The probability of a certain event is an example. Event probability

Probability event is the ratio of the number of elementary outcomes favorable to a given event to the number of all equally possible outcomes of the experience in which this event may appear. The probability of an event A is denoted by P (A) (here P is the first letter French word probabilite - probability). According to the definition
(1.2.1)
where is the number of elementary outcomes favorable to event A; - the number of all equally possible elementary outcomes of the experiment, forming a complete group of events.
This definition of probability is called classical. It arose at the initial stage of the development of the theory of probability.

The probability of an event has the following properties:
1. The probability of a reliable event is equal to one. Let's designate a valid event with a letter. For a reliable event, therefore
(1.2.2)
2. The probability of an impossible event is zero. Let's denote an impossible event by a letter. For an impossible event, therefore
(1.2.3)
3. The probability of a random event is expressed as a positive number less than one. Since for a random event the inequalities are satisfied, or, then
(1.2.4)
4. The probability of any event satisfies the inequalities
(1.2.5)
This follows from relations (1.2.2) - (1.2.4).

Example 1. The urn contains 10 balls of the same size and weight, of which 4 are red and 6 are blue. one ball is removed from the urn. What is the likelihood that the removed ball will turn out to be blue?

Solution... The event "the removed ball turned out to be blue" will be denoted by the letter A. This test has 10 equally possible elementary outcomes, of which 6 favor the event A. In accordance with formula (1.2.1), we obtain

Example 2. All natural numbers from 1 to 30 are written on identical cards and placed in the urn. After thorough mixing of the cards, one card is removed from the urn. What is the probability that the number on the taken card will be a multiple of 5?

Solution. Let us denote by A the event "the number on the taken card is a multiple of 5". In this test, there are 30 equally possible elementary outcomes, of which event A is favored by 6 outcomes (numbers 5, 10, 15, 20, 25, 30). Hence,

Example 3. Two dice are thrown, the sum of points on the upper edges is calculated. Find the probability of event B, which consists of 9 points in total on the top sides of the cubes.

Solution. In this test there are only 6 2 = 36 equally possible elementary outcomes. Event B is favored by 4 outcomes: (3; 6), (4; 5), (5; 4), (6; 3), therefore

Example 4... Chosen at random natural number not exceeding 10. What is the probability that this number is prime?

Solution. Let us denote by the letter C the event "the chosen number is prime". In this case, n = 10, m = 4 (primes 2, 3, 5, 7). Therefore, the required probability

Example 5. Two symmetrical coins are tossed. What is the likelihood that the top sides of both coins have numbers?

Solution. Let us denote by the letter D the event "there was a number on the upper side of each coin". In this test there are 4 equally possible elementary outcomes: (Г, Г), (Г, Ц), (Ц, Г), (Ц, Ц). (The entry (G, C) means that the first coin has a coat of arms, the second has a number). Event D is favored by one elementary outcome (C, C). Since m = 1, n = 4, then

Example 6. What is the probability that in a randomly chosen two-digit number the digits are the same?

Solution. Two-digit numbers are numbers from 10 to 99; there are 90 such numbers in total. The same numbers have 9 numbers (these are the numbers 11, 22, 33, 44, 55, 66, 77, 88, 99). Since in this case m = 9, n = 90, then
,
where A is the "number with the same digits" event.

Example 7. From the letters of the word differential one letter is chosen at random. What is the probability that this letter will be: a) a vowel, b) a consonant, c) a letter h?

Solution... The word differential has 12 letters, of which 5 are vowels and 7 consonants. Letters h in this word no. Let's designate events: A - "vowel letter", B - "consonant letter", C - "letter h". The number of favorable elementary outcomes: - for event A, - for event B, - for event C. Since n = 12, then
, and .

Example 8. Two dice are thrown, the number of points on the top of each die is noted. Find the probability that both dice have the same number of points.

Solution. Let's designate this event by the letter A. Events A favor 6 elementary outcomes: (1;]), (2; 2), (3; 3), (4; 4), (5; 5), (6; 6). In total, equally possible elementary outcomes that form a complete group of events, in this case n = 6 2 = 36. Hence, the required probability

Example 9. The book has 300 pages. What is the probability that a randomly opened page will have a sequence number that is a multiple of 5?

Solution. From the condition of the problem it follows that all equally possible elementary outcomes that form a complete group of events will be n = 300. Of these, m = 60 favor the onset of the specified event. Indeed, a multiple of 5 has the form 5k, where k is a natural number, and, whence ... Hence,
, where A - the "page" event has a sequence number that is a multiple of 5 ".

Example 10... Two dice are thrown, the sum of points on the upper edges is calculated. Which is more likely to get a total of 7 or 8?

Solution... Let's designate events: A - "7 points dropped out", B - "8 points dropped out". Event A is favored by 6 elementary outcomes: (1; 6), (2; 5), (3; 4), (4; 3), (5; 2), (6; 1), and event B - 5 outcomes: (2; 6), (3; 5), (4; 4), (5; 3), (6; 2). All equally possible elementary outcomes n = 6 2 = 36. Hence, and .

So, P (A)> P (B), that is, getting 7 points in total is a more likely event than getting 8 points in total.

Tasks

1. A natural number not exceeding 30 was chosen at random. What is the probability that this number is a multiple of 3?
2. In the urn a red and b blue balls, the same size and weight. What is the probability that a ball drawn at random from this urn turns out to be blue?
3. At random · chosen a number not exceeding 30. What is the probability that this number is a divisor of zo?
4. In the urn a blue and b red balls, the same size and weight. One ball is removed from this urn and set aside. This ball turned out to be red. After that, another ball is taken out of the urn. Find the probability that the second ball is also red.
5. A random number is chosen not exceeding 50. What is the probability that this number is prime?
6. Three dice are thrown, the sum of the points on the upper edges is calculated. Which is more likely to get 9 or 10 points in total?
7. Three dice are thrown, the sum of the dropped points is calculated. Which is more likely to get a total of 11 (event A) or 12 points (event B)?

Answers

1. 1/3. 2 . b/(a+b). 3 . 0,2. 4 . (b-1)/(a+b-1). 5 .0,3.6 ... p 1 = 25/216 - the probability of getting 9 points in total; p 2 = 27/216 - the probability of getting 10 points in total; p 2> p 1 7 ... P (A) = 27/216, P (B) = 25/216, P (A)> P (B).

Questions

1. What is called the probability of an event?
2. What is the probability of a certain event?
3. What is the probability of an impossible event?
4. What are the limits of the probability of a random event?
5. What are the limits of the probability of any event?
6. What definition of probability is called classical?

It is clear that each event has a certain degree of possibility of its occurrence (its realization). In order to quantitatively compare events with each other according to the degree of their possibility, it is obviously necessary to associate a certain number with each event, which is the greater, the more possible the event is. This number is called the probability of an event.

Event probability- there is a numerical measure of the degree of objective possibility of the occurrence of this event.

Consider a stochastic experiment and a random event A observed in this experiment. Let's repeat this experiment n times and let m (A) be the number of experiments in which event A happened.

Ratio (1.1)

called relative frequency events A in the series of experiments carried out.

It is easy to verify the validity of the properties:

if A and B are inconsistent (AB =), then ν (A + B) = ν (A) + ν (B) (1.2)

The relative frequency is determined only after a series of experiments and, generally speaking, can change from series to series. However, experience shows that in many cases, with an increase in the number of experiments, the relative frequency approaches a certain number. This fact of the stability of the relative frequency has been repeatedly verified and can be considered experimentally established.

Example 1.19.... If you flip one coin, no one can predict which side it will fall up. But if you throw two tons of coins, then everyone will say that about one ton will fall upwards with the coat of arms, that is, the relative frequency of the appearance of the coat of arms is approximately 0.5.

If, with an increase in the number of experiments, the relative frequency of the event ν (A) tends to a certain fixed number, then they say that event A is statistically stable, and this number is called the probability of event A.

Probability of the event A is called a certain fixed number P (A), to which the relative frequency ν (A) of this event tends with an increase in the number of experiments, that is,

This definition is called statistical definition probabilities .

Let's consider some stochastic experiment and let the space of its elementary events consist of a finite or infinite (but countable) set of elementary events ω 1, ω 2,…, ω i,…. Suppose that each elementary event ω i is assigned a certain number - p i, which characterizes the degree of possibility of the occurrence of this elementary event and satisfies the following properties:

Such a number p i is called the probability of an elementary eventω i.

Now let A be a random event observed in this experiment, and a certain set corresponds to it

In such a setting probability of event A is the sum of the probabilities of elementary events favorable to A(included in the corresponding set A):

The probability introduced in this way has the same properties as the relative frequency, namely:

And if AB = (A and B are inconsistent),

then P (A + B) = P (A) + P (B)

Indeed, according to (1.4)

In the last relation, we took advantage of the fact that no elementary event can simultaneously favor two incompatible events.

We especially note that the theory of probability does not indicate ways of determining p i, they must be sought from practical considerations or obtained from an appropriate statistical experiment.

As an example, consider the classical scheme of probability theory. To do this, consider a stochastic experiment, the space of elementary events of which consists of a finite (n) number of elements. Suppose additionally that all these elementary events are equally possible, that is, the probabilities of elementary events are p (ω i) = p i = p. Hence it follows that

Example 1.20... When a symmetrical coin is thrown, the emblem and tails are equally possible, their probabilities are equal to 0.5.

Example 1.21... When throwing a symmetrical dice, all faces are equally possible, their probabilities are equal to 1/6.

Now let event A be favored by m elementary events, they are usually called outcomes favorable to event A... Then

Received classical definition of probability: the probability P (A) of event A is equal to the ratio of the number of outcomes favorable to event A to the total number of outcomes

Example 1.22... The urn contains m white balls and n black ones. What is the probability of drawing the white ball?

Solution... There are m + n elementary events in total. They are all equally likely. Favorable event And of them m. Hence, .

The following properties follow from the definition of probability:

Property 1. The probability of a certain event is equal to one.

Indeed, if the event is reliable, then every elementary outcome of the test favors the event. In this case m = n, hence,

P (A) = m / n = n / n = 1.(1.6)

Property 2. The probability of an impossible event is zero.

Indeed, if the event is impossible, then none of the elementary outcomes of the test favors the event. In this case T= 0, therefore P (A) = m / n = 0 / n = 0. (1.7)

Property 3.The probability of a random event is positive number between zero and one.

Indeed, only a fraction of the total number of elementary test outcomes favors a random event. That is, 0≤m≤n, which means 0≤m / n≤1, therefore, the probability of any event satisfies the double inequality 0≤ P (A) ≤1. (1.8)

Comparing the definitions of probability (1.5) and relative frequency (1.1), we conclude: the definition of probability does not require tests to be performed in reality; the definition of the relative frequency assumes that tests were actually carried out... In other words, the probability is calculated before the experiment, and the relative frequency is calculated after the experiment.

However, calculating the probability requires preliminary information about the number or probabilities of elementary outcomes favorable to a given event. In the absence of such preliminary information, to determine the probability, they resort to empirical data, that is, the relative frequency of the event is determined from the results of a stochastic experiment.

Example 1.23... Technical control department found 3 custom parts in a batch of 80 randomly selected parts. The relative frequency of occurrence of non-standard parts r (A)= 3/80.

Example 1.24... By target. Produced 24 shot, and 19 hits were recorded. The relative frequency of hitting the target. r (A)=19/24.

Long-term observations have shown that if experiments are carried out under the same conditions, in each of which the number of tests is large enough, then the relative frequency exhibits the property of stability. This property is that in different experiments the relative frequency changes little (the less, the more tests are performed), fluctuating around a certain constant number. It turned out that this constant number can be taken as an approximate value of the probability.

The relationship between relative frequency and probability will be described in more detail and more precisely below. Now let us illustrate the stability property with examples.

Example 1.25... According to Swedish statistics, the relative frequency of births of girls for 1935 by months is characterized by the following numbers (the numbers are arranged in the order of months, starting with January): 0,486; 0,489; 0,490; 0.471; 0,478; 0,482; 0.462; 0,484; 0,485; 0,491; 0,482; 0,473

The relative frequency fluctuates around the number 0.481, which can be taken as approximate value the likelihood of having girls.

Note that statistics from different countries give approximately the same value for the relative frequency.

Example 1.26. Many times experiments were carried out tossing a coin, in which the number of the appearance of the "coat of arms" was counted. The results of several experiments are shown in the table.

Probability theory is a fairly extensive independent branch of mathematics. In the school course, the theory of probability is considered very superficially, however, in the exam and the GIA there are problems on this topic. However, to solve problems school course not so difficult (at least as far as arithmetic operations are concerned) - here you do not need to count derivatives, take integrals and solve complex trigonometric transformations- the main thing is to be able to handle prime numbers and fractions.

Probability theory - basic terms

The main terms of the theory of probability are trial, outcome and random event. A test in the theory of probability is called an experiment - toss a coin, draw a card, draw lots - all these are tests. The result of the test, you guessed it, is called the outcome.

But what is the randomness of an event? In the theory of probability, it is assumed that the test is carried out more than once and there are many outcomes. Many outcomes of a trial are called a random event. For example, if you flip a coin, two random events can occur - heads or tails.

Do not confuse the concepts of an outcome and a random event. The outcome is one result of one trial. A random event is a set of possible outcomes. By the way, there is such a term as an impossible event. For example, the "number 8" event on a standard game die is not possible.

How do you find the probability?

We all roughly understand what probability is, and quite often we use this word in our vocabulary. In addition, we can even draw some conclusions regarding the likelihood of a particular event, for example, if there is snow outside the window, we can most likely say that it is not summer now. However, how can this assumption be expressed numerically?

In order to introduce a formula for finding the probability, we introduce one more concept - a favorable outcome, that is, an outcome that is favorable for a particular event. The definition is rather ambiguous, of course, however, according to the condition of the problem, it is always clear which of the outcomes is favorable.

For example: There are 25 people in the class, three of them are Katya. The teacher appoints Olya on duty, and she needs a partner. What is the likelihood that Katya will become a partner?

V this example a favorable outcome - partner Katya. We will solve this problem a little later. But first, with the help of an additional definition, we introduce a formula for finding the probability.

• P = A / N, where P is the probability, A is the number of favorable outcomes, N is the total number of outcomes.

Everything school tasks revolve around this one formula, and the main difficulty usually lies in finding the outcomes. Sometimes it's easy to find them, sometimes it's not very good.

How to solve probabilities?

Problem 1

So now let's solve the problem posed above.

The number of favorable outcomes (the teacher will choose Katya) is three, because there are three Katya in the class, and there are 24 overall outcomes (25-1, because Olya has already been selected). Then the probability is: P = 3/24 = 1/8 = 0.125. Thus, the probability that Katya will be Olya's partner is 12.5%. Not difficult, right? Let's look at something a little more complicated.

Task 2

The coin was thrown twice, what is the probability of the combination: one heads and one tails?

So, consider the overall outcomes. How can coins fall - heads / heads, tails / tails, heads / tails, tails / heads? This means that the total number of outcomes is 4. How many favorable outcomes? Two - heads / tails and tails / heads. Thus, the probability of getting a heads / tails combination is:

• P = 2/4 = 0.5 or 50 percent.

Now let's consider the following problem. Masha has 6 coins in her pocket: two - 5 rubles and four - 10 rubles. Masha put 3 coins in another pocket. What is the likelihood that 5-ruble coins end up in different pockets?

For simplicity, let's designate coins with numbers - 1,2 - five-ruble coins, 3,4,5,6 - ten-ruble coins. So how can coins be in your pocket? There are 20 combinations in total:

• 123, 124, 125, 126, 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456.

At first glance, it may seem that some combinations have disappeared, for example, 231, but in our case the combinations 123, 231 and 321 are equivalent.

Now we count how many favorable outcomes we have. For them we take those combinations in which there is either the number 1 or the number 2: 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256. There are 12 of them. Thus, the probability is:

• P = 12/20 = 0.6 or 60%.

The problems in probability theory presented here are fairly straightforward, but don't think that probability theory is a simple branch of mathematics. If you decide to continue your education at a university (with the exception of humanitarian specialties), you will definitely have pairs of higher mathematics, on which you will be introduced to more complex terms of this theory, and the tasks there will be much more difficult.

V tasks of the exam in mathematics, there are also more complex probability problems (than we considered in Part 1), where you have to apply the rule of addition, multiplication of probabilities, and distinguish between joint and incompatible events.

So the theory.

Joint and incompatible events

Events are called inconsistent if the occurrence of one of them excludes the occurrence of others. That is, only one specific event can occur, or another.

For example, throwing dice, it is possible to distinguish such events as the loss of an even number of points and the loss of an odd number of points. These events are inconsistent.

Events are called joint events if the occurrence of one of them does not exclude the occurrence of the other.

For example, by throwing a dice, events such as an odd number of points and a multiple of three points can be distinguished. When three rolls, both events occur.

Sum of events

The sum (or combination) of several events is an event consisting in the occurrence of at least one of these events.

Wherein sum of two incompatible events is the sum of the probabilities of these events:

For example, the probability of getting 5 or 6 points on a dice with one roll will be because both events (roll 5, roll 6) are inconsistent and the probability of one or the second event occurring is calculated as follows:

The likelihood the sum of two joint events is equal to the sum of the probabilities of these events without taking into account their joint occurrence:

For example, in mall two identical vending machines sell coffee. The probability that the machine will run out of coffee by the end of the day is 0.3. The probability of running out of coffee in both machines is 0.12. Let us find the probability that by the end of the day the coffee will run out in at least one of the machines (that is, either in one, or in the other, or in both at once).

The probability of the first event “coffee runs out in the first machine” as well as the probability of the second event “coffee ends in the second machine” is equal to 0.3 according to the condition. The events are collaborative.

The probability of joint realization of the first two events by condition is 0.12.

This means that the probability that by the end of the day at least one of the machines will run out of coffee

Dependent and independent events

Two random events A and B are called independent if the occurrence of one of them does not change the probability of the occurrence of the other. Otherwise, events A and B are called dependent.

For example, if two dice are rolled simultaneously, the fallout on one of them, say 1, and on the second 5, are independent events.

Product of probabilities

The product (or intersection) of several events is an event consisting in the joint appearance of all these events.

If two happen independent events A and B with probabilities, respectively, P (A) and P (B), then the probability of the occurrence of events A and B is simultaneously equal to the product of probabilities:

For example, we are interested in the fallout of sixes on the dice two times in a row. Both events are independent and the probability of the realization of each of them separately is. The probability that both of these events will occur will be calculated using the above formula:.

See a selection of tasks for working out the topic.