Theme of the lesson: "Random, reliable and impossible events"
Place of the lesson in the curriculum: "Combinatorics. Random events” lesson 5/8
Lesson type: Lesson in the formation of new knowledge
Lesson Objectives:
Educational:
o introduce a definition of a random, certain and impossible event;
o teach in the process of a real situation to define the terms of probability theory: reliable, impossible, equiprobable events;
Developing:
o promote the development of logical thinking,
o cognitive interest of students,
o ability to compare and analyze,
Educational:
o fostering interest in the study of mathematics,
o development of the worldview of students.
o possession of intellectual skills and mental operations;
Teaching methods: explanatory-illustrative, reproductive, mathematical dictation.
UMC: Mathematics: textbook for 6 cells. under the editorship, etc., publishing house "Enlightenment", 2008, Mathematics, 5-6: book. for teacher / [, [ , ]. - M.: Education, 2006.
Didactic material: board posters.
Literature:
1. Mathematics: textbook. for 6 cells. general education institutions/, etc.]; ed. , ; Ros. acad. Sciences, Ros. acad. education, publishing house "Enlightenment". - 10th ed. - M.: Enlightenment, 2008.-302 p.: ill. - (Academic school textbook).
2. Mathematics, 5-b: book. for the teacher / [, ]. - M. : Education, 2006. - 191 p. : ill.
4. Solving problems in statistics, combinatorics and probability theory. 7-9 grades. / auth.- comp. . Ed. 2nd, rev. - Volgograd: Teacher, 2006. -428 p.
5. Mathematics lessons using information technology. 5-10 grades. Methodical - a manual with an electronic application / and others. 2nd ed., stereotype. - M.: Globus Publishing House, 2010. - 266 p. (Modern school).
6. Teaching mathematics in a modern school. Guidelines. Vladivostok: PIPPCRO Publishing House, 2003.
LESSON PLAN
I. Organizational moment.
II. oral work.
III. Learning new material.
IV. Formation of skills and abilities.
V. The results of the lesson.
V. Homework.
DURING THE CLASSES
1. Organizing moment
2. Updating knowledge
15*(-100) |
Oral work:
3. Explanation of new material
Teacher: Our life is largely made up of accidents. There is such a science "Probability Theory". Using its language, it is possible to describe many phenomena and situations.
Such ancient commanders as Alexander the Great or Dmitry Donskoy, preparing for battle, relied not only on the valor and skill of warriors, but also on chance.
Many people love mathematics for the eternal truths twice two is always four, the sum of even numbers is even, the area of a rectangle is equal to the product of its adjacent sides, etc. In any problems that you solve, everyone gets the same answer - you just need to make no mistakes in the decision.
Real life is not so simple and unambiguous. The outcomes of many events cannot be predicted in advance. It is impossible, for example, to say for sure which side a coin tossed will fall, when the first snow will fall next year, or how many people in the city will want to make a phone call within the next hour. Such unpredictable events are called random .
However, the case also has its own laws, which begin to manifest themselves with repeated repetition of random phenomena. If you toss a coin 1000 times, then the "eagle" will fall out about half the time, which cannot be said about two or even ten tosses. "Approximately" does not mean half. This, as a rule, may or may not be the case. The law generally does not state anything for sure, but gives a certain degree of certainty that some random event will occur.
Such regularities are studied by a special branch of mathematics - Probability theory . With its help, you can predict with a greater degree of confidence (but still not sure) both the date of the first snowfall and the number of phone calls.
Probability theory is inextricably linked with our daily life. This gives us a wonderful opportunity to establish many probabilistic laws empirically, repeatedly repeating random experiments. The materials for these experiments will most often be an ordinary coin, a dice, a set of dominoes, backgammon, roulette, or even a deck of cards. Each of these items, one way or another, is connected with games. The fact is that the case here appears in the most frequent form. And the first probabilistic tasks were associated with assessing the chances of players to win.
Modern probability theory has moved away from gambling, but their props are still the simplest and most reliable source of chance. By practicing with a roulette wheel and a die, you will learn how to calculate the probability of random events in real life situations, which will allow you to assess your chances of success, test hypotheses, and make optimal decisions not only in games and lotteries.
When solving probabilistic problems, be very careful, try to justify each step, because no other area of mathematics contains such a number of paradoxes. Like probability theory. And, perhaps, the main explanation for this is its connection with the real world in which we live.
In many games, a die is used, which has a different number of points from 1 to 6 on each side. The player rolls the die, looks at how many points have fallen (on the side that is located on top), and makes the appropriate number of moves: 1,2,3 ,4,5, or 6. Throwing a die can be considered an experience, an experiment, a test, and the result obtained can be considered an event. People are usually very interested in guessing the onset of an event, predicting its outcome. What predictions can they make when a dice is rolled?
First prediction: one of the numbers 1,2,3,4,5, or 6 will fall out. Do you think the predicted event will come or not? Of course it will definitely come.
An event that is sure to occur in a given experience is called reliable event.
Second prediction : the number 7 will fall out. Do you think the predicted event will come or not? Of course it won't, it's just impossible.
An event that cannot occur in a given experiment is called impossible event.
Third Prediction : the number 1 will fall out. Do you think the predicted event will come or not? We are not able to answer this question with complete certainty, since the predicted event may or may not occur.
Events that may or may not occur under the same conditions are called random.
Example. The box contains 5 chocolates in a blue wrapper and one in white. Without looking into the box, they randomly take out one candy. Is it possible to tell in advance what color it will be?
Exercise : describe the events that are discussed in the tasks below. As certain, impossible or random.
1. Flip a coin. The coat of arms appeared. (random)
2. The hunter shot at the wolf and hit. (random)
3. A schoolboy goes for a walk every evening. During a walk, on Monday, he met three acquaintances. (random)
4. Let's mentally carry out the following experiment: turn a glass of water upside down. If this experiment is carried out not in space, but at home or in a classroom, then water will pour out. (authentic)
5. Three shots fired at the target.” There have been five hits." (impossible)
6. Throw the stone up. The stone remains suspended in the air. (impossible)
Example Petya thought of a natural number. The event is as follows:
a) an even number is conceived; (random)
b) an odd number is conceived; (random)
c) a number is conceived that is neither even nor odd; (impossible)
d) a number that is even or odd is conceived. (authentic)
Events that under given conditions have equal chances are called equiprobable.
Random events that have equal chances are called equally possible or equiprobable .
Put the poster on the board.
At the oral exam, the student takes one of the tickets laid out in front of him. The chances of taking any of the exam tickets are equal. Equally probable is the loss of any number of points from 1 to 6 when throwing a dice, as well as heads or tails when throwing a coin.
But not all events are equally possible. The alarm clock may not ring, the light bulb burns out, the bus breaks down, but under normal conditions, such events unlikely. It is more likely that the alarm clock will ring, the light will turn on, the bus will go.
Some events chances occur more, which means they are more likely - closer to reliable. And others have less chances, they are less likely - closer to impossible.
Impossible events have no chance of happening, and certain events have every chance of happening, under certain conditions they will definitely happen.
Example Petya and Kolya compare their birthdays. The event is as follows:
a) their birthdays do not match; (random)
b) their birthdays are the same; (random)
d) both birthdays fall on holidays - New Year (January 1) and Independence Day of Russia (June 12). (random)
3. Formation of skills and abilities
Task from textbook No. 000. Which of the following random events are reliable, possible:
a) the turtle will learn to speak;
b) the water in the kettle on the stove boils;
d) you win by participating in the lottery;
e) you will not win by participating in a win-win lottery;
f) you will lose a game of chess;
g) you will meet an alien tomorrow;
h) the weather will deteriorate next week; i) you pressed the bell, but it did not ring; j) today - Thursday;
k) after Thursday there will be Friday; m) will there be Thursday after Friday?
The boxes contain 2 red, 1 yellow and 4 green balls. Three balls are drawn at random from the box. Which of the following events are impossible, random, certain:
A: Three green balls will be drawn;
B: Three red balls will be drawn;
C: balls of two colors will be drawn;
D: balls of the same color will be drawn;
E: among the drawn balls there is a blue one;
F: among the drawn ones there are balls of three colors;
G: Are there two yellow balls among the drawn balls?
Check yourself. (math dictation)
1) Indicate which of the following events are impossible, which are certain, which are random:
Football match "Spartak" - "Dynamo" will end in a draw (random)
You will win by participating in the win-win lottery ( authentic)
At midnight it will snow, and after 24 hours the sun will shine (impossible)
· There will be a math test tomorrow. (random)
· You will be elected President of the United States. (impossible)
· You will be elected president of Russia. (random)
2) You bought a TV in a store, for which the manufacturer gives a two-year warranty. Which of the following events are impossible, which are random, which are certain:
· The TV will not break within a year. (random)
The TV will not break within two years . (random)
· Within two years you will not have to pay for TV repair. (authentic)
The TV will break in the third year. (random)
3) A bus carrying 15 passengers has 10 stops to make. Which of the following events are impossible, which are random, which are certain:
· All passengers will get off the bus at different stops. (impossible)
All passengers will get off at the same stop. (random)
At every stop, at least someone will get off. (random)
There will be a stop at which no one will get off. (random)
An even number of passengers will get off at all stops. (impossible)
An odd number of passengers will get off at all stops. (impossible)
Lesson summary
Questions for students:
What events are called random?
What events are called equiprobable?
What events are considered reliable? impossible?
Which events are considered more likely? less likely?
Homework : clause 9.3
No. 000. Think of three examples each of certain, impossible events, as well as events that cannot be said to necessarily occur.
902. There are 10 red, 1 green and 2 blue pens in a box. Two pens are randomly taken out of the box. Which of the following events are impossible, certain:
A: Two red handles will be taken out; B: Two green handles will be pulled out; C: two blue handles will be pulled out; D: Two handles of different colors will be taken out;
E: Will two pencils be taken out? 03. Egor and Danila agreed: if the turntable arrow (Fig. 205) stops on a white field, then Egor will paint the fence, and if on a blue field, Danila. Which boy is more likely to paint the fence?
The events (phenomena) observed by us can be divided into the following three types: reliable, impossible and random.
credible call an event that will definitely occur if a certain set of conditions S is implemented. For example, if a vessel contains water at normal atmospheric pressure and a temperature of 20 °, then the event “the water in the vessel is in a liquid state” is certain. In this example, the specified atmospheric pressure and water temperature constitute the set of conditions S.
Impossible call an event that certainly will not occur if the set of conditions S is implemented. For example, the event “water in the vessel is in a solid state” will certainly not occur if the set of conditions of the previous example is implemented.
Random An event is called an event that, under the implementation of a set of conditions S, can either occur or not occur. For example, if a coin is thrown, then it can fall so that either a coat of arms or an inscription is on top. Therefore, the event “when tossing a coin, a “coat of arms” fell out is random. Each random event, in particular the fall of the “coat of arms”, is the result of the action of very many random causes (in our example: the force with which the coin is thrown, the shape of the coin, and many others). It is impossible to take into account the influence of all these causes on the result, since their number is very large and the laws of their action are unknown. Therefore, the theory of probability does not set itself the task of predicting whether a single event will occur or not - it simply cannot do it.
The situation is different if we consider random events that can be repeatedly observed under the same conditions S, i.e., if we are talking about massive homogeneous random events. It turns out that a sufficiently large number of homogeneous random events, regardless of their specific nature, obey certain laws, namely, probabilistic laws. It is the theory of probability that deals with the establishment of these regularities.
Thus, the subject of probability theory is the study of probabilistic regularities of massive homogeneous random events.
Methods of probability theory are widely used in various branches of natural science and technology. The theory of probability also serves to substantiate mathematical and applied statistics.
Types of random events. Events are called incompatible if the occurrence of one of them excludes the occurrence of other events in the same trial.
Example. A coin is thrown. The appearance of the "coat of arms" excludes the appearance of the inscription. The events “a coat of arms appeared” and “an inscription appeared” are incompatible.
Several events form full group, if at least one of them appears as a result of the test. In particular, if the events that form a complete group are pairwise incompatible, then one and only one of these events will appear as a result of the test. This particular case is of greatest interest to us, since it will be used below.
Example 2. Two tickets for the cash and clothing lottery were purchased. One and only one of the following events will necessarily occur: “the winnings fell on the first ticket and did not fall on the second”, “the winnings did not fall on the first ticket and fell on the second”, “the winnings fell on both tickets”, “the winnings did not win on both tickets”. fell out." These events form a complete group of pairwise incompatible events.
Example 3. The shooter fired at the target. One of the following two events is bound to happen: hit, miss. These two disjoint events form a complete group.
Events are called equally possible if there is reason to believe that neither is more possible than the other.
Example 4. The appearance of a "coat of arms" and the appearance of an inscription when a coin is tossed are equally possible events. Indeed, it is assumed that the coin is made of a homogeneous material, has a regular cylindrical shape, and the presence of a coinage does not affect the loss of one or another side of the coin.
Self-designation in capital letters of the Latin alphabet: A, B, C, .. A 1, A 2 ..
Opposites are called 2 uniquely possible so-I, forming a complete group. If one of the two opposite events is denoted by A, then other designations are A`.
Example 5. Hit and miss when firing at a target - opposite sex. own.
An event is the result of a test. What is an event? One ball is drawn at random from the urn. Removing a ball from an urn is a test. The appearance of a ball of a certain color is an event. In probability theory, an event is understood as something about which, after a certain moment of time, one and only one of the two can be said. Yes, it happened. No, it didn't happen. The possible outcome of an experiment is called an elementary event, and the set of such outcomes is simply called an event.
Unpredictable events are called random. An event is called random if, under the same conditions, it may or may not occur. Rolling a die will result in a six. I have a lottery ticket. After the publication of the results of the lottery draw, the event that interests me - winning a thousand rubles, either occurs or does not occur. Example.
Two events that, under given conditions, can occur simultaneously are called joint, and those that cannot occur simultaneously are called incompatible. A coin is thrown. The appearance of the "coat of arms" excludes the appearance of the inscription. The events “a coat of arms appeared” and “an inscription appeared” are incompatible. Example.
An event that always happens is called certain. An event that cannot happen is called impossible. Suppose, for example, a ball is drawn from an urn containing only black balls. Then the appearance of a black ball is a certain event; the appearance of a white ball is an impossible event. Examples. It won't snow next year. When you roll a die, a seven will come up. These are impossible events. Snow will fall next year. Rolling the die will result in a number less than seven. Daily sunrise. These are real events.
Problem Solving For each of the described events, determine what it is: impossible, certain, or random. 1. Of the 25 students in the class, two celebrate their birthday a) January 30; b) February 30th. 2. A literature textbook is randomly opened and the second word is found on the left page. This word begins: a) with the letter "K"; b) with the letter "b".
3. Today in Sochi the barometer shows normal atmospheric pressure. In this case: a) the water in the pan boiled at a temperature of 80º C; b) when the temperature dropped to -5º C, the water in the puddle froze. 4. Throw two dice: a) 3 points on the first dice, and 5 points on the second; b) the sum of the points on the two dice is equal to 1; c) the sum of the points rolled on the two dice is 13; d) 3 points on both dice; e) the sum of points on two dice is less than 15. Problem solving
5. You opened the book to any page and read the first noun you came across. It turned out that: a) there is a vowel in the spelling of the chosen word; b) in the spelling of the selected word there is a letter "O"; c) there are no vowels in the spelling of the chosen word; d) there is a soft sign in the spelling of the selected word. Problem solving
Probability theory, like any branch of mathematics, operates with a certain range of concepts. Most of the concepts of probability theory are defined, but some are taken as primary, not defined, as in geometry a point, a line, a plane. The primary concept of probability theory is an event. An event is something about which, after a certain point in time, one and only one of the two can be said:
- · Yes, it happened.
- · No, it didn't happen.
For example, I have a lottery ticket. After the publication of the results of the lottery draw, the event that interests me - winning a thousand rubles either occurs or does not occur. Any event occurs as a result of a test (or experience). Under the test (or experience) understand those conditions as a result of which an event occurs. For example, tossing a coin is a test, and the appearance of a “coat of arms” on it is an event. The event is usually denoted by capital Latin letters: A, B, C, .... Events in the material world can be divided into three categories - certain, impossible and random.
A certain event is one that is known in advance to occur. It is denoted by the letter W. Thus, no more than six points are reliable when throwing an ordinary dice, the appearance of a white ball when drawn from an urn containing only white balls, etc.
An impossible event is an event that is known in advance that it will not happen. It is denoted by the letter E. Examples of impossible events are drawing more than four aces from an ordinary deck of cards, the appearance of a red ball from an urn containing only white and black balls, etc.
A random event is an event that may or may not occur as a result of a test. Events A and B are called incompatible if the occurrence of one of them excludes the possibility of the occurrence of the other. So the appearance of any possible number of points when throwing a dice (event A) is inconsistent with the appearance of another number (event B). Rolling an even number of points is incompatible with rolling an odd number. Conversely, an even number of points (event A) and a number of points divisible by three (event B) will not be incompatible, because the loss of six points means the occurrence of both events A and event B, so the occurrence of one of them does not exclude the occurrence of the other. Operations can be performed on events. The union of two events C=AUB is an event C that occurs if and only if at least one of these events A and B occurs. The intersection of two events D=A?? B is an event that occurs if and only if both events A and B occur.
Translate the text into German please.Just not in the online translator.
The Golden Gate is a symbol of Kyiv, one of the oldest examples of architecture that has survived to our time. The golden gates of Kyiv were built under the famous Kiev prince Yaroslav the Wise in 1164. Initially, they were called Southern and were part of the system of defensive fortifications of the city, practically no different from other guard gates of the city. It was the Southern Gates that the first Russian Metropolitan Hilarion called "Great" in his "Sermon on Law and Grace". After the majestic Hagia Sophia was built, the “Great” gates became the main land entrance to Kyiv from the southwestern side. Realizing their significance, Yaroslav the Wise ordered to build a small Church of the Annunciation over the gates in order to pay tribute to the Christian religion that dominated the city and Russia. From that time on, all Russian chronicle sources began to call the South Gates of Kyiv the Golden Gates. The width of the gate was 7.5 m, the passage height was 12 m, and the length was about 25 m.
le sport ce n "est pas seulement des cours de gym. C" est aussi sauter toujours plus haut nager jouer au ballon danser. le sport développé ton corps et aussi ton cerveau. Quand tu prends l "escalier et non pas l" ascenseur tu fais du sport. Quand tu fais une cabane dans un arbre tu fais du sport. Quand tu te bats avec ton frere tu fais du sport. Quand tu cours, parce que tu es en retard a l "ecole, tu fais du sport.
