"Randomness is not accidental"... It sounds like a philosopher said, but in fact, the study of accidents is the destiny of the great science of mathematics. In mathematics, chance is the theory of probability. Formulas and examples of tasks, as well as the main 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 toss a coin up, it can fall heads or tails. As long as the coin is in the air, both of these possibilities are possible. That is, the probability of possible consequences correlates 1:1. If one is drawn from a deck with 36 cards, then the probability will be indicated as 1:36. It would seem that there is nothing to explore and predict, especially with the help of mathematical formulas. Nevertheless, if you repeat a certain action many times, then you can identify a certain pattern and, on its basis, predict the outcome of events in other conditions.
To summarize all of the above, the theory of probability in the classical sense studies the possibility of the occurrence of one of the possible events in a numerical sense.
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, the theory of probability 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. For a long time they studied gambling and saw certain patterns, which they decided to tell the public about.
The same technique was invented by Christian 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.
Of no small importance are the works of Jacob Bernoulli, Laplace's and Poisson's theorems. They made probability theory more like a mathematical discipline. Probability theory, formulas and examples of basic tasks got their present form thanks to Kolmogorov's axioms. As a result of all the changes, the theory of probability has become one of the mathematical branches.
Basic concepts of probability theory. Developments
The main concept of this discipline is "event". Events are of three types:
- Reliable. Those that will happen anyway (the coin will fall).
- Impossible. Events that will not happen in any scenario (the coin will remain hanging in the air).
- Random. Those that will or will not happen. They can be influenced by various factors that are very difficult to predict. If we talk about a coin, then random factors that can affect the result: the physical characteristics of the coin, its shape, initial position, throw force, etc.
All events in the examples are denoted by capital Latin letters, with the exception of R, which has a different role. For example:
- A = "students came to the lecture."
- Ā = "students didn't come to the lecture".
In practical tasks, events are usually recorded in words.
One of the most important characteristics of events is 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 probable. This happens when someone deliberately influences the outcome. For example, "marked" playing cards or dice, in which the center of gravity is shifted.
Events are also compatible and incompatible. Compatible events do not exclude the occurrence of each other. 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 appearance of one of them does not affect the appearance of the other. Incompatible events are defined by the fact that the occurrence of one precludes the occurrence of the other. 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, respectively, logical connectives "AND" and "OR" are introduced in the discipline.
The amount is determined by the fact that either event A, or B, or both can occur at the same time. In the case when they are incompatible, the last option is impossible, either A or B will drop out.
The multiplication of events consists in the appearance of A and B at the same time.
Now you can give a few examples to better remember the basics, probability theory and formulas. Examples of problem solving below.
Exercise 1: The firm is bidding for 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."
Let's try to express the following situations using actions on events:
- K = "the firm will receive all contracts."
In mathematical form, the equation will look like this: K = ABC.
- M = "the firm will not receive a single contract."
M \u003d A 1 B 1 C 1.
We complicate the task: H = "the firm will receive one contract." Since it is not known which contract the firm will receive (the first, second or third), it is necessary to record the entire range of possible events:
H \u003d 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 one. Other possible events are also recorded by the corresponding method. The symbol υ in the discipline denotes a bunch of "OR". If we translate the above example into human language, then the company will receive either the third contract, or the second, or the first. Similarly, you can write other conditions in the discipline "Probability Theory". The formulas and examples of solving problems presented above will help you do it yourself.
Actually, the probability
Perhaps, in this mathematical discipline, the probability of an event is a central concept. There are 3 definitions of probability:
- classical;
- statistical;
- geometric.
Each has its place in the study of probabilities. Probability theory, formulas and examples (Grade 9) mostly use the 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) \u003d m / n.
And, actually, an event. If the opposite of A occurs, 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 \u003d "pull out a heart suit card." 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 heart-suited card will be drawn from the deck will be 0.25.
to higher mathematics
Now it has become a little known what the theory of probability is, formulas and examples of solving tasks that come across in the school curriculum. However, the theory of probability 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. Formulas and examples (higher mathematics) are better to start learning from a small one - from a statistical (or frequency) definition of probability.
The statistical approach does not contradict the classical approach, but slightly expands it. If in the first case it was necessary to determine with what degree of 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:
If the classical formula is calculated for forecasting, then the statistical one is calculated according to the results of the experiment. Take, for example, a small task.
The department of technological control 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? Of the 100 products that were checked, 3 turned out to be of poor quality. We subtract 3 from 100, we get 97, this is the quantity of a quality product.
A bit about combinatorics
Another method of probability theory is called combinatorics. Its basic principle is that if a certain choice A can be made in m different ways, and a choice B in n different ways, then the choice of A and B can be made by multiplying.
For example, there are 5 roads from city A to city B. There are 4 routes from city B to city C. How many ways are there to get from city A to city C?
It's simple: 5x4 = 20, that is, there are twenty different ways to get from point A to point C.
Let's make the task harder. How many ways are there to play cards in solitaire? In a deck of 36 cards, this is the starting point. To find out the number of ways, you need to “subtract” one card from the starting point and multiply.
That is, 36x35x34x33x32…x2x1= the result does not fit on the calculator screen, so it can simply be denoted as 36!. Sign "!" next to the number indicates that the entire series of numbers is multiplied among themselves.
In combinatorics, there are such concepts as permutation, placement and combination. Each of them has its own formula.
An ordered set of set elements is called a layout. Placements can be repetitive, meaning one element can be used multiple times. And without repetition, when the elements are not repeated. n is all elements, m is the elements that participate in the placement. The formula for placement without repetitions 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, this looks like: P n = n!
Combinations of n elements by m are such compounds in which it is important which elements they were and what is their total number. The formula will look like:
A n m =n!/m!(n-m)!
Bernoulli formula
In the theory of probability, as well 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 appearance of A in an experiment does not depend on the appearance or non-occurrence of the same event in previous or subsequent tests.
Bernoulli equation:
P n (m) = C n m ×p m ×q n-m .
The probability (p) of the occurrence of the event (A) is unchanged for each trial. The probability that the situation will happen exactly m times in n number of experiments will be calculated by the formula that is 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. A unit is a number that is used to designate all outcomes of a situation in a discipline. Therefore, q is a number that indicates the possibility of the event not occurring.
Now you know the Bernoulli formula (probability theory). Examples of problem solving (the first level) will be considered below.
Task 2: A store visitor will make a purchase with a probability of 0.2. 6 visitors entered the store independently. What is the probability that a visitor will make a purchase?
Solution: Since it is not known how many visitors should make a purchase, one or all six, it is necessary to calculate all possible probabilities using the Bernoulli 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 (because there are 6 customers in the store). The number m will change from 0 (no customer will make a purchase) to 6 (all store visitors will purchase something). As a result, we get the solution:
P 6 (0) \u003d C 0 6 × p 0 × q 6 \u003d q 6 \u003d (0.8) 6 \u003d 0.2621.
None of the buyers will make a purchase with a probability of 0.2621.
How else is the Bernoulli formula (probability theory) used? Examples of problem solving (second level) below.
After the above example, questions arise about where C and p have gone. With respect 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 buying goods by two visitors.
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 so complicated. The Bernoulli formula, examples of which are presented above, is a direct proof of this.
Poisson formula
The Poisson equation is used to calculate unlikely random situations.
Basic formula:
P n (m)=λ m /m! × e (-λ) .
In this case, λ = n x p. Here is such a simple Poisson formula (probability theory). Examples of problem solving will be considered below.
Task 3 A: The factory produced 100,000 parts. The appearance 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 solving problems of this kind are no different from other tasks of the discipline, we substitute the necessary data into the above formula:
A = "a randomly selected part will be defective."
p = 0.0001 (according to the assignment condition).
n = 100000 (number of parts).
m = 5 (defective parts). We substitute the data in 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 essence, it can be found by the formula:
e -λ = lim n ->∞ (1-λ/n) n .
However, there are special tables that contain almost all the values of e.
De Moivre-Laplace theorem
If in the Bernoulli scheme the number of trials is sufficiently large, and the probability of the occurrence of event A in all schemes is the same, then the probability of the occurrence of event A a certain number of times in a series of trials can be found by the Laplace formula:
Р n (m)= 1/√npq x ϕ(X m).
Xm = m-np/√npq.
To better remember the Laplace formula (probability theory), examples of tasks to help below.
First we find X m , we substitute the data (they are all indicated 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 in the formula:
P 800 (267) \u003d 1 / √ (800 x 1/3 x 2/3) x 0.3988 \u003d 3/40 x 0.3988 \u003d 0.03.
So the probability that the flyer will hit exactly 267 times is 0.03.
Bayes formula
The Bayes formula (probability theory), examples of solving tasks 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 main formula is as follows:
P (A|B) = P (B|A) x P (A) / P (B).
A and B are definite events.
P(A|B) - conditional probability, that is, event A can occur, provided that event B is true.
Р (В|А) - conditional probability of event В.
So, the final part of the short course "Theory of Probability" is the Bayes formula, examples of solving problems with which are below.
Task 5: Phones from three companies were brought to the warehouse. At the same time, part of the 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%. It is necessary to find the probability that a randomly selected phone will be defective.
A = "randomly taken phone."
B 1 - the phone that the first factory made. Accordingly, introductory B 2 and B 3 will appear (for the second and third factories).
As a result, we get:
P (B 1) \u003d 25% / 100% \u003d 0.25; P (B 2) \u003d 0.6; P (B 3) \u003d 0.15 - so 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 firms:
P (A / B 1) \u003d 2% / 100% \u003d 0.02;
P (A / B 2) \u003d 0.04;
P (A / B 3) \u003d 0.01.
Now we substitute the data into the Bayes formula and get:
P (A) \u003d 0.25 x 0.2 + 0.6 x 0.4 + 0.15 x 0.01 \u003d 0.0305.
The article presents the theory of probability, formulas and examples of problem solving, but this is only the tip of the iceberg of a vast discipline. And after all that has been written, it will be logical to ask the question of whether the theory of probability is needed in life. It is difficult for a simple person to answer, it is better to ask someone who has hit the jackpot more than once with her help.
Probability theory is a mathematical science that allows, by the probabilities of some random events, to find the probabilities of other random events related in some way to the first.
A statement that an event occurs with probability, equal, for example, ½, does not yet represent in itself the final value, since we are striving for reliable knowledge. The final cognitive value are those results of the theory of probability, which allow us to assert that the probability of occurrence of any event A is very close to unity or (which is the same) the probability of not occurrence of event A is very small. In accordance with the principle of "neglecting sufficiently small probabilities", such an event is rightly considered practically certain. Below (in the section Limit theorems) it is shown that such conclusions of scientific and practical interest are usually based on the assumption that the occurrence or non-occurrence of event A depends on a large number of random, little related factors. Therefore, we can also say that the theory of probability is a mathematical science that explains the patterns that arise when a large number of random factors interact.
The subject of probability theory.
To describe a regular connection between certain conditions S and an event A, the occurrence or non-occurrence of which under given conditions can be precisely established, natural science usually uses one of the following two schemes:
a) with each implementation of the conditions S, an event A occurs. For example, all the laws of classical mechanics have this form, which state that under given initial conditions and forces acting on a body or system of bodies, the movement will occur in a uniquely defined way.
b) Under conditions S, the event A has a certain probability P (A / S) equal to p. So, for example, the laws of radioactive radiation state that for each radioactive substance there is a certain probability that some number N of atoms will decay from a given amount of substance in a given period of time.
Let us call the frequency of event A in a given series of n trials (that is, of n repeated implementations of conditions S) the ratio h = m/n of the number m of those trials in which A occurred to their total number n. The fact that the event A under conditions S has a certain probability equal to p is manifested in the fact that in almost every sufficiently long series of trials the frequency of the event A is approximately equal to p.
Statistical regularities, that is, regularities described by a scheme of type (b), were first discovered on the example of gambling games like dice. The statistical regularities of birth and death have also been known for a very long time (for example, the probability of a newborn being a boy is 0.515). Late 19th century and 1st half of the 20th century. marked by the discovery of a large number of statistical regularities in physics, chemistry, biology, etc.
The possibility of applying the methods of probability theory to the study of statistical regularities related to very distant fields of science is based on the fact that the probabilities of events always satisfy some simple relations, which will be discussed below (see section Basic Concepts of Probability Theory). The study of the properties of the probabilities of events on the basis of these simple relations is the subject of probability theory.
Basic concepts of probability theory.
The basic concepts of probability theory as a mathematical discipline are defined most simply within the framework of the so-called elementary probability theory. Each trial T considered in elementary probability theory is such that it ends with one and only one of the events E1, E2,..., ES (one or the other, depending on the case). These events are called trial outcomes. Each outcome Ek is associated with a positive number pk - the probability of this outcome. The numbers pk must add up to one. Then the events A are considered, which consist in the fact that "either Ei, or Ej, ..., or Ek occurs." The outcomes Ei, Ej,..., Ek are called favorable A, and by definition, the probability P (A) of the event A is assumed to be equal to the sum of the probabilities of the favorable outcomes:
P(A) = pi + ps + … + pk. (one)
The special case p1 = p2 =... ps = 1/S leads to the formula
P(A) = r/s. (2)
Formula (2) expresses the so-called classical definition of probability, according to which the probability of any event A is equal to the ratio of the number r of outcomes favoring A to the number s of all "equally possible" outcomes. The classical definition of probability only reduces the notion of "probability" to the notion of "equipossibility", which remains without a clear definition.
Example. When throwing two dice, each of the 36 possible outcomes can be designated (i, j), where i is the number of points that falls on the first die, j - on the second. The outcomes are assumed to be equally probable. Event A - "the sum of points is 4", is favored by three outcomes (1; 3), (2; 2), (3; 1). Therefore, P(A) = 3/36 = 1/12.
Based on any data of events, two new events can be defined: their union (sum) and combination (product). An event B is called a union of events A 1, A 2,..., Ar,-, if it has the form: "either A1, or A2,..., or Ar occurs".
An event C is called a combination of events A1, A.2,..., Ar, if it has the form: "A1, A2,..., and Ar occur". The combination of events is denoted by the sign È, and the combination - by the sign Ç. Thus, they write:
B = A1 È A2 È … È Ar, C = A1 Ç A2 Ç … Ç Ar.
Events A and B are called incompatible if their simultaneous implementation is impossible, that is, if there is not a single favorable outcome of the test for both A and B.
The two main theorems of V. t., the theorems of addition and multiplication of probabilities, are connected with the introduced operations of combining and superimposing events.
The theorem of addition of probabilities. If the events A1, A2,..., Ar are such that every two of them are incompatible, then the probability of their union is equal to the sum of their probabilities.
So, in the above example with throwing two dice, event B - "the sum of points does not exceed 4", is the union of three incompatible events A2, A3, A4, consisting in the fact that the sum of points is 2, 3, 4, respectively. The probabilities of these events 1/36; 2/36; 3/36. By the addition theorem, the probability P(B) is equal to
1/36 + 2/36 + 3/36 = 6/36 = 1/6.
The conditional probability of an event B under condition A is determined by the formula
which, as can be shown, is in full agreement with the properties of frequencies. Events A1, A2,..., Ar are called independent if the conditional probability of each of them, provided that any of the others have occurred, is equal to its "unconditional" probability
Probability multiplication theorem. The probability of combining the events A1, A2,..., Ar is equal to the probability of the event A1, multiplied by the probability of the event A2, taken under the condition that A1 has occurred,..., multiplied by the probability of the event Ar, provided that A1, A2,.. ., Ar-1 have arrived. For independent events, the multiplication theorem leads to the formula:
P (A1 Ç A2 Ç … Ç Ar) = P (A1) Ї P (A2) Ї … Ї P (Ar), (3)
that is, the probability of combining independent events is equal to the product of the probabilities of these events. Formula (3) remains valid if some of the events in both parts of it are replaced by opposite ones.
Example. Fires 4 shots at the target with a hit probability of 0.2 on a single shot. Target hits for different shots are assumed to be independent events. What is the probability of hitting the target exactly three times?
Each test outcome can be indicated by a sequence of four letters [e.g., (y, n, n, y) means that the first and fourth shots were hit (success), and the second and third hits were not (failure)]. In total there will be 2Ї2Ї2Ї2 = 16 outcomes. In accordance with the assumption of the independence of the results of individual shots, formula (3) and a note to it should be used to determine the probabilities of these outcomes. So, the probability of the outcome (y, n. n, n) should be set equal to 0.2 0.8 0.8 0.8 = 0.1024; here 0.8 \u003d 1-0.2 - the probability of a miss with a single shot. The event "the target is hit three times" is favored by the outcomes (y, y, y, n), (y, y, n, y), (y, n, y, y). (n, y, y, y), the probability of each is the same:
0.2Ї0.2Ї0.2Ї0.8 \u003d ...... \u003d 0.8 0.2 0.2 0.2 \u003d 0.0064;
therefore, the desired probability is equal to
4Ї0.0064 = 0.0256.
Generalizing the reasoning of the analyzed example, one of the basic formulas of probability theory can be deduced: if the events A1, A2,..., An are independent and each have probability p, then the probability of occurrence of exactly m of them is equal to
Pn (m) = Cnmpm (1 - p) n-m; (four)
here Cnm denotes the number of combinations of n elements by m. For large n, calculations using formula (4) become difficult. Let the number of shots in the previous example be 100, and the question is to find the probability x that the number of hits lies in the range from 8 to 32. Applying formula (4) and the addition theorem gives an exact, but practically unsuitable expression for the desired probability
An approximate value of the probability x can be found using Laplace's theorem
and the error does not exceed 0.0009. The found result shows that the event 8 £ m £ 32 is almost certain. This is the simplest but typical example of using the limit theorems of probability theory.
The basic formulas of the elementary probability theory also include the so-called total probability formula: if the events A1, A2,..., Ar are pairwise incompatible and their union is a certain event, then for any event B its probability is equal to the sum
The probabilities multiplication theorem is especially useful when considering compound tests. A trial T is said to be composed of trials T1, T2,..., Tn-1, Tn, if each outcome of trial T is a combination of some outcomes Ai, Bj,..., Xk, Yl of the corresponding trials T1, T2,... , Tn-1, Tn. From one reason or another, the probabilities are often known
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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:
- you called to 1st Door
- you called to 2nd Door
- you called to 3rd Door
And now consider all the options where a friend can be:
a. Per 1st door
b. Per 2nd door
in. Per 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 Maybe 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 Door
2) Call 2nd 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.
- 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.
- 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:
- If the experiment is carried out once (once a coin is tossed, the doorbell rings once, etc.), then the events are always independent.
- 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?
Solution:
Consider all possible options:
- eagle eagle
- tails eagle
- tails-eagle
- 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.
Solution:
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.
- What is the probability of drawing a white ball?
- We added more black balls to the box. What is the probability of drawing a white ball now?
Solution:
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?
Solution:
Let's count the number favorable outcomes.
NOT a red marker, that means green, blue, yellow, or black.
| 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:
- Eagle-eagle-eagle
- Eagle-head-tails
- Head-tails-eagle
- Head-tails-tails
- tails-eagle-eagle
- Tails-heads-tails
- Tails-tails-heads
- 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:
- Eagle-eagle-eagle
- Eagle-head-tails
- Head-tails-eagle
- Head-tails-tails
- tails-eagle-eagle
- Tails-heads-tails
- Tails-tails-heads
- 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?
Solution:
Example 6
A die is thrown twice, what is the probability that a total of 8 will come up?
Solution.
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:
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.
- What is the probability of drawing clubs in a row (we put the first card drawn back into the deck and shuffle)?
- 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 Ace The sequence in which the cards will be drawn does not matter.
Answers:
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. AVERAGE 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 the 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 that the toss of a coin will land on heads? And what is the probability of a tails?
- What is the probability that an even number will come up when a dice is thrown? And with what - odd?
- 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:
- 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: .
- 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. - 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. 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, simple, yellow, and the rest are orange. What is the probability of not drawing green?
Solution:
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:
- A die is thrown twice. What is the probability that it will come up both times?
- A coin is tossed times. What is the probability of getting heads first 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 that the multiplication rule works: .
- The probability of an eagle is equal. Tails probability too. We multiply:
- 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?
Solution .
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?
Solution .
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:
- What is the probability that two coin tosses come up with the same side both times?
- A die is thrown twice. What is the probability that the sum will drop points?
Solutions:
Another example:
We toss a coin once. What is the probability that heads will come up at least once?
Solution:
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.
Well, the topic is over. If you are reading these lines, then you are very cool.
Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!
Now the most important thing.
You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.
The problem is that this may not be enough ...
For what?
For the successful passing of the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.
I will not convince you of anything, I will just say one thing ...
People who have received a good education earn much more than those who have not received 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 much 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 exam and be ultimately ... happier?
FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.
On the exam, you will not be asked theory.
You will need solve problems on time.
And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.
It's like in sports - you need to repeat many times to win for sure.
Find a collection anywhere you want necessarily with solutions, detailed analysis and decide, decide, decide!
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“Understood” and “I know how to solve” are completely different skills. You need both.
Find problems and solve!
Probability theory is a branch of mathematics that studies the patterns of random phenomena: random events, random variables, their properties and operations on them.
For a long time, the theory of probability did not have a clear definition. It was formulated only in 1929. The emergence of probability theory as a science is attributed to the Middle Ages and the first attempts at the mathematical analysis of gambling (toss, dice, roulette). The French mathematicians of the 17th century Blaise Pascal and Pierre de Fermat discovered the first probabilistic patterns that arise when throwing dice while studying the prediction of winnings in gambling.
The theory of probability arose as a science from the belief that certain regularities underlie massive random events. Probability theory studies these patterns.
Probability theory deals with the study of events, the occurrence of which is not known for certain. It allows you to judge the degree of probability of the occurrence of some events compared to others.
For example: it is impossible to unambiguously determine the result of a coin tossing heads or tails, but with repeated tossing, approximately the same number of heads and tails falls out, which means that the probability that heads or tails will fall ", is equal to 50%.
test in this case, the implementation of a certain set of conditions is called, that is, in this case, the tossing of a coin. The challenge can be played an unlimited number of times. In this case, the complex of conditions includes random factors.
The test result is event. The event happens:
- Reliable (always occurs as a result of testing).
- Impossible (never happens).
- Random (may or may not occur as a result of the test).
For example, when tossing a coin, an impossible event - the coin will end up on the edge, a random event - the loss of "heads" or "tails". The specific test result is called elementary event. As a result of the test, only elementary events occur. The totality of all possible, different, specific test outcomes is called elementary event space.
Basic concepts of the theory
Probability- the degree of possibility of the occurrence of the event. When the reasons for some possible event to actually occur outweigh the opposite reasons, then this event is called probable, otherwise - unlikely or improbable.
Random value- this is a value that, as a result of the test, can take one or another value, and it is not known in advance which one. For example: the number of fire stations per day, the number of hits with 10 shots, etc.
Random variables can be divided into two categories.
- Discrete random variable such a quantity is called, which, as a result of the test, can take certain values \u200b\u200bwith a certain probability, forming a countable set (a set whose elements can be numbered). This set can be either finite or infinite. For example, the number of shots before the first hit on the target is a discrete random variable, because this value can take on an infinite, although countable, number of values.
- Continuous random variable is a quantity that can take any value from some finite or infinite interval. Obviously, the number of possible values of a continuous random variable is infinite.
Probability space- the concept introduced by A.N. Kolmogorov in the 1930s to formalize the concept of probability, which gave rise to the rapid development of probability theory as a rigorous mathematical discipline.
The probability space is a triple (sometimes framed in angle brackets: , where
This is an arbitrary set, the elements of which are called elementary events, outcomes or points;
- sigma-algebra of subsets called (random) events;
- probabilistic measure or probability, i.e. sigma-additive finite measure such that .
De Moivre-Laplace theorem- one of the limiting theorems of probability theory, established by Laplace in 1812. She states that the number of successes in repeating the same random experiment with two possible outcomes is approximately normally distributed. It allows you to find an approximate value of the probability.
If, for each of the independent trials, the probability of occurrence of some random event is equal to () and is the number of trials in which it actually occurs, then the probability of the validity of the inequality is close (for large ) to the value of the Laplace integral.
Distribution function in probability theory- a function characterizing the distribution of a random variable or a random vector; the probability that a random variable X will take on a value less than or equal to x, where x is an arbitrary real number. Under certain conditions, it completely determines a random variable.
Expected value- the average value of a random variable (this is the probability distribution of a random variable, considered in probability theory). In English literature, it is denoted by, in Russian -. In statistics, the notation is often used.
Let a probability space and a random variable defined on it be given. That is, by definition, a measurable function. Then, if there is a Lebesgue integral of over space , then it is called the mathematical expectation, or mean value, and is denoted by .
Variance of a random variable- a measure of the spread of a given random variable, i.e. its deviation from the mathematical expectation. Designated in Russian literature and in foreign. In statistics, the designation or is often used. The square root of the variance is called the standard deviation, standard deviation, or standard spread.
Let be a random variable defined on some probability space. Then
where the symbol denotes the mathematical expectation.
In probability theory, two random events are called independent if the occurrence of one of them does not change the probability of the occurrence of the other. Similarly, two random variables are called dependent if the value of one of them affects the probability of the values of the other.
The simplest form of the law of large numbers is Bernoulli's theorem, which states that if the probability of an event is the same in all trials, then as the number of trials increases, the frequency of the event tends to the probability of the event and ceases to be random.
The law of large numbers in probability theory states that the arithmetic mean of a finite sample from a fixed distribution is close to the theoretical mean expectation of that distribution. Depending on the type of convergence, a weak law of large numbers is distinguished, when convergence in probability takes place, and a strong law of large numbers, when convergence almost certainly takes place.
The general meaning of the law of large numbers is that the joint action of a large number of identical and independent random factors leads to a result that, in the limit, does not depend on chance.
Methods for estimating probability based on the analysis of a finite sample are based on this property. A good example is the prediction of election results based on a survey of a sample of voters.
Central limit theorems- a class of theorems in probability theory stating that the sum of a sufficiently large number of weakly dependent random variables that have approximately the same scale (none of the terms dominates, does not make a decisive contribution to the sum) has a distribution close to normal.
Since many random variables in applications are formed under the influence of several weakly dependent random factors, their distribution is considered normal. In this case, the condition must be observed that none of the factors is dominant. Central limit theorems in these cases justify the application of the normal distribution.
Events that occur in reality or in our imagination can be divided into 3 groups. These are certain events that are bound to happen, impossible events, and random events. Probability theory studies random events, i.e. events that may or may not occur. This article will briefly present the theory of probability formulas and examples of solving problems in the theory of probability, which will be in the 4th task of the Unified State Examination in mathematics (profile level).
Why do we need the theory of probability
Historically, the need to study these problems arose in the 17th century in connection with the development and professionalization of gambling and the emergence of casinos. It was a real phenomenon that required its study and research.
Playing cards, dice, roulette created situations where any of a finite number of equally probable events could occur. There was a need to give numerical estimates of the possibility of the occurrence of an event.
In the 20th century, it became clear that this seemingly frivolous science plays an important role in understanding the fundamental processes occurring in the microcosm. The modern theory of probability was created.
Basic concepts of probability theory
The object of study of probability theory is events and their probabilities. If the event is complex, then it can be broken down into simple components, the probabilities of which are easy to find.
The sum of events A and B is called event C, which consists in the fact that either event A, or event B, or events A and B happened at the same time.
The product of events A and B is the event C, which consists in the fact that both the event A and the event B happened.
Events A and B are said to be incompatible if they cannot happen at the same time.
An event A is said to be impossible if it cannot happen. Such an event is denoted by the symbol .
An event A is called certain if it will definitely occur. Such an event is denoted by the symbol .
Let each event A be assigned a number P(A). This number P(A) is called the probability of the event A if the following conditions are satisfied with such a correspondence.
An important special case is the situation when there are equally probable elementary outcomes, and arbitrary of these outcomes form events A. In this case, the probability can be introduced by the formula . The probability introduced in this way is called the classical probability. It can be proved that properties 1-4 hold in this case.
Problems in the theory of probability, which are found on the exam in mathematics, are mainly related to classical probability. Such tasks can be very simple. Particularly simple are problems in the theory of probability in demonstration versions. It is easy to calculate the number of favorable outcomes, the number of all outcomes is written directly in the condition.
We get the answer according to the formula.
An example of a task from the exam in mathematics to determine the probability
There are 20 pies on the table - 5 with cabbage, 7 with apples and 8 with rice. Marina wants to take a pie. What is the probability that she will take the rice cake?
Solution.
There are 20 equiprobable elementary outcomes in total, that is, Marina can take any of the 20 pies. But we need to estimate the probability that Marina will take the rice patty, that is, where A is the choice of the rice patty. This means that we have a total of 8 favorable outcomes (choosing rice pies). Then the probability will be determined by the formula:
Independent, Opposite, and Arbitrary Events
However, more complex tasks began to appear in the open bank of tasks. Therefore, let us draw the reader's attention to other questions studied in probability theory.
Events A and B are called independent if the probability of each of them does not depend on whether the other event occurred.
Event B consists in the fact that event A did not occur, i.e. event B is opposite to event A. The probability of the opposite event is equal to one minus the probability of the direct event, i.e. .
Addition and multiplication theorems, formulas
For arbitrary events A and B, the probability of the sum of these events is equal to the sum of their probabilities without the probability of their joint event, i.e. .
For independent events A and B, the probability of the product of these events is equal to the product of their probabilities, i.e. in this case .
The last 2 statements are called the theorems of addition and multiplication of probabilities.
Not always counting the number of outcomes is so simple. In some cases, it is necessary to use combinatorics formulas. The most important thing is to count the number of events that meet certain conditions. Sometimes such calculations can become independent tasks.
In how many ways can 6 students be seated in 6 empty seats? The first student will take any of the 6 places. Each of these options corresponds to 5 ways to place the second student. For the third student there are 4 free places, for the fourth - 3, for the fifth - 2, the sixth will take the only remaining place. To find the number of all options, you need to find the product, which is denoted by the symbol 6! and read "six factorial".
In the general case, the answer to this question is given by the formula for the number of permutations of n elements. In our case, .
Consider now another case with our students. In how many ways can 2 students be seated in 6 empty seats? The first student will take any of the 6 places. Each of these options corresponds to 5 ways to place the second student. To find the number of all options, you need to find the product.
In the general case, the answer to this question is given by the formula for the number of placements of n elements by k elements
In our case .
And the last one in this series. How many ways are there to choose 3 students out of 6? The first student can be chosen in 6 ways, the second in 5 ways, and the third in 4 ways. But among these options, the same three students occur 6 times. To find the number of all options, you need to calculate the value: . In the general case, the answer to this question is given by the formula for the number of combinations of elements by elements:
In our case .
Examples of solving problems from the exam in mathematics to determine the probability
Task 1. From the collection, ed. Yashchenko.
There are 30 pies on a plate: 3 with meat, 18 with cabbage and 9 with cherries. Sasha randomly chooses one pie. Find the probability that he ends up with a cherry.
.
Answer: 0.3.
Problem 2. From the collection, ed. Yashchenko.
In each batch of 1000 light bulbs, an average of 20 defective ones. Find the probability that a light bulb chosen at random from a batch is good.
Solution: The number of serviceable light bulbs is 1000-20=980. Then the probability that a light bulb taken at random from the batch will be serviceable is:
Answer: 0.98.
The probability that student U. correctly solves more than 9 problems on a math test is 0.67. The probability that U. correctly solves more than 8 problems is 0.73. Find the probability that U. correctly solves exactly 9 problems.
If we imagine a number line and mark points 8 and 9 on it, then we will see that the condition "U. correctly solve exactly 9 problems” is included in the condition “U. correctly solve more than 8 problems", but does not apply to the condition "W. correctly solve more than 9 problems.
However, the condition "U. correctly solve more than 9 problems" is contained in the condition "U. correctly solve more than 8 problems. Thus, if we designate events: “W. correctly solve exactly 9 problems" - through A, "U. correctly solve more than 8 problems" - through B, "U. correctly solve more than 9 problems ”through C. Then the solution will look like this:
Answer: 0.06.
In the geometry exam, the student answers one question from the list of exam questions. The probability that this is a trigonometry question is 0.2. The probability that this is an Outer Corners question is 0.15. There are no questions related to these two topics at the same time. Find the probability that the student will get a question on one of these two topics on the exam.
Let's think about what events we have. We are given two incompatible events. That is, either the question will relate to the topic "Trigonometry", or to the topic "External angles". According to the probability theorem, the probability of incompatible events is equal to the sum of the probabilities of each event, we must find the sum of the probabilities of these events, that is:
Answer: 0.35.
The room is illuminated by a lantern with three lamps. The probability of one lamp burning out in a year is 0.29. Find the probability that at least one lamp does not burn out within a year.
Let's consider possible events. We have three light bulbs, each of which may or may not burn out independently of any other light bulb. These are independent events.
Then we will indicate the variants of such events. We accept the notation: - the light bulb is on, - the light bulb is burned out. And immediately next we calculate the probability of an event. For example, the probability of an event in which three independent events “the light bulb burned out”, “the light bulb is on”, “the light bulb is on” occurred: .
