Presented to date in the open bank of USE problems in mathematics (mathege.ru), the solution of which is based on only one formula, which is a classical definition of probability.
The easiest way to understand the formula is with examples.
Example 1 There are 9 red balls and 3 blue ones in the basket. The balls differ only in color. At random (without looking) we get one of them. What is the probability that the ball chosen in this way will be blue?
Comment. In problems in probability theory, something happens (in this case, our action of pulling the ball) that can have a different result - an outcome. It should be noted that the result can be viewed in different ways. "We pulled out a ball" is also a result. "We pulled out the blue ball" is the result. "We drew this particular ball out of all possible balls" - this least generalized view of the result is called the elementary outcome. It is the elementary outcomes that are meant in the formula for calculating the probability.
Solution. Now we calculate the probability of choosing a blue ball.
Event A: "the chosen ball turned out to be blue"
Total number of all possible outcomes: 9+3=12 (number of all balls we could draw)
Number of outcomes favorable for event A: 3 (the number of such outcomes in which event A occurred - that is, the number of blue balls)
P(A)=3/12=1/4=0.25
Answer: 0.25
Let us calculate for the same problem the probability of choosing a red ball.
The total number of possible outcomes will remain the same, 12. The number of favorable outcomes: 9. The desired probability: 9/12=3/4=0.75
The probability of any event always lies between 0 and 1.
Sometimes in everyday speech (but not in probability theory!) The probability of events is estimated as a percentage. The transition between mathematical and conversational assessment is done by multiplying (or dividing) by 100%.
So,
In this case, the probability is zero for events that cannot happen - improbable. For example, in our example, this would be the probability of drawing a green ball from the basket. (The number of favorable outcomes is 0, P(A)=0/12=0 if counted according to the formula)
Probability 1 has events that will absolutely definitely happen, without options. For example, the probability that "the chosen ball will be either red or blue" is for our problem. (Number of favorable outcomes: 12, P(A)=12/12=1)
We've looked at a classic example that illustrates the definition of probability. All similar USE tasks according to probability theory are solved by applying this formula.
Instead of red and blue balls, there can be apples and pears, boys and girls, learned and unlearned tickets, tickets containing and not containing a question on a certain topic (prototypes , ), defective and high-quality bags or garden pumps (prototypes , ) - the principle remains the same.
They differ slightly in the formulation of the problem of the USE probability theory, where you need to calculate the probability of an event occurring on a certain day. ( , ) As in the previous tasks, you need to determine what is an elementary outcome, and then apply the same formula.
Example 2 The conference lasts three days. On the first and second days, 15 speakers each, on the third day - 20. What is the probability that the report of Professor M. will fall on the third day, if the order of the reports is determined by lottery?
What is the elementary outcome here? - Assigning a professor's report to one of all possible serial numbers for a speech. 15+15+20=50 people participate in the draw. Thus, Professor M.'s report can receive one of 50 numbers. This means that there are only 50 elementary outcomes.
What are the favorable outcomes? - Those in which it turns out that the professor will speak on the third day. That is, the last 20 numbers.
According to the formula, the probability P(A)= 20/50=2/5=4/10=0.4
Answer: 0.4
The drawing of lots here is the establishment of a random correspondence between people and ordered places. In Example 2, matching was considered in terms of which of the places a particular person could take. You can approach the same situation from the other side: which of the people with what probability could get to a particular place (prototypes , , , ):
Example 3 5 Germans, 8 Frenchmen and 3 Estonians participate in the draw. What is the probability that the first (/second/seventh/last - it doesn't matter) will be a Frenchman.
The number of elementary outcomes is the number of all possible people who could get to this place by lot. 5+8+3=16 people.
Favorable outcomes - the French. 8 people.
Desired probability: 8/16=1/2=0.5
Answer: 0.5
The prototype is slightly different. There are tasks about coins () and dice () that are somewhat more creative. Solutions to these problems can be found on the prototype pages.
Here are some examples of coin tossing or dice tossing.
Example 4 When we toss a coin, what is the probability of getting tails?
Outcomes 2 - heads or tails. (it is believed that the coin never falls on the edge) Favorable outcome - tails, 1.
Probability 1/2=0.5
Answer: 0.5.
Example 5 What if we flip a coin twice? What is the probability that it will come up heads both times?
The main thing is to determine which elementary outcomes we will consider when tossing two coins. After tossing two coins, one of the following results can occur:
1) PP - both times it came up tails
2) PO - first time tails, second time heads
3) OP - the first time heads, the second time tails
4) OO - heads up both times
There are no other options. This means that there are 4 elementary outcomes. Only the first one is favorable, 1.
Probability: 1/4=0.25
Answer: 0.25
What is the probability that two tosses of a coin will land on tails?
The number of elementary outcomes is the same, 4. Favorable outcomes are the second and third, 2.
Probability of getting one tail: 2/4=0.5
In such problems, another formula may come in handy.
If at one toss of a coin we have 2 possible outcomes, then for two tosses of results there will be 2 2=2 2 =4 (as in example 5), for three tosses 2 2 2=2 3 =8, for four: 2·2·2·2=2 4 =16, … for N throws of possible outcomes there will be 2·2·...·2=2 N .
So, you can find the probability of getting 5 tails out of 5 coin tosses.
The total number of elementary outcomes: 2 5 =32.
Favorable outcomes: 1. (RRRRRR - all 5 times tails)
Probability: 1/32=0.03125
The same is true for the dice. With one throw, there are 6 possible results. So, for two throws: 6 6=36, for three 6 6 6=216, etc.
Example 6 We throw a dice. What is the probability of getting an even number?
Total outcomes: 6, according to the number of faces.
Favorable: 3 outcomes. (2, 4, 6)
Probability: 3/6=0.5
Example 7 Throw two dice. What is the probability that the total rolls 10? (round to hundredths)
There are 6 possible outcomes for one die. Hence, for two, according to the above rule, 6·6=36.
What outcomes will be favorable for a total of 10 to fall out?
10 must be decomposed into the sum of two numbers from 1 to 6. This can be done in two ways: 10=6+4 and 10=5+5. So, for cubes, options are possible:
(6 on the first and 4 on the second)
(4 on the first and 6 on the second)
(5 on the first and 5 on the second)
In total, 3 options. Desired probability: 3/36=1/12=0.08
Answer: 0.08
Other types of B6 problems will be discussed in one of the following "How to Solve" articles.
1. Specify correct definition. The sum of two events is called:
a) A new event, consisting in the fact that both events occur at the same time;
b) A new event, consisting in the fact that either the first, or the second, or both occur; +
- Specify correct definition. The product of two events is called:
a) A new event, consisting in the fact that both events occur at the same time;+
b) A new event, consisting in the fact that either the first or the second occurs, or both together;
c) A new event, consisting in the fact that one thing happens but another does not happen.
- Specify correct definition. The probability of an event is:
a) The product of the number of outcomes that favor the occurrence of the event by the total number of outcomes;
b) The sum of the number of outcomes that favor the occurrence of the event and the total number of outcomes;
c) The ratio of the number of outcomes that favor the occurrence of an event to the total number of outcomes; +
- Specify correct statement. Probability of an impossible event:
b) equal to zero;+
c) is equal to one;
- Specify correct statement. Probability of certain event:
a) greater than zero and less than one;
b) equal to zero;
c) is equal to one;+
- Specify correct property. Probability of a random event:
a) greater than zero and less than one; +
b) equal to zero;
c) is equal to one;
- Specify correct statement:
a) The probability of the sum of events is equal to the sum of the probabilities of these events;
b) The probability of the sum of independent events is equal to the sum of the probabilities of these events;
c) The probability of the sum of incompatible events is equal to the sum of the probabilities of these events; +
- Specify correct statement:
a) The probability of producing events is equal to the product of the probabilities of these events;
b) The probability of producing independent events is equal to the product of the probabilities of these events; +
c) The probability of producing incompatible events is equal to the product of the probabilities of these events;
- Specify correct definition.Event is:
a) Elementary outcome;
b) The space of elementary outcomes;
c) A subset of the set of elementary outcomes.+
- Specify right answer. What events are called hypotheses?
a) any pairwise incompatible events;
b) pairwise incompatible events, the combination of which forms a reliable event; +
c) the space of elementary events.
- Specify right Answer Bayes formulas define:
a) prior probability of the hypothesis,
b) the posterior probability of the hypothesis,
c) the probability of the hypothesis.+
- Specify correct property. The distribution function of the random variable X is:
a) non-increasing; b) non-decreasing; +c) arbitrary form.
- Specify correct
a) independent +; b) dependent; c) everyone.
- Specify correct property. The equality is valid for random variables:
a) independent; + b) dependent; c) everyone.
- Specify correct conclusion. From the fact that the correlation moment for two random variables X and Y is equal to zero, it follows:
a) there is no functional relationship between X and Y;
b) X and Y are independent;+
c) there is no linear correlation between X and Y;
- Specify right answer. A discrete random variable is given by:
a) indicating its probabilities;
b) indicating its distribution law;+
c) putting each elementary outcome in correspondence
real number.
- Specify correct definition. The mathematical expectation of a random variable is:
a) the initial moment of the first order;+
b) the central moment of the first order;
c) an arbitrary moment of the first order.
- Specify correct definition. The variance of a random variable is:
a) the initial moment of the second order;
b) second-order central moment;+
c) an arbitrary moment of the second order.
- Specify faithful formula. The formula for calculating the standard deviation of a random variable:
a) +; b) ; in) .
- Specify correct definition. The distribution mode is:
a) the value of a random variable at which the probability is 0.5;
b) the value of a random variable at which either the probability or the density function reaches its maximum value;+
c) the value of a random variable at which the probability is 0.
- Specify faithful formula. The dispersion of a random variable is calculated by the formula:
- Specify faithful formula. The density of the normal distribution of a random variable is determined by the formula:
- Specify right answer The mathematical expectation of a random variable distributed according to the normal distribution law is equal to:
- Specify right answer. The mathematical expectation of a random variable distributed according to the exponential distribution law is:
- Specify right answer. The variance of a random variable distributed according to the exponential distribution law is equal to:
- Specify faithful formula. For a uniform distribution, the mathematical expectation is determined by the formula:
- Specify faithful formula. For a uniform distribution, the dispersion is determined by the formula:
- Specify incorrect statement. Sample variance properties:
a) if all options are increased by the same number of times, then the variance will increase by the same number of times.
b) the variance of the constant is zero.
c) if all options are increased by the same number, then the sample variance will not change.+
- Specify correct statement. Parameter estimation is called:
a) Representation of observations as independent random variables having the same distribution law.
b) the totality of the results of observations;
c) any function of the results of observation.+
- Specify correct statement. Distribution parameter estimates have the following property:
a) unbiased;+
b) significance;
c) importance.
- Specify not correct statement.
a) The maximum likelihood method is used to obtain estimates;
b) The sample variance is a biased estimate for the variance;
c) Unbiased, inconsistent, effective estimates are used as statistical estimates of parameters.+
- Specify incorrect statement. The following properties are true for the distribution function of a two-dimensional random variable:
a) ; b) ; c) +.
- Specify incorrect statement:
a) One-dimensional (marginal) distributions of individual components can always be found from a multidimensional distribution function.
b) One-dimensional (marginal) distributions of individual components can always be used to find a multidimensional distribution function.
c) One-dimensional (marginal) distribution densities of individual components can always be found from a multidimensional density function.
- Specify correct statement. The variance of the difference of two random variables is determined by the formula:
a); b)+; in) .
- Specify incorrect statement. Joint Density Formula:
- Specify incorrect statement. Random variables X and Y are called independent if:
a) The distribution law of the random variable X does not depend on the value of the random variable Y.
c) the correlation coefficient between the random variables X and Y is equal to zero.
- Specify right answer. The formula is:
a) an analogue of the Bayes formula for continuous random variables;
b) an analog of the total probability formula for continuous random variables;+
c) an analogue of the formula for the product of the probabilities of independent events for continuous random variables.
- Specify incorrect definition:
a) The initial moment of the order of a two-dimensional random variable (X, Y) is the expectation of the product by, i.e.
b) The central moment of the order of a two-dimensional random variable (X,Y) is the mathematical expectation of the product centered on, i.e.)
c) The correlation moment of a two-dimensional random variable (X, Y) is the mathematical expectation of the product by, i.e. +
- Specify right answer. The dispersion of a random variable distributed according to the normal distribution law is equal to:
- Specify incorrect statement. The simplest tasks mathematical statistics are:
a) sampling and grouping of statistical data obtained as a result of the experiment;
b) determination of distribution parameters, the form of which is known in advance;
c) obtaining an estimate of the probability of the event under study.
Basic concepts on the topic:
1. Trial, elementary outcome, trial outcome, event.
2. Certain event, impossible event, random event.
3. Joint events, incompatible events, equivalent events, equally possible events, the only possible events.
4. Complete group of events, opposite events.
5. Elementary event, composite event.
6. The sum of several events, the product of several events. Their geometric interpretation
1. In the problem “Two shots are fired at the target. Find the probability that the target will be hit once" by the test is:
1) * two shots are fired at the target;
2) the target will be hit once;
3) the target will be hit twice.
2. Throw a coin. Event: A - “the coat of arms will fall out”. The event - “a number will come up” is:
1) random;
2) reliable;
3) impossible;
4) * opposite.
3. A dice is rolled. Let's denote the events: A - "loss of 6 points", B - "loss of 4 points", D - "loss of 2 points", C - "loss of an even number of points". Then the event C is
1)
;
2)
;
3)*
;
4)
.
4. The student must pass two exams. Event A - "the student passed the first exam", event B - "the student passed the second exam", event C - "the student passed both exams". Then the event C is
1)*
;
2)
;
3)
;
4)
.
5. From the letters of the word "TASK" one letter is randomly selected. The event - "the letter K is selected" is
1) random;
2) reliable;
3)* impossible;
4) opposite.
6. From the letters of the word "WORLD" one letter is randomly selected. The event - "the letter M is selected" is
1)* random;
2) reliable;
3) impossible.
7. The event - "a white ball is drawn from an urn containing only white balls" is
1) random;
2) * reliable;
3) impossible.
8. Two students take an exam. Events: A - "the first student will pass the exam", B - "the second student will pass the exam" are
1) incompatible;
2) reliable;
3) impossible;
4)*joint.
9. Events are called incompatible if
4) * the onset of one excludes the possibility of the appearance of the other.
10. Events are called the only possible ones if
1) the occurrence of one does not exclude the possibility of the appearance of another;
2) in the implementation of a set of conditions, each of them has an equal opportunity to occur;
3) * during the test, at least one of them will definitely occur;
Topic 2. Classical definition of probability
Basic concepts on the topic:
1. The probability of an event, the classical definition of the probability of a random event.
2. An outcome favorable to the event.
3. Geometric definition of probability.
4. Relative frequency of the event.
5. Statistical definition of probability.
6. Properties of probability.
7. Methods for counting the number of elementary outcomes: permutations, combinations, placements.
Application of all these concepts on practical examples.
Sample test tasks offered in this topic:
1. Events are called equally likely if
1) they are incompatible;
2) * in the implementation of a set of conditions, each of them has an equal opportunity to occur;
3) during the test, at least one of them will definitely occur;
4) the occurrence of one excludes the possibility of the appearance of the other.
2. Test - "throw two coins." Event - "at least one of the coins will have a coat of arms." The number of elementary outcomes that favor this event is equal to:
4) four.
3. Test - "throw two coins." Event - "a coat of arms will fall on one of the coins." The number of all elementary, equally possible, the only possible, incompatible outcomes is equal to:
4)* four.
4. There are 12 balls in the urn, they do not differ in anything except the color. Among these balls, 5 are black and 7 are white. The event is "a white ball is randomly drawn." For this event, the number of favorable outcomes is:
5. There are 12 balls in the urn, they do not differ in anything except the color. Among these balls, 5 are black and 7 are white. The event is "a white ball is randomly drawn." For this event, the number of all outcomes is:
6. The probability of an event takes any value from the interval:
3)
;
4)
;
5)*
.
7. The subscriber forgot the last two digits of the telephone number and, knowing only that they are different, dialed them at random. In how many ways can he do this?
1)
;
2)*
;
Exercise
Demo option
1. and are independent events. Then the following statement is true: a) they are mutually exclusive events
b) 
G) 
e) 
2. , , - event probabilities , , 0 " style="margin-left:55.05pt;border-collapse:collapse;border:none">
3. Probabilities of events and https://pandia.ru/text/78/195/images/image012_30.gif" width="105" height="28 src=">.gif" width="55" height="24"> there is:
a) 1.25 b) 0.3886 c) 0.25 d) 0.8614
d) there is no correct answer
4. Prove equality using truth tables or show that it is false.
Section 2. Probabilities of combining and crossing events, conditional probability, total probability and Bayesian formulas.
Exercise: Choose the correct answer and mark the corresponding letter in the table.
Demo option
1. Throw two dice at the same time. What is the probability that the sum of the rolled points is not greater than 6?
a) ; b) ; in) ; G) ;
d) there is no correct answer
2. Each letter of the word "CRAFT" is written on a separate card, then the cards are mixed. We take out three cards at random. What is the probability of getting the word "WOOD"?
a) ; b) ; in) ; G) ;
d) there is no correct answer
3. Among the second-year students, 50% never missed classes, 40% missed classes no more than 5 days per semester, and 10% missed classes for 6 or more days. Among the students who did not miss classes, 40% received the highest score, among those who missed no more than 5 days - 30%, and among the rest - 10% received the highest score. The student received the highest score on the exam. Find the probability that he missed classes for more than 6 days.
a) https://pandia.ru/text/78/195/images/image024_14.gif" width="17 height=53" height="53">; c) ; d) ; e) no correct answer
Test on the course of probability theory and mathematical statistics.
Section 3. Discrete random variables and their numerical characteristics.
Exercise: Choose the correct answer and mark the corresponding letter in the table.
Demo option
1 . Discrete random variables X and Y are given by their own laws
distribution
Random variable Z = X+Y. Find Probability 
a) 0.7; b) 0.84; c) 0.65; d) 0.78; d) there is no correct answer
2. X, Y, Z are independent discrete random variables. The X value is distributed according to the binomial law with parameters n=20 and p=0.1. The Y value is distributed over geometric law with parameter p=0.4. The value of Z is distributed according to the Poisson law with the parameter =2. Find the variance of a random variable U= 3X+4Y-2Z
a) 16.4 b) 68.2; c) 97.3; d) 84.2; d) there is no correct answer
3. Two-dimensional random vector (X, Y) is given by the distribution law
event, event 
. What is the probability of event A+B?
a) 0.62; b) 0.44; c) 0.72; d) 0.58; d) there is no correct answer
Test on the course of probability theory and mathematical statistics.
Section 4. Continuous random variables and their numerical characteristics.
Exercise: Choose the correct answer and mark the corresponding letter in the table.
Option demo
1. Independent continuous random variables X and Y are uniformly distributed on the segments: X at https://pandia.ru/text/78/195/images/image032_6.gif" width="32" height="23">.
Random variable Z = 3X +3Y +2. Find D(Z)
a) 47.75; b) 45.75; c) 15.25; d) 17.25; d) there is no correct answer
2 ..gif" width="97" height="23">
a) 0.5; b) 1; c) 0; d) 0.75; d) there is no correct answer
3. A continuous random variable X is given by its probability density https://pandia.ru/text/78/195/images/image036_7.gif" width="99" height="23 src=">.
a) 0.125; b) 0.875; c) 0.625; d) 0.5; d) there is no correct answer
4. Random variable X is normally distributed with parameters 8 and 3. Find
a) 0.212; b) 0.1295; c) 0.3413; d) 0.625; d) there is no correct answer
Test on the course of probability theory and mathematical statistics.
Section 5. Introduction to mathematical statistics.
Exercise: Choose the correct answer and mark the corresponding letter in the table.
Demo option
1. The following mathematical expectation estimates are proposed https://pandia.ru/text/78/195/images/image041_6.gif" width="98" height="22">:
A) https://pandia.ru/text/78/195/images/image043_5.gif" width="205" height="40">
C) https://pandia.ru/text/78/195/images/image045_4.gif" width="205" height="40">
E) 0 "style="margin-left:69.2pt;border-collapse:collapse;border:none">
2. The variance of each measurement in the previous problem is . Then the most efficient of the unbiased estimates obtained in the first problem is the estimate
3. Based on the results of independent observations of a random variable X obeying the Poisson law, construct an estimate of the unknown parameter by the method of moments 425 " style="width:318.65pt;margin-left:154.25pt;border-collapse:collapse; border:none">
a) 2.77; b) 2.90; c) 0.34; d) 0.682; d) there is no correct answer
4. Half-width of 90% confidence interval constructed to estimate the unknown mathematical expectation of a normally distributed random variable X for sample size n=120, sample mean https://pandia.ru/text/78/195/images/image052_3.gif" width="19 "height="16">=5, yes
a) 0.89; b) 0.49; c) 0.75; d) 0.98; d) there is no correct answer
Validation Matrix - test demo
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OPTION 1
1. In a random experiment, two dice are thrown. Find the probability of getting 5 points in total. Round the result to the nearest hundredth.
2. In a random experiment, a symmetrical coin is thrown three times. Find the probability that heads come up exactly twice.
3. On average, out of 1,400 garden pumps sold, 7 leak. Find the probability that one randomly selected pump does not leak.
4. The competition of performers is held in 3 days. There are 50 entries in total, one from each country. There are 34 performances on the first day, the rest are distributed equally among the remaining days. The order of performances is determined by a draw. What is the probability that the performance of the representative of Russia will take place on the third day of the competition?
5. The taxi company has 50 cars; 27 of them are black with yellow inscriptions on the sides, the rest are yellow with black inscriptions. Find the probability that a car will arrive at a random call yellow color with black lettering.
6. Groups perform at the rock festival - one from each of the declared countries. The order of performance is determined by lot. What is the probability that a group from Germany will perform after a group from France and after a group from Russia? Round the result to the nearest hundredth.
7. What is the probability that a randomly selected natural number 41 to 56 is divisible by 2?
8. There are only 20 tickets in the collection of tickets in mathematics, 11 of them contain a question on logarithms. Find the probability that a student will get a logarithm question in a ticket randomly selected in the exam.
9. The figure shows a labyrinth. The spider crawls into the maze at the "Entrance" point. The spider cannot turn around and crawl back. At each fork, the spider chooses a path that has not yet crawled. Considering the choice of the further path as random, determine with what probability the spider will come to the exit.
10. To enter the institute for the specialty "Translator", the applicant must score at least 79 points on the Unified State Examination in each of the three subjects - mathematics, Russian and a foreign language. To enter the specialty "Customs", you need to score at least 79 points in each of the three subjects - mathematics, Russian language and social studies.
The probability that applicant B. will receive at least 79 points in mathematics is 0.9, in Russian - 0.7, in foreign language- 0.8 and in social studies - 0.9.
OPTION 2
1. There are three sellers in the store. Each of them is busy with a client with a probability of 0.3. Find the probability that, at a random time, all three salespeople are busy at the same time (assume that the customers enter independently of each other).
2. In a random experiment, a symmetrical coin is tossed three times. Find the probability that the outcome of the RPP will come (all three times it comes up tails).
3. The factory produces bags. On average, for every 200 quality bags, there are four bags with hidden defects. Find the probability that the purchased bag will be of high quality. Round the result to the nearest hundredth.
4. The competition of performers is held in 3 days. There are 55 entries in total, one from each country. There are 33 performances on the first day, the rest are distributed equally among the remaining days. The order of performances is determined by a draw. What is the probability that the performance of the representative of Russia will take place on the third day of the competition?
5. There are 10 digits on the telephone keypad, from 0 to 9. What is the probability that a randomly pressed number will be less than 4?
6. Biathlete shoots at targets 9 times. The probability of hitting the target with one shot is 0.8. Find the probability that the biathlete hit the targets the first 3 times and missed the last 6. Round the result to the nearest hundredth.
7. Two factories produce the same glass for car headlights. The first factory produces 30 of these glasses, the second - 70. The first factory produces 4 defective glasses, and the second - 1. Find the probability that a glass randomly bought in a store will be defective.
8. There are only 25 tickets in the collection of chemistry tickets, 6 of them contain a question on hydrocarbons. Find the probability that a student will get a question on hydrocarbons in a ticket randomly selected in the exam.
9. To enter the institute for the specialty "Translator", the applicant must score at least 69 points on the Unified State Examination in each of the three subjects - mathematics, Russian language and a foreign language. To enter the specialty "Management", you need to score at least 69 points in each of the three subjects - mathematics, Russian language and social studies.
The probability that applicant T. will receive at least 69 points in mathematics is 0.6, in Russian - 0.6, in a foreign language - 0.5 and in social studies - 0.6.
Find the probability that T. will be able to enter one of the two specialties mentioned.
10. The figure shows a labyrinth. The spider crawls into the maze at the "Entrance" point. The spider cannot turn around and crawl back. At each fork, the spider chooses a path that has not yet crawled. Considering the choice of the further path as random, determine with what probability the spider will come to the exit.
OPTION 3
1. 60 athletes participate in the gymnastics championship: 14 from Hungary, 25 from Romania, the rest from Bulgaria. The order in which the gymnasts perform is determined by lot. Find the probability that the athlete who competes first is from Bulgaria.
2. Automatic production line for batteries. The probability that a finished battery is defective is 0.02. Before packaging, each battery goes through a control system. The probability that the system will reject a bad battery is 0.97. The probability that the system will mistakenly reject a good battery is 0.02. Find the probability that a randomly selected battery will be rejected.
3. To enter the institute for the specialty " International relationships”, the applicant must score at least 68 points on the exam in each of the three subjects - mathematics, Russian and a foreign language. To enroll in the specialty "Sociology", you need to score at least 68 points in each of the three subjects - mathematics, Russian language and social studies.
The probability that applicant V. will receive at least 68 points in mathematics is 0.7, in Russian - 0.6, in a foreign language - 0.6 and in social studies - 0.7.
Find the probability that B. will be able to enter one of the two specialties mentioned.
4. The figure shows a labyrinth. The spider crawls into the maze at the "Entrance" point. The spider cannot turn around and crawl back. At each fork, the spider chooses a path that has not yet crawled. Considering the choice of the further path as random, determine with what probability the spider will come to the exit.
5. What is the probability that a randomly chosen natural number from 52 to 67 is divisible by 4?
6. On the geometry exam, the student gets one question from the list of exam questions. The probability that this is an inscribed circle question is 0.1. The probability that this is a trigonometry question is 0.35. 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.
7. Seva, Slava, Anya, Andrey, Misha, Igor, Nadya and Karina cast lots for who to start the game. Find the probability that a boy will start the game.
8. 5 scientists from Spain, 4 from Denmark and 7 from Holland came to the seminar. The order of reports is determined by a draw. Find the probability that the report of a scientist from Denmark will be the twelfth.
9. There are only 25 tickets in the collection of tickets on philosophy, 8 of them contain a question on Pythagoras. Find the probability that a student will not get a question on Pythagoras in a ticket randomly chosen at the exam.
10. There are two payment machines in the store. Each of them can be faulty with a probability of 0.09, regardless of the other automaton. Find the probability that at least one automaton is serviceable.
OPTION 4
1. Groups perform at the rock festival - one from each of the declared countries. The order of performance is determined by lot. What is the probability that a band from the USA will perform after a band from Vietnam and after a band from Sweden? Round the result to the nearest hundredth.
2. The probability that student T. correctly solves more than 8 problems on the history test is 0.58. The probability that T. correctly solves more than 7 problems is 0.64. Find the probability that T. correctly solves exactly 8 problems.
3. The factory produces bags. On average, for every 60 quality bags, there are six bags with hidden defects. Find the probability that the purchased bag will be of high quality. Round the result to the nearest hundredth.
4. Sasha had four sweets in his pocket - “Mishka”, “Vzlyotnaya”, “Squirrel” and “Roasting”, as well as the keys to the apartment. Taking out the keys, Sasha accidentally dropped one candy from his pocket. Find the probability that the take-off candy is lost.
5. The figure shows a labyrinth. The spider crawls into the maze at the "Entrance" point. The spider cannot turn around and crawl back. At each fork, the spider chooses a path that has not yet crawled. Considering the choice of the further path as random, determine with what probability the spider will come to the exit.
6. In a random experiment, three dice are thrown. Find the probability of getting 15 points in total. Round the result to the nearest hundredth.
7. Biathlete shoots at targets 10 times. The probability of hitting the target with one shot is 0.7. Find the probability that the biathlete hit the targets the first 7 times and missed the last 3. Round the result to the nearest hundredth.
8. 5 scientists from Switzerland, 7 from Poland and 2 from Great Britain came to the seminar. The order of reports is determined by a draw. Find the probability that the thirteenth is the report of a scientist from Poland.
9. To enter the institute for the specialty "International Law", the applicant must score at least 68 points on the Unified State Examination in each of the three subjects - mathematics, Russian language and a foreign language. To enroll in the specialty "Sociology", you need to score at least 68 points in each of the three subjects - mathematics, Russian language and social studies.
The probability that applicant B. will receive at least 68 points in mathematics is 0.6, in Russian - 0.8, in a foreign language - 0.5 and in social studies - 0.7.
Find the probability that B. will be able to enter one of the two specialties mentioned.
10. There are two identical coffee machines in the mall. The probability that the machine will run out of coffee by the end of the day is 0.25. The probability that both machines will run out of coffee is 0.14. Find the probability that by the end of the day there will be coffee left in both vending machines.