How to calculate the probability of numbers falling out. Calculation of the probability of combining (logical sum) of events

“Accidents are not accidental”... It sounds like something a philosopher said, but in fact, studying accidents is the destiny great science mathematics. In mathematics, chance is dealt with by probability theory. Formulas and examples of tasks, as well as the basic definitions of this science will be presented in the article.

What is probability theory?

Probability theory is one of the mathematical disciplines that studies random events.

To make it a little clearer, let's give a small example: if you throw a coin up, it can land on heads or tails. While the coin is in the air, both of these probabilities are possible. That is, the probability of possible consequences is 1:1. If one is drawn from a deck of 36 cards, then the probability will be indicated as 1:36. It would seem that there is nothing to explore and predict here, especially with the help of mathematical formulas. However, if you repeat a certain action many times, you can identify a certain pattern and, based on it, predict the outcome of events in other conditions.

To summarize all of the above, probability theory in the classical sense studies the possibility of the occurrence of one of the possible events in a numerical value.

From the pages of history

The theory of probability, formulas and examples of the first tasks appeared in the distant Middle Ages, when attempts to predict the outcome of card games first arose.

Initially, probability theory had nothing to do with mathematics. It was justified by empirical facts or properties of an event that could be reproduced in practice. The first works in this area as a mathematical discipline appeared in the 17th century. The founders were Blaise Pascal and Pierre Fermat. Long time they studied gambling and saw certain patterns, which they decided to tell society about.

The same technique was invented by Christiaan Huygens, although he was not familiar with the results of the research of Pascal and Fermat. The concept of “probability theory”, formulas and examples, which are considered the first in the history of the discipline, were introduced by him.

The works of Jacob Bernoulli, Laplace's and Poisson's theorems are also of no small importance. They made probability theory more like a mathematical discipline. Probability theory, formulas and examples of basic tasks received their current form thanks to Kolmogorov’s axioms. As a result of all the changes, probability theory became one of the mathematical branches.

Basic concepts of probability theory. Events

The main concept of this discipline is “event”. There are three types of events:

Reliable. Those that will happen anyway (the coin will fall).
Impossible. Events that will not happen under any circumstances (the coin will remain hanging in the air).
Random. The ones that will happen or won't happen. They can be influenced by various factors that are very difficult to predict. If we talk about a coin, then there are random factors that can affect the result: the physical characteristics of the coin, its shape, its original position, the force of the throw, etc.

All events in the examples are indicated in capital Latin letters, with the exception of P, which has a different role. For example:

A = “students came to lecture.”
Ā = “students did not come to the lecture.”

IN practical tasks Events are usually recorded in words.

One of the most important characteristics events - their equal possibility. That is, if you toss a coin, all variants of the initial fall are possible until it falls. But events are also not equally possible. This happens when someone deliberately influences an outcome. For example, "labeled" playing cards or dice in which the center of gravity is shifted.

Events can also be compatible and incompatible. Compatible events do not exclude each other's occurrence. For example:

A = “the student came to the lecture.”
B = “the student came to the lecture.”

These events are independent of each other, and the occurrence of one of them does not affect the occurrence of the other. Incompatible events are defined by the fact that the occurrence of one excludes the occurrence of another. If we talk about the same coin, then the loss of “tails” makes it impossible for the appearance of “heads” in the same experiment.

Actions on events

Events can be multiplied and added; accordingly, logical connectives “AND” and “OR” are introduced in the discipline.

The amount is determined by the fact that either event A or B, or two, can occur simultaneously. If they are incompatible, the last option is impossible; either A or B will be rolled.

Multiplication of events consists in the appearance of A and B at the same time.

Now we can give several examples to better remember the basics, probability theory and formulas. Examples of problem solving below.

Exercise 1: The company takes part in a competition to receive contracts for three types of work. Possible events that may occur:

A = “the firm will receive the first contract.”
A 1 = “the firm will not receive the first contract.”
B = “the firm will receive a second contract.”
B 1 = “the firm will not receive a second contract”
C = “the firm will receive a third contract.”
C 1 = “the firm will not receive a third contract.”

Using actions on events, we will try to express the following situations:

K = “the company will receive all contracts.”

In mathematical form, the equation will have the following form: K = ABC.

M = “the company will not receive a single contract.”

M = A 1 B 1 C 1.

Let’s complicate the task: H = “the company will receive one contract.” Since it is not known which contract the company will receive (first, second or third), it is necessary to record the entire series of possible events:

H = A 1 BC 1 υ AB 1 C 1 υ A 1 B 1 C.

And 1 BC 1 is a series of events where the firm does not receive the first and third contract, but receives the second. Other possible events were recorded using the appropriate method. The symbol υ in the discipline denotes the connective “OR”. If we translate the above example into human language, the company will receive either the third contract, or the second, or the first. In a similar way, you can write down other conditions in the discipline “Probability Theory”. The formulas and examples of problem solving presented above will help you do this yourself.

Actually, the probability

Perhaps, in this mathematical discipline, the probability of an event is the central concept. There are 3 definitions of probability:

classic;
statistical;
geometric.

Each has its place in the study of probability. Probability theory, formulas and examples (grade 9) mainly use classic definition, which sounds like this:

The probability of situation A is equal to the ratio of the number of outcomes that favor its occurrence to the number of all possible outcomes.

The formula looks like this: P(A)=m/n.

A is actually an event. If a case opposite to A appears, it can be written as Ā or A 1 .

m is the number of possible favorable cases.

n - all events that can happen.

For example, A = “draw a card of the heart suit.” There are 36 cards in a standard deck, 9 of them are of hearts. Accordingly, the formula for solving the problem will look like:

P(A)=9/36=0.25.

As a result, the probability that a card of the heart suit will be drawn from the deck will be 0.25.

Toward higher mathematics

Now it has become a little known what probability theory is, formulas and examples of solving problems that come across in school curriculum. However, probability theory is also found in higher mathematics, which is taught in universities. Most often they operate with geometric and statistical definitions of the theory and complex formulas.

The theory of probability is very interesting. It is better to start studying formulas and examples (higher mathematics) small - with the statistical (or frequency) definition of probability.

The statistical approach does not contradict the classical one, but slightly expands it. If in the first case it was necessary to determine with what probability an event will occur, then in this method it is necessary to indicate how often it will occur. Here a new concept of “relative frequency” is introduced, which can be denoted by W n (A). The formula is no different from the classic one:

If classic formula calculated for prediction, then statistical - according to the results of the experiment. Let's take a small task for example.

The technological control department checks products for quality. Among 100 products, 3 were found to be of poor quality. How to find the frequency probability of a quality product?

A = “the appearance of a quality product.”

W n (A)=97/100=0.97

Thus, the frequency of a quality product is 0.97. Where did you get 97 from? Out of 100 products that were checked, 3 were found to be of poor quality. We subtract 3 from 100 and get 97, this is the amount of quality goods.

A little about combinatorics

Another method of probability theory is called combinatorics. Its basic principle is that if a certain choice A can be made m different ways, and the choice of B is in n different ways, then the choice of A and B can be done by multiplication.

For example, there are 5 roads leading from city A to city B. There are 4 paths from city B to city C. In how many ways can you get from city A to city C?

It's simple: 5x4=20, that is, in twenty different ways you can get from point A to point C.

Let's complicate the task. How many ways are there to lay out cards in solitaire? There are 36 cards in the deck - this is the starting point. To find out the number of ways, you need from starting point“subtract” one card at a time and multiply.

That is, 36x35x34x33x32...x2x1= the result does not fit on the calculator screen, so it can simply be designated 36!. Sign "!" next to the number indicates that the entire series of numbers is multiplied together.

In combinatorics there are such concepts as permutation, placement and combination. Each of them has its own formula.

An ordered set of elements of a set is called an arrangement. Placements can be repeated, that is, one element can be used several times. And without repetition, when elements are not repeated. n are all elements, m are elements that participate in the placement. The formula for placement without repetition will look like:

A n m =n!/(n-m)!

Connections of n elements that differ only in the order of placement are called permutations. In mathematics it looks like: P n = n!

Combinations of n elements of m are those compounds in which it is important what elements they were and what their total. The formula will look like:

A n m =n!/m!(n-m)!

Bernoulli's formula

In probability theory, as in every discipline, there are works of outstanding researchers in their field who have taken it to a new level. One of these works is the Bernoulli formula, which allows you to determine the probability of a certain event occurring under independent conditions. This suggests that the occurrence of A in an experiment does not depend on the occurrence or non-occurrence of the same event in earlier or subsequent trials.

Bernoulli's equation:

P n (m) = C n m ×p m ×q n-m.

The probability (p) of the occurrence of event (A) is constant for each trial. The probability that the situation will occur exactly m times in n number of experiments will be calculated by the formula presented above. Accordingly, the question arises of how to find out the number q.

If event A occurs p number of times, accordingly, it may not occur. Unit is a number that is used to designate all outcomes of a situation in a discipline. Therefore, q is a number that denotes the possibility of an event not occurring.

Now you know Bernoulli's formula (probability theory). We will consider examples of problem solving (first level) below.

Task 2: A store visitor will make a purchase with probability 0.2. 6 visitors independently entered the store. What is the likelihood that a visitor will make a purchase?

Solution: Since it is unknown how many visitors should make a purchase, one or all six, it is necessary to calculate all possible probabilities, using Bernoulli's formula.

A = “the visitor will make a purchase.”

In this case: p = 0.2 (as indicated in the task). Accordingly, q=1-0.2 = 0.8.

n = 6 (since there are 6 customers in the store). The number m will vary from 0 (not a single customer will make a purchase) to 6 (all visitors to the store will purchase something). As a result, we get the solution:

P 6 (0) = C 0 6 ×p 0 ×q 6 =q 6 = (0.8) 6 = 0.2621.

None of the buyers will make a purchase with probability 0.2621.

How else is Bernoulli's formula (probability theory) used? Examples of problem solving (second level) below.

After the above example, questions arise about where C and r went. Relative to p, a number to the power of 0 will be equal to one. As for C, it can be found by the formula:

C n m = n! /m!(n-m)!

Since in the first example m = 0, respectively, C = 1, which in principle does not affect the result. Using the new formula, let's try to find out what is the probability of two visitors purchasing goods.

P 6 (2) = C 6 2 ×p 2 ×q 4 = (6×5×4×3×2×1) / (2×1×4×3×2×1) × (0.2) 2 × (0.8) 4 = 15 × 0.04 × 0.4096 = 0.246.

The theory of probability is not that complicated. Bernoulli's formula, examples of which are presented above, is direct proof of this.

Poisson's formula

Poisson's equation is used to calculate low probability random situations.

Basic formula:

P n (m)=λ m /m! × e (-λ) .

In this case λ = n x p. Here is a simple Poisson formula (probability theory). We will consider examples of problem solving below.

Task 3: The factory produced 100,000 parts. Occurrence of a defective part = 0.0001. What is the probability that there will be 5 defective parts in a batch?

As you can see, marriage is an unlikely event, and therefore the Poisson formula (probability theory) is used for calculation. Examples of problem solving this kind are no different from other tasks in the discipline; we substitute the necessary data into the given formula:

A = “a randomly selected part will be defective.”

p = 0.0001 (according to the task conditions).

n = 100000 (number of parts).

m = 5 (defective parts). We substitute the data into the formula and get:

R 100000 (5) = 10 5 /5! X e -10 = 0.0375.

Just like the Bernoulli formula (probability theory), examples of solutions using which are written above, the Poisson equation has an unknown e. In fact, it can be found by the formula:

e -λ = lim n ->∞ (1-λ/n) n .

However, there are special tables that contain almost all values of e.

De Moivre-Laplace theorem

If in the Bernoulli scheme the number of trials is sufficiently large, and the probability of occurrence of event A in all schemes is the same, then the probability of occurrence of event A a certain number of times in a series of tests can be found by Laplace’s formula:

Р n (m)= 1/√npq x ϕ(X m).

X m = m-np/√npq.

To better remember Laplace’s formula (probability theory), examples of problems are below to help.

First, let's find X m, substitute the data (they are all listed above) into the formula and get 0.025. Using tables, we find the number ϕ(0.025), the value of which is 0.3988. Now you can substitute all the data into the formula:

P 800 (267) = 1/√(800 x 1/3 x 2/3) x 0.3988 = 3/40 x 0.3988 = 0.03.

Thus, the probability that the flyer will work exactly 267 times is 0.03.

Bayes formula

The Bayes formula (probability theory), examples of solving problems with the help of which will be given below, is an equation that describes the probability of an event based on the circumstances that could be associated with it. The basic formula is as follows:

P (A|B) = P (B|A) x P (A) / P (B).

A and B are definite events.

P(A|B) is a conditional probability, that is, event A can occur provided that event B is true.

P (B|A) - conditional probability of event B.

So, the final part of the short course “Probability Theory” is the Bayes formula, examples of solutions to problems with which are below.

Task 5: Phones from three companies were brought to the warehouse. At the same time, the share of phones that are manufactured at the first plant is 25%, at the second - 60%, at the third - 15%. It is also known that the average percentage of defective products at the first factory is 2%, at the second - 4%, and at the third - 1%. You need to find the probability that a randomly selected phone will be defective.

A = “randomly picked phone.”

B 1 - the phone that the first factory produced. Accordingly, introductory B 2 and B 3 will appear (for the second and third factories).

As a result we get:

P (B 1) = 25%/100% = 0.25; P(B 2) = 0.6; P (B 3) = 0.15 - thus we found the probability of each option.

Now you need to find the conditional probabilities of the desired event, that is, the probability of defective products in companies:

P (A/B 1) = 2%/100% = 0.02;

P(A/B 2) = 0.04;

P (A/B 3) = 0.01.

Now let’s substitute the data into the Bayes formula and get:

P (A) = 0.25 x 0.2 + 0.6 x 0.4 + 0.15 x 0.01 = 0.0305.

The article presents probability theory, formulas and examples of problem solving, but this is only the tip of the iceberg of a vast discipline. And after everything that has been written, it would be logical to ask the question of whether the theory of probability is needed in life. To the common man It’s difficult to answer, it’s better to ask someone who has used it to win the jackpot more than once.

I understand that everyone wants to know in advance how the sporting event will end, who will win and who will lose. With this information, you can place bets on sporting events. But is it even possible, and if so, how to calculate the probability of an event?

Probability is a relative value, therefore it cannot speak with certainty about any event. This value allows you to analyze and evaluate the need to place a bet on a particular competition. Determining probabilities is a whole science that requires careful study and understanding.

Probability coefficient in probability theory

In sports betting, there are several options for the outcome of the competition:

first team victory;
victory of the second team;
draw;
total

Each outcome of the competition has its own probability and frequency with which this event will occur, provided that the initial characteristics are maintained. As we said earlier, it is impossible to accurately calculate the probability of any event - it may or may not coincide. Thus, your bet can either win or lose.

There cannot be a 100% accurate prediction of the results of the competition, since many factors influence the outcome of the match. Naturally, bookmakers do not know the outcome of the match in advance and only assume the result, making decisions using their analysis system and offering certain odds for betting.

How to calculate the probability of an event?

Let’s assume that the bookmaker’s odds are 2.1/2 – we get 50%. It turns out that the coefficient is 2 equal to probability 50%. Using the same principle, you can get a break-even probability coefficient - 1/probability.

Many players think that after several repeated defeats, a win will definitely happen - this is a mistaken opinion. The probability of winning a bet does not depend on the number of losses. Even if you flip several heads in a row in a coin game, the probability of flipping tails remains the same - 50%.

First level

Probability theory. Problem Solving (2019)

What is probability?

The first time I encountered this term, I would not have understood what it was. Therefore, I will try to explain clearly.

Probability is the chance that the event we want will happen.

For example, you decided to go to a friend’s house, you 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 staircase, and in front of you there are doors to choose from.

What is the chance (probability) that if you ring the first doorbell, your friend will answer the door for you? There are only apartments, and a friend lives only behind one of them. With an equal chance we can choose any door.

But what is this chance?

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

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

You called 1st door
You called 2nd door
You called 3rd door

Now let’s look at all the options where a friend could be:

A. Behind 1st the door
b. Behind 2nd the door
V. Behind 3rd the door

Let's compare all the options in table form. A checkmark indicates options when your choice coincides with a friend's location, a cross - when it does not coincide.

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

A favorable outcomes of all . That is, you will guess once by ringing the doorbell once, i.e. .

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

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

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

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

The word “outcomes” probably caught your eye. Since mathematicians call various actions (in our case, such an action is a doorbell) experiments, the result of such experiments is usually called the outcome.

Well, there are favorable and unfavorable outcomes.

Let's go back to our example. Let's say we rang one of the doors, but it was opened for us stranger. We didn't guess right. 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 1st door
2) Call 2nd door

The friend, despite all this, is definitely behind one of them (after all, he wasn’t behind the one we called):

a) Friend for 1st the door
b) Friend for 2nd the door

Let's draw the table again:

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

Why not?

The situation we 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 influence the following actions. After all, if after the first ring the doorbell was answered by a friend, what would be the probability that he was behind one of the other two? Right, .

But if there are dependent events, then there must also be independent? That's right, they do happen.

A textbook example is tossing a coin.

Toss a coin once. What is the probability of getting heads, for example? That's right - because there are all the options (either heads or tails, we will neglect the probability of the coin landing on its edge), but it only suits us.
But it came up heads. Okay, let's throw it again. What is the probability of getting heads now? Nothing has changed, everything is the same. How many options? Two. How many are we happy with? One.

And let it come up heads at least a thousand times in a row. The probability of getting heads at once will be the same. There are always options, and favorable ones.

It is easy to distinguish dependent events from independent ones:

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

Let's practice determining probability a little.

Example 1.

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

Solution:

Let's consider all possible options:

Eagle-eagle
Heads-tails
Tails-Heads
Tails-tails

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

If the condition simply asks you to find the probability, then the answer must be given in the form of a decimal fraction. If it were specified that the answer should be given as a percentage, then we would multiply by.

Answer:

Example 2.

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

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

Solution:

How many possible outcomes are there? .

That is, if you take one candy, it will be one of those available in the box.

How many favorable outcomes?

Because the box contains only chocolates with nuts.

Answer:

Example 3.

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

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

Solution:

a) There are only balls in the box. Of them are white.

The probability is:

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

Answer:

Total probability

The probability of all possible events is equal to ().

Let's say there are red and green balls in a box. 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 markers in the box: green, red, blue, yellow, black.

What is the probability of drawing NOT a red marker?

Solution:

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 take out a red marker) is .

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

Answer:

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

Rule for multiplying the probabilities of independent events

You already know what independent events are.

What 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 if we flip a coin once, we will see heads twice?

We have already considered - .

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

Total possible options:

Eagle-eagle-eagle
Heads-heads-tails
Heads-tails-heads
Heads-tails-tails
Tails-heads-heads
Tails-heads-tails
Tails-tails-heads
Tails-tails-tails

I don’t know about you, but I made mistakes several times when compiling this list. Wow! And only option (the first) suits us.

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

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

In other words,

Let's look at the example of the same ill-fated coin.

Probability of getting heads in a challenge? . Now we flip the coin once.

What is the probability of getting heads in a row?

This rule doesn't only work if we are asked to find the probability that the same event will happen several times in a row.

If we wanted to find the sequence TAILS-HEADS-TAILS for consecutive tosses, we would do the same.

The probability of getting tails is , heads - .

Probability of getting the sequence TAILS-HEADS-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 toss it once.
Possible options:

Eagle-eagle-eagle
Heads-heads-tails
Heads-tails-heads
Heads-tails-tails
Tails-heads-heads
Tails-heads-tails
Tails-tails-heads
Tails-tails-tails

So, incompatible events are a certain, given sequence of events. - these are incompatible events.

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

You need to understand that heads or tails are two independent events.

If we want to determine the probability of a sequence (or any other) occurring, 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 tosses?

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

Total options suit 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 certain, inconsistent, sequences of events.

There is a great rule to help you avoid getting confused when to multiply and when to add:

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

Should fall out:
(heads AND tails AND tails) OR (tails AND heads AND tails) OR (tails AND tails AND heads).
This is how 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 pencil And?

Solution:

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

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

Answer:

Example 6.

If a die is thrown twice, what is the probability of getting a total of 8?

Solution.

How can we get points?

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

The probability of getting one (any) face is .

We calculate the probability:

Answer:

Training.

I think now you understand when you need to calculate probabilities, when to add them, and when to multiply them. Is not it? Let's practice a little.

Tasks:

Let's take a card deck containing cards including spades, hearts, 13 clubs and 13 diamonds. From to Ace of each suit.

What is the probability of drawing clubs in a row (we put the first card pulled out back into the deck and shuffle it)?
What is the probability of drawing a black card (spades or clubs)?
What is the probability of drawing a picture (jack, queen, king or ace)?
What is the probability of drawing two pictures in a row (we remove the first card drawn from the deck)?
What is the probability, taking two cards, to collect a combination - (jack, queen or king) and an ace? The sequence in which the cards are drawn does not matter.

Answers:

In a deck of cards of each value, it means:
Events are dependent, since after the first card pulled out, the number of cards in the deck decreased (as did the number of “pictures”). There are total jacks, queens, kings and aces in the deck initially, which means the probability of drawing a “picture” with the first card:
Since we remove the first card from the deck, it means that there are already cards left in the deck, including pictures. Probability of drawing a picture with the second card:
Since we are interested in the situation when we take out a “picture” AND a “picture” from the deck, we need to multiply the probabilities:
Answer:
After the first card pulled out, the number of cards in the deck will decrease. Thus, two options suit us:
1) The first card is Ace, the second is Jack, Queen or King
2) We take out a jack, queen or king with the first card, and an ace with the second. (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 great! Now you will crack probability theory problems in the Unified State Exam like nuts!

PROBABILITY THEORY. AVERAGE LEVEL

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

So we roll the dice and we want it to come up or. And we get it.

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

If it happened, the event would also be favorable. In total, only two favorable events can happen.

How many are unfavorable? Since there are total possible events, it means that the unfavorable ones are events (this is if or falls out).

Definition:

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

Indicates probability Latin letter(apparently from English word probability - probability).

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

And in percentage: .

Examples (decide for yourself):

What is the probability of getting heads when tossing a coin? What is the probability of landing heads?
What is the probability when throwing dice Will the number come up even? Which one is odd?
In a box of simple, blue and red pencils. We draw one pencil at random. What is the probability of getting a simple one?

Solutions:

How many options are there? Heads and tails - just two. How many of them are favorable? Only one is an eagle. So the probability
It's the same with tails: .
Total options: (how many sides does the cube have, so many various options). Favorable ones: (these are all even numbers:).
Probability. Of course, it’s the same with odd numbers.
Total: . Favorable: . Probability: .

Total probability

All pencils in the box 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 the same number of favorable events as there are total events (all events are favorable). So the probability is equal to or.

Such an event is called reliable.

If a box contains green and red pencils, what is the probability of drawing green or red? Yet again. Let's note this: the probability of pulling out green is equal, and red is equal.

In sum, these probabilities are exactly equal. That is, 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, plain, yellow, and the rest are orange. What is the probability of not drawing green?

Solution:

We remember that all probabilities add up. And the probability of getting 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 equal to minus the probability that the event will occur.

Independent events and the multiplication rule

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

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

Heads-Heads, Tails-Heads, Heads-Tails, Tails-Tails. What else?

Total options. Of these, only one suits us: Eagle-Eagle. In total, the probability is equal.

Fine. Now let's flip a coin once. Do the math yourself. Happened? (answer).

You may have noticed that with the addition of each subsequent throw, the probability decreases by half. General rule 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 throw a coin several times, each time a new throw is made, the result of which does not depend on all previous throws. We can just as easily throw two different coins at the same time.

More examples:

The dice are thrown twice. What is the probability of getting it both times?
The coin is tossed once. What is the probability that it will come up heads the first time, and then tails twice?
The player rolls two dice. What is the probability that the sum of the numbers on them will be equal?

Answers:

The events are independent, which means the multiplication rule works: .
The probability of heads is equal. The probability of tails is the same. Multiply:
12 can only be obtained if two -ki are rolled: .

Incompatible events and the addition rule

Events that complement each other to the point of full probability are called incompatible. As the name suggests, they cannot happen simultaneously. For example, if we flip a coin, it can come up either heads or tails.

Example.

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

Solution .

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

Favorable events in all: green + red. This means that the probability of drawing green or red is equal.

The same probability can be represented in this form: .

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

Mixed type problems

Example.

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

Solution .

This means that if the first result is heads, the second must be tails, and vice versa. It turns out that there are two pairs of independent events, 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 is going to happen using the conjunctions “AND” or “OR”. For example, in this case:

It should come up (heads and tails) or (tails and heads).

Where there is a conjunction “and” there will be multiplication, and where there is “or” there will be addition:

Try it yourself:

What is the probability that if a coin is tossed twice, the coin will land on the same side both times?
The dice are thrown twice. What is the probability of getting a total of points?

Solutions:

(Heads fell and tails fell) or (tails fell and tails fell): .
What are the options? And. Then:
Dropped (and) or (and) or (and): .

Another example:

Toss a coin once. What is the probability that heads will appear at least once?

Solution:

Oh, how I don’t want to go through the options... Heads-tails-tails, Eagle-heads-tails,... But there’s no need! Let's remember about total probability. Do you remember? What is the probability that the eagle will never fall out? It’s simple: heads fly all the time, that’s why.

PROBABILITY THEORY. BRIEFLY ABOUT THE MAIN THINGS

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.

Total probability

The probability of all possible events is equal to ().

The probability that an event will not occur is equal to 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 event

Incompatible events

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

The probabilities of incompatible events add up.

Having described what should happen, using the conjunctions “AND” or “OR”, instead of “AND” we put a multiplication sign, and instead of “OR” we put an addition sign.

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For successful passing the Unified State Exam, for admission to college on a budget and, MOST IMPORTANTLY, for life.

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People who received a good education, earn much more than those who did not receive it. This is statistics.

But this is not the main thing.

The main thing is that they are MORE HAPPY (there are such studies). Perhaps because many more opportunities open up before them and life becomes brighter? Don't know...

But think for yourself...

What does it take to be sure to be better than others on the Unified State Exam and ultimately be... happier?

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There is a whole class of experiments for which the probabilities of their possible outcomes can be easily assessed directly from the conditions of the experiment itself. To do this, it is necessary that the different outcomes of the experiment have symmetry and, therefore, be objectively equally possible.

Consider, for example, the experience of throwing a die, i.e. a symmetrical cube, on the sides of which is marked different number points: from 1 to 6.

Due to the symmetry of the cube, there is reason to consider all six possible outcomes of the experiment to be equally possible. This is what gives us the right to assume that when throwing a die multiple times, all six sides will appear approximately equally often. This assumption, for a properly made bone, is indeed justified by experience; when throwing a die multiple times, each of its sides appears in approximately one sixth of all cases of throwing, and the deviation of this fraction from 1/6 is less than larger number experiments have been carried out. Bearing in mind that the probability of a reliable event is assumed to be equal to one, it is natural to assign a probability equal to 1/6 to the loss of each individual face. This number characterizes some objective properties of this random phenomenon, namely the property of symmetry of the six possible outcomes of the experiment.

For any experiment in which the possible outcomes are symmetrical and equally possible, a similar technique can be applied, which is called direct calculation of probabilities.

The symmetry of the possible outcomes of an experiment is usually observed only in artificially organized experiments, such as gambling. Since the theory of probability received its initial development precisely in gambling schemes, the technique of directly calculating probabilities, which historically arose along with the emergence of the mathematical theory of random phenomena, for a long time was considered fundamental and was the basis of the so-called “classical” theory of probability. At the same time, experiments that did not have symmetry of possible outcomes were artificially reduced to the “classical” scheme.

Despite the limited scope practical applications of this scheme, it is still of some interest, since it is precisely through experiments that have symmetry of possible outcomes, and through events associated with such experiments, that it is easiest to become acquainted with the basic properties of probabilities. We will deal with these kinds of events, which allow for direct calculation of probabilities, first of all.

Let us first introduce some auxiliary concepts.

1. Complete group of events.

Several events in a given experiment are said to form a complete group of events if at least one of them must necessarily appear as a result of the experience.

Examples of events that form a complete group:

3) the appearance of 1,2,3,4,5,6 points when throwing a die;

4) the appearance of a white ball and the appearance of a black ball when one ball is taken out of an urn containing 2 white and 3 black balls;

5) no typos, one, two, three or more than three typos when checking a page of printed text;

6) at least one hit and at least one miss with two shots.

2. Incompatible events.

Several events are said to be incompatible in a given experience if no two of them can occur together.

Examples of incompatible events:

1) loss of the coat of arms and loss of numbers when throwing a coin;

2) hit and miss when fired;

3) the appearance of 1,3, 4 points with one throw of the dice;

4) exactly one failure, exactly two failures, exactly three failures of a technical device in ten hours of operation.

3. Equally possible events.

Several events in a given experiment are called equally possible if, according to the conditions of symmetry, there is reason to believe that none of these events is objectively more possible than the other.

Examples of equally possible events:

1) loss of the coat of arms and loss of numbers when throwing a coin;

2) the appearance of 1,3, 4, 5 points when throwing a dice;

3) the appearance of a card of diamonds, hearts, clubs when a card is removed from the deck;

4) the appearance of a ball with No. 1, 2, 3 when taking one ball from an urn containing 10 renumbered balls.

There are groups of events that have all three properties: they form a complete group, are incompatible and equally possible; for example: the appearance of a coat of arms and numbers when throwing a coin; the appearance of 1, 2, 3, 4, 5, 6 points when throwing a die. The events that form such a group are called cases (otherwise known as “chances”).

If any experience in its structure has symmetry of possible outcomes, then the cases represent an exhaustive system of equally possible and mutually exclusive outcomes of the experience. Such experience is said to be “reduced to a pattern of cases” (otherwise known as a “pattern of urns”).

The scheme of cases predominantly takes place in artificially organized experiments, in which the same possibility of experimental outcomes is ensured in advance and consciously (as, for example, in gambling). For such experiments, it is possible to directly calculate probabilities based on an assessment of the proportion of so-called “favorable” cases in the total number of cases.

A case is called favorable (or “favorable”) for a certain event if the occurrence of this case entails the occurrence of this event.

For example, when throwing a dice, six cases are possible: the appearance of 1, 2, 3, 4, 5, 6 points. Of these, the event - the appearance of an even number of points - is favorable in three cases: 2, 4, 6 and the remaining three are unfavorable.

If experience is reduced to a pattern of cases, then the probability of an event in a given experiment can be estimated by the relative proportion of favorable cases. The probability of an event is calculated as the ratio of the number of favorable cases to the total number of cases:

where P(A) is the probability of the event; – total number cases; – number of cases favorable to the event.

Since the number of favorable cases is always between 0 and (0 for an impossible event and for a certain event), the probability of an event calculated using formula (2.2.1) is always a rational proper fraction:

Formula (2.2.1), the so-called “classical formula” for calculating probabilities, has long appeared in the literature as a definition of probability. Currently, when defining (explaining) probability, they usually proceed from other principles, directly connecting the concept of probability with the empirical concept of frequency; formula (2.2.1) is preserved only as a formula for directly calculating probabilities, suitable if and only if experience is reduced to a scheme of cases, i.e. has symmetry of possible outcomes.

To increase your chances of winning, a player must understand how a bookmaker works.

Bookmaker odds represent the probability of an event with a certain percentage of markup (margin), which varies between 1.5-10% in different offices. If margins didn't exist, all bookmakers would go out of business within hours.

The player must understand what the odds are and bet only on prices that are profitable for themselves. Therefore, he needs to be able to convert odds into probabilities and vice versa.

Formula for converting a coefficient into a percentage of the probability of an event:

V=1/odf*100%

Conversion of probability into odds is calculated using the formula:

K=100%/probability

Example

The bookmaker's odds for the match between Real Madrid and Liverpool are:

2.25 (Win1) – 3.7 (draw) – 3.09 (Win2)

Converting probability coefficients

V(P1) = 1/2.25*100%= 44.4%

V(draw) = 1/3.7*100%= 27%

V(P2) = 1/3.09*100%= 32.4%

We add up the probabilities of this match and get the total probability

V = 44.4%+27%+32.4%= 103.8%

Many will wonder why the probability is more than one hundred percent. The answer is simply simple, everything over 100% is the bookmaker’s margin. In our case it is 3.8%.

Odds for equally probable events should ideally be K(P1) = K(P2) = 2.0 (50%), however, due to the bookmaker’s margin, they will be underestimated. For example, if the bookmaker’s markup is 7%, then the odds will be 1.86, if 2%, then the odds will be 1.96.

The key to success for a successful player is to always bet best odds. Bookmakers employ traders who can also make mistakes in their calculations. Skilled players make a good living from such miscalculations.

For example, the bookmaker estimates the victory of Juventus over Roma with a probability of 60% (1.66), and after carefully analyzing the match, you calculated the probability of 67% (1.49). If your calculations are correct, then the bookmaker gives an inflated (valuable) odds for this outcome of this event. The player should definitely take advantage of this opportunity by betting on Juventus to win. Such odds are called value odds and in long-term play they will certainly bring profit to the player.

If your probability was less than 60%, this would mean that the bookmaker underestimated the odds on this outcome. Placing bets on obviously low odds is strictly prohibited!

To find value bets, a player must be able to correctly analyze the probability of an outcome, although there are many reputable services that provide such services for a fee.