) Call to 1st a door
2) Call 2nd a door

A friend, with all this, is definitely behind one of them (after all, he was not behind the one we called):

a) a friend 1st door
b) a friend for 2nd door

Let's draw the table again:

As you can see, there are all options, of which - favorable. That is, the probability is equal.

Why not?

The situation we have considered is example of dependent events. The first event is the first doorbell, the second event is the second doorbell.

And they are called dependent because they affect the following actions. After all, if a friend opened the door after the first ring, what would be the probability that he was behind one of the other two? Correctly, .

But if there are dependent events, then there must be independent? True, there are.

A textbook example is tossing a coin.

1. We toss a coin. What is the probability that, for example, heads will come up? That's right - because the options for everything (either heads or tails, we will neglect the probability of a coin to stand on edge), but only suits us.
2. But the tails fell out. Okay, let's do it again. What is the probability of coming up heads now? Nothing has changed, everything is the same. How many options? Two. How much are we satisfied with? One.

And let tails fall out at least a thousand times in a row. The probability of falling heads at once will be the same. There are always options, but favorable ones.

Distinguishing dependent events from independent events is easy:

1. If the experiment is carried out once (once a coin is tossed, the doorbell rings once, etc.), then the events are always independent.
2. If the experiment is carried out several times (a coin is tossed once, the doorbell is rung several times), then the first event is always independent. And then, if the number of favorable or the number of all outcomes changes, then the events are dependent, and if not, they are independent.

Let's practice a little to determine the probability.

Example 1

The coin is tossed twice. What is the probability of getting heads up twice in a row?

Decision:

Consider all possible options:

1. eagle eagle
2. tails eagle
3. tails-eagle
4. Tails-tails

As you can see, all options. Of these, we are satisfied only. That is the probability:

If the condition asks simply to find the probability, then the answer must be given as a decimal fraction. If it were indicated that the answer must be given as a percentage, then we would multiply by.

Answer:

Example 2

In a box of chocolates, all candies are packed in the same wrapper. However, from sweets - with nuts, cognac, cherries, caramel and nougat.

What is the probability of taking one candy and getting a candy with nuts. Give your answer in percentage.

Decision:

How many possible outcomes are there? .

That is, taking one candy, it will be one of those in the box.

And how many favorable outcomes?

Because the box contains only chocolates with nuts.

Answer:

Example 3

In a box of balls. of which are white and black.

1. What is the probability of drawing a white ball?
2. We added more black balls to the box. What is the probability of drawing a white ball now?

Decision:

a) There are only balls in the box. of which are white.

The probability is:

b) Now there are balls in the box. And there are just as many whites left.

Answer:

Full Probability

The probability of all possible events is ().

For example, in a box of red and green balls. What is the probability of drawing a red ball? Green ball? Red or green ball?

Probability of drawing a red ball

Green ball:

Red or green ball:

As you can see, the sum of all possible events is equal to (). Understanding this point will help you solve many problems.

Example 4

There are felt-tip pens in the box: green, red, blue, yellow, black.

What is the probability of drawing NOT a red marker?

Decision:

Let's count the number favorable outcomes.

NOT a red marker, that means green, blue, yellow, or black.

Probability of all events. And the probability of events that we consider unfavorable (when we pull out a red felt-tip pen) is .

Thus, the probability of drawing NOT a red felt-tip pen is -.

Answer:

The probability that an event will not occur is minus the probability that the event will occur.

Rule for multiplying the probabilities of independent events

You already know what independent events are.

And if you need to find the probability that two (or more) independent events will occur in a row?

Let's say we want to know what is the probability that by tossing a coin once, we will see an eagle twice?

We have already considered - .

What if we toss a coin? What is the probability of seeing an eagle twice in a row?

Total possible options:

1. Eagle-eagle-eagle
2. Eagle-head-tails
3. Head-tails-eagle
4. Head-tails-tails
5. tails-eagle-eagle
6. Tails-heads-tails
7. Tails-tails-heads
8. Tails-tails-tails

I don't know about you, but I made this list wrong once. Wow! And only option (the first) suits us.

For 5 rolls, you can make a list of possible outcomes yourself. But mathematicians are not as industrious as you.

Therefore, they first noticed, and then proved, that the probability of a certain sequence of independent events decreases each time by the probability of one event.

In other words,

Consider the example of the same, ill-fated, coin.

Probability of coming up heads in a trial? . Now we are tossing a coin.

What is the probability of getting tails in a row?

This rule does not only work if we are asked to find the probability that the same event will occur several times in a row.

If we wanted to find the TAILS-EAGLE-TAILS sequence on consecutive flips, we would do the same.

The probability of getting tails - , heads - .

The probability of getting the sequence TAILS-EAGLE-TAILS-TAILS:

You can check it yourself by making a table.

The rule for adding the probabilities of incompatible events.

So stop! New definition.

Let's figure it out. Let's take our worn out coin and flip it once.
Possible options:

1. Eagle-eagle-eagle
2. Eagle-head-tails
3. Head-tails-eagle
4. Head-tails-tails
5. tails-eagle-eagle
6. Tails-heads-tails
7. Tails-tails-heads
8. Tails-tails-tails

So here are incompatible events, this is a certain, given sequence of events. are incompatible events.

If we want to determine what is the probability of two (or more) incompatible events, then we add the probabilities of these events.

You need to understand that the loss of an eagle or tails is two independent events.

If we want to determine what is the probability of a sequence) (or any other) falling out, then we use the rule of multiplying probabilities.
What is the probability of getting heads on the first toss and tails on the second and third?

But if we want to know what is the probability of getting one of several sequences, for example, when heads come up exactly once, i.e. options and, then we must add the probabilities of these sequences.

Total options suits us.

We can get the same thing by adding up the probabilities of occurrence of each sequence:

Thus, we add probabilities when we want to determine the probability of some, incompatible, sequences of events.

There is a great rule to help you not get confused when to multiply and when to add:

Let's go back to the example where we tossed a coin times and want to know the probability of seeing heads once.
What is going to happen?

Should drop:
(heads AND tails AND tails) OR (tails AND heads AND tails) OR (tails AND tails AND heads).
And so it turns out:

Let's look at a few examples.

Example 5

There are pencils in the box. red, green, orange and yellow and black. What is the probability of drawing red or green pencils?

Decision:

What is going to happen? We have to pull out (red OR green).

Now it’s clear, we add up the probabilities of these events:

Answer:

Example 6

A die is thrown twice, what is the probability that a total of 8 will come up?

Decision.

How can we get points?

(and) or (and) or (and) or (and) or (and).

The probability of falling out of one (any) face is .

We calculate the probability:

Answer:

Workout.

I think now it has become clear to you when you need to how to count the probabilities, when to add them, and when to multiply them. Is not it? Let's get some exercise.

Tasks:

Let's take a deck of cards in which the cards are spades, hearts, 13 clubs and 13 tambourines. From to Ace of each suit.

1. What is the probability of drawing clubs in a row (we put the first card drawn back into the deck and shuffle)?
2. What is the probability of drawing a black card (spades or clubs)?
3. What is the probability of drawing a picture (jack, queen, king or ace)?
4. What is the probability of drawing two pictures in a row (we remove the first card drawn from the deck)?
5. What is the probability, taking two cards, to collect a combination - (Jack, Queen or King) and Ace The sequence in which the cards will be drawn does not matter.

Answers:

1. In a deck of cards of each value, it means:
2. The events are dependent, since after the first card drawn, the number of cards in the deck has decreased (as well as the number of “pictures”). Total jacks, queens, kings and aces in the deck initially, which means the probability of drawing the “picture” with the first card:

   Since we are removing the first card from the deck, it means that there is already a card left in the deck, of which there are pictures. Probability of drawing a picture with the second card:

   Since we are interested in the situation when we get from the deck: “picture” AND “picture”, then we need to multiply the probabilities:

   Answer:

3. After the first card is drawn, the number of cards in the deck will decrease. Thus, we have two options:
   1) With the first card we take out Ace, the second - jack, queen or king
   2) With the first card we take out a jack, queen or king, the second - an ace. (ace and (jack or queen or king)) or ((jack or queen or king) and ace). Don't forget about reducing the number of cards in the deck!

If you were able to solve all the problems yourself, then you are a great fellow! Now tasks on the theory of probability in the exam you will click like nuts!

PROBABILITY THEORY. MIDDLE LEVEL

Consider an example. Let's say we throw a die. What kind of bone is this, do you know? This is the name of a cube with numbers on the faces. How many faces, so many numbers: from to how many? Before.

So we roll a die and want it to come up with an or. And we fall out.

In probability theory they say what happened favorable event(not to be confused with good).

If it fell out, the event would also be auspicious. In total, only two favorable events can occur.

How many bad ones? Since all possible events, then the unfavorable of them are events (this is if it falls out or).

Definition:

Probability is the ratio of the number of favorable events to the number of all possible events.. That is, the probability shows what proportion of all possible events are favorable.

They denote the probability with a Latin letter (apparently, from the English word probability - probability).

It is customary to measure the probability as a percentage (see topics and). To do this, the probability value must be multiplied by. In the dice example, probability.

And in percentage: .

Examples (decide for yourself):

1. What is the probability that the toss of a coin will land on heads? And what is the probability of a tails?
2. What is the probability that an even number will come up when a dice is thrown? And with what - odd?
3. In a drawer of plain, blue and red pencils. We randomly draw one pencil. What is the probability of pulling out a simple one?

Solutions:

1. How many options are there? Heads and tails - only two. And how many of them are favorable? Only one is an eagle. So the probability

   Same with tails: .

2. Total options: (how many sides a cube has, so many different options). Favorable ones: (these are all even numbers :).
   Probability. With odd, of course, the same thing.
3. Total: . Favorable: . Probability: .

Full Probability

All pencils in the drawer are green. What is the probability of drawing a red pencil? There are no chances: probability (after all, favorable events -).

Such an event is called impossible.

What is the probability of drawing a green pencil? There are exactly as many favorable events as there are total events (all events are favorable). So the probability is or.

Such an event is called certain.

If there are green and red pencils in the box, what is the probability of drawing a green or a red one? Yet again. Note the following thing: the probability of drawing green is equal, and red is .

In sum, these probabilities are exactly equal. I.e, the sum of the probabilities of all possible events is equal to or.

Example:

In a box of pencils, among them are blue, red, green, simple, yellow, and the rest are orange. What is the probability of not drawing green?

Decision:

Remember that all probabilities add up. And the probability of drawing green is equal. This means that the probability of not drawing green is equal.

Remember this trick: The probability that an event will not occur is minus the probability that the event will occur.

Independent events and the multiplication rule

You flip a coin twice and you want it to come up heads both times. What is the probability of this?

Let's go through all the possible options and determine how many there are:

Eagle-Eagle, Tails-Eagle, Eagle-Tails, Tails-Tails. What else?

The whole variant. Of these, only one suits us: Eagle-Eagle. So, the probability is equal.

Good. Now let's flip a coin. Count yourself. Happened? (answer).

You may have noticed that with the addition of each next throw, the probability decreases by a factor. The general rule is called multiplication rule:

The probabilities of independent events change.

What are independent events? Everything is logical: these are those that do not depend on each other. For example, when we toss a coin several times, each time a new toss is made, the result of which does not depend on all previous tosses. With the same success, we can throw two different coins at the same time.

More examples:

1. A die is thrown twice. What is the probability that it will come up both times?
2. A coin is tossed times. What is the probability of getting heads first and then tails twice?
3. The player rolls two dice. What is the probability that the sum of the numbers on them will be equal?

Answers:

1. The events are independent, which means that the multiplication rule works: .
2. The probability of an eagle is equal. Tails probability too. We multiply:
3. 12 can only be obtained if two -ki fall out: .

Incompatible events and the addition rule

Incompatible events are events that complement each other to full probability. As the name implies, they cannot happen at the same time. For example, if we toss a coin, either heads or tails can fall out.

Example.

In a box of pencils, among them are blue, red, green, simple, yellow, and the rest are orange. What is the probability of drawing green or red?

Decision .

The probability of drawing a green pencil is equal. Red - .

Auspicious events of all: green + red. So the probability of drawing green or red is equal.

The same probability can be represented in the following form: .

This is the addition rule: the probabilities of incompatible events add up.

Mixed tasks

Example.

The coin is tossed twice. What is the probability that the result of the rolls will be different?

Decision .

This means that if heads come up first, tails should be second, and vice versa. It turns out that there are two pairs of independent events here, and these pairs are incompatible with each other. How not to get confused about where to multiply and where to add.

There is a simple rule for such situations. Try to describe what should happen by connecting the events with the unions "AND" or "OR". For example, in this case:

Must roll (heads and tails) or (tails and heads).

Where there is a union "and", there will be multiplication, and where "or" is addition:

Try it yourself:

1. What is the probability that two coin tosses come up with the same side both times?
2. A die is thrown twice. What is the probability that the sum will drop points?

Solutions:

1. (Heads up and heads up) or (tails up and tails up): .
2. What are the options? and. Then:
   Rolled (and) or (and) or (and): .

Another example:

We toss a coin once. What is the probability that heads will come up at least once?

Decision:

Oh, how I don’t want to sort through the options ... Head-tails-tails, Eagle-heads-tails, ... But you don’t have to! Let's talk about full probability. Remembered? What is the probability that the eagle will never drop? It's simple: tails fly all the time, that means.

PROBABILITY THEORY. BRIEFLY ABOUT THE MAIN

Probability is the ratio of the number of favorable events to the number of all possible events.

Independent events

Two events are independent if the occurrence of one does not change the probability of the other occurring.

Full Probability

The probability of all possible events is ().

The probability that an event will not occur is minus the probability that the event will occur.

Rule for multiplying the probabilities of independent events

The probability of a certain sequence of independent events is equal to the product of the probabilities of each of the events

Incompatible events

Incompatible events are those events that cannot possibly occur simultaneously as a result of an experiment. A number of incompatible events form a complete group of events.

The probabilities of incompatible events add up.

Having described what should happen, using the unions "AND" or "OR", instead of "AND" we put the sign of multiplication, and instead of "OR" - addition.

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When a coin is tossed, it can be said that it will land heads up, or probability of this is 1/2. Of course, this does not mean that if a coin is tossed 10 times, it will necessarily land on heads 5 times. If the coin is "fair" and if it is tossed many times, then heads will come up very close half the time. Thus, there are two kinds of probabilities: experimental and theoretical .

Experimental and theoretical probability

If we toss a coin a large number of times - say 1000 - and count how many times it comes up heads, we can determine the probability that it will come up heads. If heads come up 503 times, we can calculate the probability of it coming up:
503/1000, or 0.503.

This is experimental definition of probability. This definition of probability stems from observation and study of data and is quite common and very useful. For example, here are some probabilities that were determined experimentally:

1. The chance of a woman developing breast cancer is 1/11.

2. If you kiss someone who has a cold, then the probability that you will also get a cold is 0.07.

3. A person who has just been released from prison has an 80% chance of going back to prison.

If we consider the toss of a coin and taking into account that it is equally likely to come up heads or tails, we can calculate the probability of coming up heads: 1 / 2. This is the theoretical definition of probability. Here are some other probabilities that have been theoretically determined using mathematics:

1. If there are 30 people in a room, the probability that two of them have the same birthday (excluding the year) is 0.706.

2. During a trip, you meet someone and during the course of the conversation you discover that you have a mutual acquaintance. Typical reaction: "That can't be!" In fact, this phrase does not fit, because the probability of such an event is quite high - just over 22%.

Therefore, the experimental probability is determined by observation and data collection. Theoretical probabilities are determined by mathematical reasoning. Examples of experimental and theoretical probabilities, such as those discussed above, and especially those that we do not expect, lead us to the importance of studying probability. You may ask, "What is true probability?" Actually, there is none. It is experimentally possible to determine the probabilities within certain limits. They may or may not coincide with the probabilities that we obtain theoretically. There are situations in which it is much easier to define one type of probability than another. For example, it would be sufficient to find the probability of catching a cold using theoretical probability.

Calculation of experimental probabilities

Consider first the experimental definition of probability. The basic principle we use to calculate such probabilities is as follows.

Principle P (experimental)

If in an experiment in which n observations are made, the situation or event E occurs m times in n observations, then the experimental probability of the event is said to be P (E) = m/n.

Example 1 Sociological survey. An experimental study was conducted to determine the number of left-handers, right-handers and people in whom both hands are equally developed. The results are shown in the graph.

a) Determine the probability that the person is right-handed.

b) Determine the probability that the person is left-handed.

c) Determine the probability that the person is equally fluent in both hands.

d) Most PBA tournaments have 120 players. Based on this experiment, how many players can be left-handed?

Decision

a) The number of people who are right-handed is 82, the number of left-handers is 17, and the number of those who are equally fluent in both hands is 1. The total number of observations is 100. Thus, the probability that a person is right-handed is P
P = 82/100, or 0.82, or 82%.

b) The probability that a person is left-handed is P, where
P = 17/100 or 0.17 or 17%.

c) The probability that a person is equally fluent with both hands is P, where
P = 1/100 or 0.01 or 1%.

d) 120 bowlers and from (b) we can expect 17% to be left handed. From here
17% of 120 = 0.17.120 = 20.4,
that is, we can expect about 20 players to be left-handed.

Example 2 Quality control . It is very important for a manufacturer to keep the quality of their products at a high level. In fact, companies hire quality control inspectors to ensure this process. The goal is to release the minimum possible number of defective products. But since the company produces thousands of items every day, it cannot afford to inspect each item to determine if it is defective or not. To find out what percentage of products are defective, the company tests far fewer products.
The USDA requires that 80% of the seeds that growers sell germinate. To determine the quality of the seeds that the agricultural company produces, 500 seeds are planted from those that have been produced. After that, it was calculated that 417 seeds germinated.

a) What is the probability that the seed will germinate?

b) Do the seeds meet government standards?

Decision a) We know that out of 500 seeds that were planted, 417 sprouted. The probability of seed germination P, and
P = 417/500 = 0.834, or 83.4%.

b) Since the percentage of germinated seeds exceeded 80% on demand, the seeds meet the state standards.

Example 3 TV ratings. According to statistics, there are 105,500,000 TV households in the United States. Every week, information about viewing programs is collected and processed. Within one week, 7,815,000 households were tuned in to CBS' hit comedy series Everybody Loves Raymond and 8,302,000 households were tuned in to NBC's hit Law & Order (Source: Nielsen Media Research). What is the probability that one home's TV is tuned to "Everybody Loves Raymond" during a given week? to "Law & Order"?

Solution The probability that the TV in one household is set to "Everybody Loves Raymond" is P, and
P = 7.815.000/105.500.000 ≈ 0.074 ≈ 7.4%.
The possibility that the household TV was set to "Law & Order" is P, and
P = 8.302.000/105.500.000 ≈ 0.079 ≈ 7.9%.
These percentages are called ratings.

theoretical probability

Suppose we are doing an experiment, such as tossing a coin or dart, drawing a card from a deck, or testing items on an assembly line. Each possible outcome of such an experiment is called Exodus . The set of all possible outcomes is called outcome space . Event it is a set of outcomes, that is, a subset of the space of outcomes.

Example 4 Throwing darts. Suppose that in the "throwing darts" experiment, the dart hits the target. Find each of the following:

b) Outcome space

Decision
a) Outcomes are: hitting black (H), hitting red (K) and hitting white (B).

b) There is an outcome space (hit black, hit red, hit white), which can be written simply as (B, R, B).

Example 5 Throwing dice. A die is a cube with six sides, each of which has one to six dots.


Suppose we are throwing a die. Find
a) Outcomes
b) Outcome space

Decision
a) Outcomes: 1, 2, 3, 4, 5, 6.
b) Outcome space (1, 2, 3, 4, 5, 6).

We denote the probability that an event E occurs as P(E). For example, "the coin will land on tails" can be denoted by H. Then P(H) is the probability that the coin will land on tails. When all outcomes of an experiment have the same probability of occurring, they are said to be equally likely. To see the difference between events that are equally likely and events that are not equally likely, consider the target shown below.

For target A, black, red, and white hit events are equally likely, since black, red, and white sectors are the same. However, for target B, the zones with these colors are not the same, that is, hitting them is not equally likely.

Principle P (Theoretical)

If an event E can happen in m ways out of n possible equiprobable outcomes from the outcome space S, then theoretical probability event, P(E) is
P(E) = m/n.

Example 6 What is the probability of rolling a 3 by rolling a die?

Decision There are 6 equally likely outcomes on the die and there is only one possibility of throwing the number 3. Then the probability P will be P(3) = 1/6.

Example 7 What is the probability of rolling an even number on the die?

Decision The event is the throwing of an even number. This can happen in 3 ways (if you roll 2, 4 or 6). The number of equiprobable outcomes is 6. Then the probability P(even) = 3/6, or 1/2.

We will be using a number of examples related to a standard 52-card deck. Such a deck consists of the cards shown in the figure below.

Example 8 What is the probability of drawing an ace from a well-shuffled deck of cards?

Decision There are 52 outcomes (the number of cards in the deck), they are equally likely (if the deck is well mixed), and there are 4 ways to draw an ace, so according to the P principle, the probability
P(drawing an ace) = 4/52, or 1/13.

Example 9 Suppose we choose without looking one marble from a bag of 3 red marbles and 4 green marbles. What is the probability of choosing a red ball?

Decision There are 7 equally likely outcomes to get any ball, and since the number of ways to draw a red ball is 3, we get
P(choosing a red ball) = 3/7.

The following statements are results from the P principle.

Probability Properties

a) If the event E cannot happen, then P(E) = 0.
b) If the event E is bound to happen then P(E) = 1.
c) The probability that event E will occur is a number between 0 and 1: 0 ≤ P(E) ≤ 1.

For example, in tossing a coin, the event that the coin lands on its edge has zero probability. The probability that a coin is either heads or tails has a probability of 1.

Example 10 Suppose that 2 cards are drawn from a deck with 52 cards. What is the probability that both of them are spades?

Decision The number of ways n of drawing 2 cards from a well-shuffled 52-card deck is 52 C 2 . Since 13 of the 52 cards are spades, the number m of ways to draw 2 spades is 13 C 2 . Then,
P(stretching 2 peaks) \u003d m / n \u003d 13 C 2 / 52 C 2 \u003d 78/1326 \u003d 1/17.

Example 11 Suppose 3 people are randomly selected from a group of 6 men and 4 women. What is the probability that 1 man and 2 women will be chosen?

Decision Number of ways to choose three people from a group of 10 people 10 C 3 . One man can be chosen in 6 C 1 ways and 2 women can be chosen in 4 C 2 ways. According to the fundamental principle of counting, the number of ways to choose the 1st man and 2 women is 6 C 1 . 4C2. Then, the probability that 1 man and 2 women will be chosen is
P = 6 C 1 . 4 C 2 / 10 C 3 \u003d 3/10.

Example 12 Throwing dice. What is the probability of throwing a total of 8 on two dice?

Decision There are 6 possible outcomes on each dice. The outcomes are doubled, that is, there are 6.6 or 36 possible ways in which the numbers on two dice can fall. (It's better if the cubes are different, say one is red and the other is blue - this will help visualize the result.)

Pairs of numbers that add up to 8 are shown in the figure below. There are 5 possible ways to get the sum equal to 8, hence the probability is 5/36.

INTRODUCTION

Many things are incomprehensible to us, not because our concepts are weak;
but because these things do not enter the circle of our concepts.
Kozma Prutkov

The main goal of studying mathematics in secondary specialized educational institutions is to give students a set of mathematical knowledge and skills necessary for studying other program disciplines that use mathematics to one degree or another, for the ability to perform practical calculations, for the formation and development of logical thinking.

In this paper, all the basic concepts of the section of mathematics "Fundamentals of Probability Theory and Mathematical Statistics", provided for by the program and the State Educational Standards of Secondary Vocational Education (Ministry of Education of the Russian Federation. M., 2002), are consistently introduced, the main theorems are formulated, most of which are not proved . The main tasks and methods for their solution and technologies for applying these methods to solving practical problems are considered. The presentation is accompanied by detailed comments and numerous examples.

Methodical instructions can be used for initial acquaintance with the studied material, when taking notes of lectures, for preparing for practical exercises, for consolidating the acquired knowledge, skills and abilities. In addition, the manual will be useful for undergraduate students as a reference tool that allows you to quickly restore in memory what was previously studied.

At the end of the work, examples and tasks are given that students can perform in self-control mode.

Methodological instructions are intended for students of correspondence and full-time forms of education.

BASIC CONCEPTS

Probability theory studies the objective regularities of mass random events. It is a theoretical basis for mathematical statistics, dealing with the development of methods for collecting, describing and processing the results of observations. Through observations (tests, experiments), i.e. experience in the broad sense of the word, there is a knowledge of the phenomena of the real world.

In our practical activities, we often encounter phenomena, the outcome of which cannot be predicted, the result of which depends on chance.

A random phenomenon can be characterized by the ratio of the number of its occurrences to the number of trials, in each of which, under the same conditions of all trials, it could occur or not occur.

Probability theory is a branch of mathematics in which random phenomena (events) are studied and regularities are revealed when they are massively repeated.

Mathematical statistics is a branch of mathematics that has as its subject the study of methods for collecting, systematizing, processing and using statistical data to obtain scientifically sound conclusions and make decisions.

At the same time, statistical data is understood as a set of numbers that represent the quantitative characteristics of the features of the studied objects that are of interest to us. Statistical data are obtained as a result of specially designed experiments and observations.

Statistical data in its essence depend on many random factors, so mathematical statistics is closely related to probability theory, which is its theoretical basis.

I. PROBABILITY. THEOREMS OF ADDITION AND PROBABILITY MULTIPLICATION

1.1. Basic concepts of combinatorics

In the section of mathematics called combinatorics, some problems are solved related to the consideration of sets and the compilation of various combinations of elements of these sets. For example, if we take 10 different numbers 0, 1, 2, 3,:, 9 and make combinations of them, we will get different numbers, for example 143, 431, 5671, 1207, 43, etc.

We see that some of these combinations differ only in the order of the digits (for example, 143 and 431), others in the numbers included in them (for example, 5671 and 1207), and others also differ in the number of digits (for example, 143 and 43).

Thus, the obtained combinations satisfy various conditions.

Depending on the compilation rules, three types of combinations can be distinguished: permutations, placements, combinations.

Let's first get acquainted with the concept factorial.

The product of all natural numbers from 1 to n inclusive is called n-factorial and write.

Calculate: a) ; b) ; in) .

Decision. a) .

b) as well as , then you can take it out of brackets

Then we get

in) .

Permutations.

A combination of n elements that differ from each other only in the order of the elements is called a permutation.

Permutations are denoted by the symbol P n , where n is the number of elements in each permutation. ( R- the first letter of the French word permutation- permutation).

The number of permutations can be calculated using the formula

or with factorial:

Let's remember that 0!=1 and 1!=1.

Example 2. In how many ways can six different books be arranged on one shelf?

Decision. The desired number of ways is equal to the number of permutations of 6 elements, i.e.

Accommodations.

Placements from m elements in n in each, such compounds are called that differ from each other either by the elements themselves (at least one), or by the order from the location.

Locations are denoted by the symbol , where m is the number of all available elements, n is the number of elements in each combination. ( BUT- first letter of the French word arrangement, which means "placement, putting in order").

At the same time, it is assumed that nm.

The number of placements can be calculated using the formula

,

those. the number of all possible placements from m elements by n is equal to the product n consecutive integers, of which the greater is m.

We write this formula in factorial form:

Example 3. How many options for the distribution of three vouchers to a sanatorium of various profiles can be made for five applicants?

Decision. The desired number of options is equal to the number of placements of 5 elements by 3 elements, i.e.

.

Combinations.

Combinations are all possible combinations of m elements by n, which differ from each other by at least one element (here m and n- natural numbers, and nm).

Number of combinations from m elements by n are denoted ( With- the first letter of the French word combination- combination).

In general, the number of m elements by n equal to the number of placements from m elements by n divided by the number of permutations from n elements:

Using factorial formulas for placement and permutation numbers, we get:

Example 4. In a team of 25 people, you need to allocate four to work in a certain area. In how many ways can this be done?

Decision. Since the order of the chosen four people does not matter, this can be done in ways.

We find by the first formula

.

In addition, when solving problems, the following formulas are used that express the main properties of combinations:

(by definition, and are assumed);

.

1.2. Solving combinatorial problems

Task 1. 16 subjects are studied at the faculty. On Monday, you need to put 3 subjects in the schedule. In how many ways can this be done?

Decision. There are as many ways to schedule three items out of 16 as there are placements of 16 elements of 3 each.

Task 2. Out of 15 objects, 10 objects must be selected. In how many ways can this be done?

Task 3. Four teams participated in the competition. How many options for the distribution of seats between them are possible?

.

Problem 4. In how many ways can a patrol of three soldiers and one officer be formed if there are 80 soldiers and 3 officers?

Decision. Soldier on patrol can be selected

ways, and officers ways. Since any officer can go with each team of soldiers, there are only ways.

Task 5. Find if it is known that .

Since , we get

,

,

By definition of combination it follows that , . That. .

1.3. The concept of a random event. Event types. Event Probability

Any action, phenomenon, observation with several different outcomes, realized under a given set of conditions, will be called test.

The result of this action or observation is called event .

If an event under given conditions can occur or not occur, then it is called random . In the event that an event must certainly occur, it is called authentic , and in the case when it certainly cannot happen, - impossible.

The events are called incompatible if only one of them can appear each time.

The events are called joint if, under the given conditions, the occurrence of one of these events does not exclude the occurrence of the other in the same test.

The events are called opposite , if under the test conditions they, being its only outcomes, are incompatible.

Events are usually denoted by capital letters of the Latin alphabet: A, B, C, D, : .

A complete system of events A 1 , A 2 , A 3 , : , A n is a set of incompatible events, the occurrence of at least one of which is mandatory for a given test.

If a complete system consists of two incompatible events, then such events are called opposite and are denoted by A and .

Example. There are 30 numbered balls in a box. Determine which of the following events are impossible, certain, opposite:

got a numbered ball (BUT);

draw an even numbered ball (AT);

drawn a ball with an odd number (WITH);

got a ball without a number (D).

Which of them form a complete group?

Decision . BUT- certain event; D- impossible event;

In and With- opposite events.

The complete group of events is BUT and D, V and With.

The probability of an event is considered as a measure of the objective possibility of the occurrence of a random event.

1.4. The classical definition of probability

The number, which is an expression of the measure of the objective possibility of the occurrence of an event, is called probability this event and is denoted by the symbol P(A).

Definition. Probability of an event BUT is the ratio of the number of outcomes m that favor the occurrence of a given event BUT, to the number n all outcomes (incompatible, unique and equally possible), i.e. .

Therefore, in order to find the probability of an event, it is necessary, after considering the various outcomes of the test, to calculate all possible incompatible outcomes n, choose the number of outcomes we are interested in m and calculate the ratio m to n.

The following properties follow from this definition:

The probability of any trial is a non-negative number not exceeding one.

Indeed, the number m of the desired events lies within . Dividing both parts into n, we get

2. The probability of a certain event is equal to one, because .

3. The probability of an impossible event is zero because .

Problem 1. There are 200 winners out of 1000 tickets in the lottery. One ticket is drawn at random. What is the probability that this ticket wins?

Decision. The total number of different outcomes is n=1000. The number of outcomes favoring the winning is m=200. According to the formula, we get

.

Task 2. In a batch of 18 parts, there are 4 defective ones. 5 pieces are chosen at random. Find the probability that two out of these 5 parts are defective.

Decision. Number of all equally possible independent outcomes n is equal to the number of combinations from 18 to 5 i.e.

Let's calculate the number m favorable for the event A. Among the 5 parts taken at random, there should be 3 quality and 2 defective ones. The number of ways to select two defective parts from 4 available defective parts is equal to the number of combinations from 4 to 2:

The number of ways to select three quality parts from 14 available quality parts is equal to

.

Any group of quality parts can be combined with any group of defective parts, so the total number of combinations m is

The desired probability of the event A is equal to the ratio of the number of outcomes m that favor this event to the number n of all equally possible independent outcomes:

.

The sum of a finite number of events is an event consisting in the occurrence of at least one of them.

The sum of two events is denoted by the symbol A + B, and the sum n events symbol A 1 +A 2 + : +A n .

The theorem of addition of probabilities.

The probability of the sum of two incompatible events is equal to the sum of the probabilities of these events.

Corollary 1. If the event А 1 , А 2 , : , А n form a complete system, then the sum of the probabilities of these events is equal to one.

Corollary 2. The sum of the probabilities of opposite events and is equal to one.

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Problem 1. There are 100 lottery tickets. It is known that 5 tickets get a win of 20,000 rubles, 10 - 15,000 rubles, 15 - 10,000 rubles, 25 - 2,000 rubles. and nothing for the rest. Find the probability that the purchased ticket will win at least 10,000 rubles.

Decision. Let A, B, and C be events consisting in the fact that a prize equal to 20,000, 15,000 and 10,000 rubles falls on the purchased ticket. since the events A, B and C are incompatible, then

Task 2. The correspondence department of the technical school receives tests in mathematics from cities A, B and With. The probability of receipt of control work from the city BUT equal to 0.6, from the city AT- 0.1. Find the probability that the next control work will come from the city With.

The simplest example of a connection between two events is a causal relationship, when the occurrence of one of the events necessarily leads to the occurrence of the other, or vice versa, when the occurrence of one excludes the possibility of the occurrence of the other.

To characterize the dependence of some events on others, the concept is introduced conditional probability.

Definition. Let be BUT and AT- two random events of the same test. Then the conditional probability of the event BUT or the probability of event A, provided that event B has occurred, is called the number.

Denoting the conditional probability , we obtain the formula

, .

Task 1. Calculate the probability that a second boy will be born in a family with one boy child.

Decision. Let the event BUT consists in the fact that there are two boys in the family, and the event AT- that one boy.

Consider all possible outcomes: boy and boy; boy and girl; girl and boy; girl and girl.

Then , and by the formula we find

.

Event BUT called independent from the event AT if the occurrence of the event AT has no effect on the probability of an event occurring BUT.

Probability multiplication theorem

The probability of the simultaneous occurrence of two independent events is equal to the product of the probabilities of these events:

The probability of the occurrence of several events that are independent in the aggregate is calculated by the formula

Problem 2. The first urn contains 6 black and 4 white balls, the second urn contains 5 black and 7 white balls. One ball is drawn from each urn. What is the probability that both balls are white.

A and AT there is an event AB. Hence,

b) If the first element works, then an event occurs (the opposite of the event BUT- the failure of this element); if the second element works - event AT. Find the probabilities of events and :

Then the event consisting in the fact that both elements will work is, and, therefore,