Basics of probability theory presentation on the topic
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Probability theory
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Probability theory Introduction Basic combinatorial objects Elements of probability theory
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Basic combinatorial objects Multiplication rule Combinations Permutation Placement Addition rule Problems in which all possible combinations made according to a certain rule are counted are called combinatorial. The branch of mathematics dealing with their solution is called combinatorics.
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Elements of probability theory Basic concepts of probability theory Repetition of tests Addition and multiplication theorems of probabilities
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Basic concepts of probability theory Classical probability formula Statistical and geometric probabilities Random events. Operations on events
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Probability addition and multiplication theorems Probability addition theorem Probability multiplication theorem. Conditional probability Formula for total probability. Bayes formula
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Repetition of tests Asymptotic formulas Bernoulli formula
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Introduction Probability theory arose as a science from the belief that mass random events are based on deterministic patterns; probability theory studies these patterns. Mathematical statistics is a science that studies methods for processing the results of observations of mass random phenomena that have statistical stability in order to identify these patterns
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Multiplication rule If you need to perform some K actions one after another, where 1 action can be performed in 1 ways, 2 action - in 2 ways, and so on until the Kth action, which can be performed in a k ways, then all K actions together can be performed in a 1 · a 2 · a 3 ...a k ways. 4 boys and 4 girls sit on 8 chairs in a row, with boys sitting in even-numbered seats and girls in odd-numbered seats. In how many ways can this be done? The first boy can sit in any of the four even-numbered seats, the second - in any of the remaining three seats, the third - in any of the remaining two seats. The last boy is given only one opportunity. According to the multiplication rule, boys can occupy four places in 4 · 3 · 2 · 1=24 ways. Girls have the same opportunities. Thus, according to the multiplication rule, boys and girls can occupy all the chairs in 24 · 24 = 576 ways.
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Addition rule If two actions are mutually exclusive, and one of them can be performed in m ways, and the other in n ways, then any one of these actions can be performed in m+n ways. This rule can be easily extended to any finite number of actions.
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Placements Theorem: the number of placements from n to m is equal to An arrangement of n elements to m is any ordered subset of m elements of a set consisting of n different elements Example problem
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1) The magazine has 10 pages, you need to place 4 photographs on the pages. In how many ways can this be done if no page of the newspaper should contain more than one photograph? 2) How many four-digit numbers can you write using all ten digits without repetition? back
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Permutations A permutation of n elements is any ordered set that contains once all n different elements of the given set. Theorem: The number of permutations of n different elements is equal to n! Sample task
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Write down all possible permutations for the numbers 3,5,7 3,5,7 ; 3,7,5; 5,3,7; 5,7,3; 7,3,5; 7,5,3 2) In how many ways can nine different books be arranged on a shelf so that certain four books stand next to each other? back
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Combinations A combination of n elements by m is any subset of m elements that belong to a set consisting of n different elements Theorem: The number of combinations of n by m is equal to Corollary: The number of combinations of n elements by nm is equal to the number of combinations of n elements by m Example tasks
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1) There are 10 white and 5 black balls. In how many ways can you choose 7 balls so that among them there are 3 black ones? Solution: among the selected balls there are 4 white and 3 black. Methods for choosing old balls Methods for choosing black balls According to the multiplication rule, the required number of ways is 2) In how many ways can a group of 12 people be divided into two subgroups, one of which should have no more than 5 people, and the second no more than 9 people? The choice of the first subgroup uniquely determines the second; according to the addition rule, the required number of ways is equal to: Subgroup of 3 people Subgroup of 4 people Subgroup of 5 people back
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Random events. Operations on events An event is a phenomenon that occurs as a result of the implementation of a specific set of conditions. The implementation of a set of conditions is called an experience or test. Event is the result of a test. A random event is an event that may or may not happen as a result of some trial (when tossing a coin, it may or may not come up on heads). A certain event is an event that is certain to occur as a result of a test (removing a white ball from a box of white balls). An event that cannot occur as a result of a given test (removing a black ball from a box of white balls) is considered impossible. Further
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Random events Event A is said to be favorable to event B if the occurrence of event A entails the occurrence of event B. Events A and B are called non-joint if, as a result of this test, the appearance of one of them excludes the appearance of the other (test: shooting at a target; A - knocking out an even number of points; B - not an even number). Events A and B are called joint if, as a result of this test, the appearance of one of them does not exclude the appearance of the other (A - the teacher entered the classroom; B - the student entered). back next
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Random events Two events A are called opposite if the non-occurrence of one of them as a result of a test entails the appearance of the other (negation of A). If a group of events is such that at least one of them must occur as a result of the test and any two of them are incompatible, then this group of events is called a complete group of events. Events are called equally possible if, according to the test conditions, there is no reason to consider any of them more possible than any other (A - heads; B - tails). back next
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Operations on events The sum of several events is an event consisting of the occurrence of at least one of them as a result of a test. Example: there are red, black and white balls in a box. A- removing the black ball B- removing the red ball C- removing the white ball A+B – black or red ball is removed B+C – red or white ball is removed A+C – black or white ball is removed back further
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Operations on events The product of several events is an event consisting in the joint occurrence of all these events as a result of a test. Example: the following events occur: A - a “queen” is taken out from a deck of cards B – a card of the spades suit is taken out A∙B – event – a “queen of spades” card is taken out back
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Classic probability formula The probability of an event is a numerical measure of the objective possibility of its occurrence. If there is a complete group of pairwise incompatible and equally possible events, then the probability P(A) of the occurrence of event A is calculated as the ratio of the number of outcomes favorable to the occurrence of the event to the number of all outcomes of the test. N – the number of all outcomes of the test M – the number of outcomes favorable to event A Property of probability: 1) The probability of a reliable event is 1 2) The probability of an impossible event is 0 3) The probability of event A satisfies the double inequality Example of a problem
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1) There are 4 black and 6 white balls in a box, 1 ball is taken out, what is the probability that the ball will be white or black? N =10; M=6; A- Removing the white ball N =10; M=4; A- Removing a black ball 2) There are 10 balls in a box, 2 black, 4 white, 4 red, 1 ball is removed. What is the probability that he: A- black; B - white; C - red; D - green N =10; M=2 N=10; M=4 N=10; M=0 N=10; M=4 back
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Statistical and geometric probabilities It was noticed that when experiments are repeated many times, the relative frequency of occurrence of an event in these experiments tends to become stable. The relative frequency of occurrence of an event is understood as the ratio M / N, where N is the number of experiments; M is the number of occurrence of the event. With an increase in experiments, the relative frequency of the occurrence of an event will differ almost as little as desired from a certain constant number, which is taken as the probability of an event in a separate experiment. The relative frequency of occurrence of an event is called statistical probability. With an increase in the number of experiments, the relative frequency tends to the probability P(G) = 0.5. With a sufficiently large number of experiments, the relative frequency can be considered an approximate value of the probability. The geometric probability of an event is the ratio of the measure of the region favorable to the occurrence of the event to the measure of the entire region.
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Probability addition theorem The probability of the occurrence of one of two incompatible events is equal to the sum of the probabilities of these events: P(A+B)=P(A)+P(B) The probability of the occurrence of one of several pairwise incompatible events is equal to the sum of the probabilities of these events: The sum of the pairwise probabilities of incompatible events forming a complete group is equal to 1. Further
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Probability addition theorem The sum of the probabilities of opposite events is equal to 1 The probability of the occurrence of at least one of two joint events is equal to the sum of the probabilities of these events without the probability of their joint occurrence: back
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Probability multiplication theorem. Conditional probability Conditional probability is the probability of event B, calculated under the assumption that event A has already occurred. The probability of the joint occurrence of two events is equal to the product of the probability of one of them by the conditional probability of the other, calculated under the assumption that the first event has already occurred: Two events are called independent if the occurrence of either of them does not change the probability of the occurrence of the other: The probability of the joint occurrence of two independent events is equal to the product their probabilities: further
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Probability multiplication theorem. Conditional probability The probability of the joint occurrence of a finite number of events is equal to the product of the probability of one of them by the conditional probabilities of all the others, and the conditional probability of each subsequent event is calculated under the assumption that all previous ones have already occurred: P(A 1 A 2 A 3 ...A n) = P (A 1 )P A1 (A 2 )P A1A2 (A 3 )…P A1A2A3…A n -1 (A n ); Р А1А2А3…А n -1 (А n ) – probability of occurrence of event А n , calculated under the assumption that events А 1 А 2 А 3 …А n -1 occurred. The probability of the joint occurrence of several events, independent in the aggregate, is equal to the product of the probabilities of these events: The probability of the occurrence of at least one of the events A 1 A 2 A 3 ...A n, independent in the aggregate, is equal to the difference between one and the product of the probabilities of opposite events back
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Total probability formula. Bayes formula The probability of an event A, which can occur only if one of the events H1, H2, H3,...,Hn, forming a complete group of pairwise incompatible events, occurs, is equal to the sum of the products of the probabilities of each of the events H1, H2, H 3 ,…,H n to the corresponding conditional probability of event A: Total probability formula below
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Total probability formula. Bayes formula Consider the events B 1, B 2, B 3,..., B n which form a complete group of events and upon the occurrence of each of them B i, event A can occur with some conditional probability. Then the probability of the occurrence of event A is equal to the sum of the products of the probabilities of each of the events to the corresponding conditional probability of event A No matter how many probabilities there are: back next
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Total probability formula. Bayes' formula Let's consider an event A that can occur subject to the occurrence of one of the incompatible events, B 1, B 2, B 3,..., B n, which form a complete group of events. If event A has already occurred, then the probability of events can be overestimated using Bayes' formula, the formula for the probability of hypotheses: back
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Bernoulli formula The probability that in n independent trials in each of which the probability of an event occurring is equal to P, P( 0
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Asymptotic formulas If the number of tests is large, then using the Bernoulli formula will be inappropriate due to the need to perform cumbersome calculations. The Moivre-Laplace theorem, which gives an asymptotic formula, allows one to calculate the probability approximately. Theorem: If the probability of the occurrence of event A in each of n independent trials is equal to p and different from zero and one, and the number of trials is sufficiently large, then the probability P n (m) that event A will occur m times in n trials is approximately equal to the value functions further
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Asymptotic formulas. Poisson distribution If the probability of an event in an individual trial is close to zero, then another asymptotic formula is used - the Poisson formula. Theorem: If the probability p of the occurrence of event A in each trial is constant, but close to zero, the number of independent trials n is sufficiently large, and the product np = , then the probability P n (m) that event A will occur m times in n independent trials, approximately equal to back
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1) The magazine has 10 pages, you need to place 4 photographs on the pages. In how many ways can this be done if no page of the newspaper should contain more than one photograph? 2) How many four-digit numbers can you write using all ten digits without repetition? back
Technological lesson map with presentation “Principles of Probability Theory”, 8th grade
Beginnings of probability theory.
Lesson-research with a presentation and a printed notebook Author: Elena Mikhailovna Krasina, mathematics teacher at Chekhov Secondary School - 3 with in-depth study of individual subjects. Description: a lesson on discovering new knowledge is intended for mathematics teachers working in grades 7-8. This is the first lesson in the Probability Theory course. The development will enable the teacher to use historical material, interesting examples from cinema and the surrounding life, as well as an experiment to develop cognitive interest in a new branch of mathematics and create positive motivation for learning. Purpose: to review the basic concepts and formula for calculating the probability of random events. Objectives: Educational: - classify events and formulate definitions of each type; — formulate the classical definition of probability; — create an algorithm for calculating the probability of a random event, learn how to apply this algorithm to solve problems. Developmental: - create conditions for the development of independent work skills, intellectual qualities, the ability to analyze, generalize, and highlight the main thing. Educational: - create conditions for the development of cognitive interest in the subject and self-confidence; - create positive motivation for learning. Planned results: Subject:
- know the classification of events and definitions;
— know the classical definition of probability and the algorithm for finding the probability of a random event; — be able to determine the type of event, find the probability of a random event using an algorithm. Personal: - be able to listen and engage in dialogue; — participate in collective discussion of problems; - conduct an experiment and interpret its results. Meta-subject: - be able to set a goal; — define tasks; — plan the sequence of actions; — analyze the results of activities and make adjustments; - carry out self-assessment and self-control. Educational technology: Technology of the active method. Equipment for the lesson: interactive whiteboard, presentation, printed notebook (printed for each student, please note that 4 different versions of the first page are needed to obtain different keywords at the update stage), dice for the experiment. Additional sources: the work uses statements by Robert Bringhurst, Gottfried Wilhelm Leibniz, a video fragment of the animated film “12 Months”, a fragment of the “News” program, the song “According to the Theory of Probability” performed by Vadim Mulerman, music by Igor Krutoy.
Lesson progress:
I. Organizational moment. Motivation for learning activities. Teacher:
There are many different sayings about mathematics.
But the saying that resonates with me is this: “ Mathematics is not there to force hard work on anyone.
On the contrary, it exists only for...” Robert Bringhurst There is a missing word at the end of the statement.
How would you finish the sentence? Student: Joy, pleasure. Teacher: Do you enjoy your math lessons? When does this happen? Student: When you manage to solve a complex problem, when you understand a new topic. It’s a pleasure to have learned something new and important. Teacher: We finished the previous topic and wrote a test. What is ahead of us today? Student: A lesson in discovering new knowledge. Teacher: I hope that today we will “absorb knowledge” with pleasure. II. Updating the basic knowledge of students, forming a lesson topic, setting a goal. Teacher: What needs to be done before we turn to a new topic? Student: Repeat the knowledge that we will need today to discover new knowledge. Teacher: For work we will use a fragment of a printed notebook that I prepared for you. Open your notebook and complete the task in the updating section. The result will be the keyword. I will ask those who make up the words first to come out and pin them on the board. Example of tasks for obtaining the keyword “Probability”
Students attach the words “Probability”, “Event”, “Reliable”, “Random”, “Outcome” to the board. Message of the lesson topic and purpose: Teacher: We have received the basic concepts of a section of mathematics that is still unknown to you. But maybe someone has heard of him? Student: “Probability theory” Teacher: What does probability theory study? Where can we find the answer to this question? Student: For example, in a textbook, on the Internet, in a theoretical block of a printed notebook. Probability theory is a mathematical science that studies the patterns of mass random events. Teacher: Complete your notes with the missing words.
Teacher: Where does learning a new topic begin? Student: We consider the basic definitions, concepts, notations, etc. Teacher: Let’s formulate the topic and purpose of the lesson, based on the fact that this is the first lesson in the topic. Student: Topic: “The beginnings of probability theory.” Purpose: to review the basic concepts and classical definition of probability. III. Studying new material 1. Historical background Teacher: Let’s turn to the historical background (viewing a story about the origins of the “Theory of Probability”). 2. Classification of types of events Teacher: The theory of probability is based on such a concept as an event. Now I will test with a dice and an event will occur (a dice is thrown and we see what comes up). Please note that the event is indicated by a capital letter. Its outcome, that is, the result of the test, is written in curly brackets. A = {dropping an even number} What should I write in the line of event B? Do your test, roll the dice and write down the event that occurs - this is event C.
Teacher: Every day is full of events. There are events that we dream about when we fall asleep at night. It happens that some of them are not destined to happen. Sometimes something unexpected happens to us. Sometimes we accidentally find ourselves inside an event. But now we need to look at events through the eyes of mathematicians and classify them.
Let's return to our concepts on the board. How many types of events have we identified? Student: Two. Reliable and random. Teacher: Attention to the screen, there is a series of events in front of you. We need to distribute them according to known species. Students classify events by type (reliable and random). Teacher: We were forced to miss two events. Why? Student: We could not classify these events as either reliable or random. Teacher: So our classification is incomplete. Suggest a name for these events. Student: Unreal, not happening, non-existent, impossible. Teacher: Complete your supporting summary in the “Types of Events” section. Student: 1. An event that is sure to occur is called reliable. 2. An event that will never occur is called impossible. 3. An event that may or may not occur is called random. 4. Any test result is called an outcome.
Teacher: Let's check if you understand these concepts well. Let's turn to the fairy tale (viewing the fragment “12 months”). Tell me the events that you saw in this fragment. Students: After 31, December 32 will come, etc.; Snowdrops will bloom in December. Teacher: How could you call these events? Students: Impossible. Teacher: Maybe someone has a different opinion? I suggest watching another story. This is Channel 1 news about an event that happened in 2012 in Belgrade. What conclusion can be drawn from what we saw? Students: We were wrong about the second event. Since such an event happened in real life, it can no longer be called impossible. This is a random event. Teacher: Recently, by historical standards, giving a girl a bouquet of snowdrops was a sign of attention, a symbol of spring. Today, these spring flowers are listed in the Red Book as an endangered species. And such a gift is a crime against nature. If we do not take care of primroses, then perhaps your children in math class will call such an event as “snowdrops appeared in the spring” as impossible. Every year, on April 19, many countries around the world celebrate a beautiful spring holiday that has already become traditional - “Snowdrop Day”.
Teacher: Why do you think you made a mistake in the case of snowdrops? Students: This is a rare phenomenon, and we did not know about it. We may not have enough knowledge about what is happening in the world. Teacher: “A rare event”, “frequently occurring”, “sometimes can be found” - from a mathematical point of view, could this be an opportunity to make a comparative assessment? What can we compare using mathematics? Student: We can compare numbers. Teacher: The success rate of an event began to be called “probability.” And as we know, the share can be expressed as a number. Want to know how the great mathematicians of the past did it? 3. Conducting an experiment Teacher: I suggest you go back a few centuries and repeat their experiment. Remember the historical video essay. What game prompted scientists to explore? Pupils: Dice game. Teacher: An analogue of a dice can be a dice. In children's games you have met him more than once. Today we use it in experimental work. Experiment with dice. Work in pairs. The 1st row conducts 1 experiment, the 2nd row - 2nd experiment and the 3rd row - 3rd experiment. Work plan on the presentation screen and in a printed notebook. After finishing the experiment, write down your result on the board where your row number is written.
Teacher: Exactly the same experiment by the French scientist Georges Buffon. But he was not limited by the lesson time and did this 4040 times. During the experiment, Buffon's coat of arms fell out 2048 times. How to find what part of all outcomes is the outcome “the coat of arms fell”? Do the calculation using a calculator, round to the nearest tenth and answer what is the frequency of the “coat of arms falling” event. Student: Let's find the ratio of 2048 to 4040. After rounding we get 0.5. Therefore, the frequency of the “coat of arms dropped” event in this experiment is 0.5. Teacher: The English mathematician Karl Pearson conducted the same experiment at the beginning of the twentieth century. He tossed a coin 24,000 times. Pearson counted how many times the coin would come up heads. During the experiment, he found that it fell out in 12,012 cases. Based on the fact that we have already analyzed the results of Georges Buffon's experiment, draw a conclusion. Student: Let’s find the ratio of 1202 to 2400 and round it up. Therefore, the frequency of the “heads up” event in this experiment is 0.5.
Teacher: Why did you, when conducting the same experiment, get different answers and such serious discrepancies? Student: We conducted a small number of experiments. Teacher: Therefore, the more tests, the closer the probability obtained experimentally to reality. Problematic questions: Is the experimental method convenient for calculating the probability of an event? Why? What knowledge could help us? Student: Inconvenient because it requires a lot of testing and a lot of time. If there was a formula or algorithm, then this would be the best option for finding the probability. 4. Classical definition of probability Teacher: We work in a printed notebook. Based on the definition, we will write a formula and create an algorithm class=”aligncenter” width=”458″ height=”292″[/img] P(A)
=
m/n 5. Algorithm for calculating probability. Step 1. I will determine the number n =... - the number of all possible outcomes. Step 2. Determine the number m =... - the number of favorable outcomes. Step 3. Find the ratio: P(A)
=
m/n
Frontal discussion of the nuances of the topic: Probability of an impossible event P(A)
= 0 Probability of a reliable event
P(B)
= 1 Probability of a random event
P(C) belongs to the interval (0;1). IV. Consolidation 1. Frontal work: solving an experimental task analytically, working in a printed notebook. Teacher: Experimentally, you were unable to come to a consensus. Let's solve the problem analytically. Explains row 1: Step 1. n
= 6 Step 2.
m
= 1 Step 3.
Р(А
) =
1/6 Explains row 2: Step 1. n
= 6 Step 2.
m
= 3 Step 3.
Р(А)
=
0.5 Explains row 3: Step 1. n
= 6 Step 2.
m
= 4 Step 3.
P(A)
=
2/3 Teacher: Let’s solve problems using an algorithm. Problem 1: There are 25 tickets in the exam, Sergey did not learn 3 of them. Find the probability that he will come across the learned ticket. (Answer: 0.88) Problem 2: On a plate there are 5 pies with meat, 4 with cabbage and 3 with cherries. Natasha chooses one pie at random. Find the probability that he ends up with a cherry. (Answer: 0.25) 2. Pair work: solve by options with commenting out loud. Self-test according to the example. Teacher: Option 1 decides, draws up and tells option 2. Problem: The taxi company currently has 9 black, 4 yellow and 7 green cars available. One of the cars, which happened to be closest to the customer, responded to the call. Find the probability that a yellow taxi will come to him. (Answer: 0.2) Teacher: Option 2 solves, draws up and tells option 1 Task: On average, out of 100 flashlights that go on sale, eight are faulty. Find the probability that a flashlight chosen at random in a store will turn out to be working. (Answer: 0.92) V. Reflection 1) What new did you learn in class today? 2) Which stages were the most interesting? 3) What do you remember about today’s lesson? 4) What caused the difficulty? Teacher: Please note that you have a sticker on your notebook. I will ask you to rate your interest in this topic. Are you ready to study it only at the compulsory level? Analyze, maybe you are ready to solve problems in probability theory at an advanced level in order to successfully cope with them at the Unified State Exam, at the Olympiad? Or would you like to engage in research work on these issues (probability theory and statistics are promising areas in different fields of science)? Before leaving the office, attach your sticker to the axis of mathematical satisfaction for the selected period. This will help you understand what your needs are in this topic and select the appropriate material. VI. Homework 1. Problems in a notebook with a printed base.
2. Additional task: FIPI website (Bank of tasks for preparing for the OGE, page with tasks on probability theory) Conclusion: Strange as it may seem, in Ancient Greece music was part of the triad of sciences, not art.
“Music is the mysterious arithmetic of the soul,” said Leibniz. Interesting, associative scientific concepts and theories are reflected in songs and creativity. I invite you to listen to what associations arose for the author of the song “According to the Theory of Probability” performed by Vadim Mulerman. Download Technological map of the lesson “Beginnings of Probability Theory”
Presentation on the topic: The beginnings of probability theory
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