An event is the result of a test. What is an event? One ball is taken at random from the urn. Retrieving 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 point in time, one and only one of two things can be said. Yes, it happened. No, it didn't happen. A possible outcome of an experiment is called an elementary event, and a 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. When rolling the dice, the result will be a six. I have a lottery ticket. After the results of the lottery are published, the event that interests me - winning a thousand rubles - either happens or does not happen. 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 tossed. 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 occurs is called reliable. An event that cannot happen is called impossible. For example, suppose a ball is drawn from an urn containing only black balls. Then the appearance of the black ball is a reliable event; the appearance of a white ball is an impossible event. Examples. There will be no snow next year. When rolling the dice, the result will be a seven. These are impossible events. There will be snow next year. When you roll the dice, you will get a number less than seven. Daily sunrise. These are reliable events.
Problem solving For each of the described events, determine what it is: impossible, reliable or random. 1. Of the 25 students in the class, two celebrate their birthday on a) January 30; b) February 30. 2. The literature textbook randomly opens and the second word is found on the left page. This word begins: a) with the letter “K”; b) starting with the letter “Ъ”.
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. Two dice are thrown: a) the first dice shows 3 points, and the second - 5 points; b) the sum of the points rolled on the two dice is 1; c) the sum of the points rolled on the two dice is 13; d) both dice got 3 points; 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) the spelling of the selected word contains a vowel; b) the spelling of the selected word contains the letter “O”; c) there are no vowels in the spelling of the selected word; d) there is a soft sign in the spelling of the selected word. Problem solving
Just not in an online translator.
The Golden Gate is a symbol of Kyiv, one of the oldest examples of architecture that has survived to this day. The Golden Gate of Kyiv was 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 South Gate that the first Russian Metropolitan Hilarion called “Great” in his “Sermon on Law and Grace.” After the majestic Church of Hagia Sophia was built, the “Great” Gate became the main land entrance to Kyiv from the southwestern side. Realizing their significance, Yaroslav the Wise ordered the construction of a small Church of the Annunciation over the gates in order to pay tribute to the dominant Christian religion in the city and in Rus'. From that time on, all Russian chronicle sources began to call the Southern Gate of Kyiv the Golden Gate. The width of the gate was 7.5 m, the height of the passage 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.
The purpose of the lesson:
- Introduce the concept of reliable, impossible and random events.
- Develop knowledge and skills to determine the type of events.
- Develop: computing skill; attention; ability to analyze, reason, draw conclusions; group work skills.
During the classes
1) Organizational moment.
Interactive exercise: children must solve examples and decipher words; based on the results, they are divided into groups (reliable, impossible and random) and determine the topic of the lesson.
1 card.
| 0,5 | 1,6 | 12,6 | 5,2 | 7,5 | 8 | 5,2 | 2,08 | 0,5 | 9,54 | 1,6 |
2 card
| 0,5 | 2,1 | 14,5 | 1,9 | 2,1 | 20,4 | 14 | 1,6 | 5,08 | 8,94 | 14 |
3 card
| 5 | 2,4 | 6,7 | 4,7 | 8,1 | 18 | 40 | 9,54 | 0,78 |
2) Updating the learned knowledge.
Game “Clap”: even number - clap, odd number - stand up.
Task: from the given series of numbers 42, 35, 8, 9, 7, 10, 543, 88, 56, 13, 31, 77, ... determine even and odd.
3) Studying a new topic.
There are cubes on your tables. Let's take a closer look at them. What do you see?
Where are dice used? How?
Work in groups.
Conducting an experiment.
What predictions can you make when throwing a die?
First prediction: one of the numbers 1,2,3,4,5 or 6 will appear.
An event that is sure to occur in a given experience is called reliable.
Second prediction: the number 7 will appear.
Do you think the predicted event will happen or not?
This is impossible!
An event that cannot occur in a given experience is called impossible.
Third prediction: the number 1 will appear.
Will this event happen?
An event that may or may not occur in a given experience is called random.
4) Consolidation of the studied material.
I. Determine the type of event
-Tomorrow it will snow red.
It will snow heavily tomorrow.
Tomorrow, even though it is July, it will snow.
Tomorrow, even though it is July, there will be no snow.
Tomorrow it will snow and there will be a blizzard.
II. Add a word to this sentence in such a way that the event becomes impossible.
Kolya received an A in history.
Sasha did not complete a single task on the test.
Oksana Mikhailovna (history teacher) will explain a new topic.
III. Give examples of impossible, random and reliable events.
IV. Work from the textbook (in groups).
Describe the events discussed in the tasks below as reliable, impossible or random.
No. 959. Petya came up with a natural number. The event is as follows:
a) an even number is intended;
b) an odd number is intended;
c) a number is conceived that is neither even nor odd;
d) a number is conceived that is even or odd.
No. 960. You opened this textbook to any page and chose the first noun that came up. The event is as follows:
a) there is a vowel in the spelling of the selected word;
b) the spelling of the selected word contains the letter “o”;
c) there are no vowels in the spelling of the selected word;
d) there is a soft sign in the spelling of the selected word.
Solve No. 961, No. 964.
Discussion of solved tasks.
5) Reflection.
1. What events did you learn about in the lesson?
2. Indicate which of the following events is certain, which is impossible and which is random:
a) there will be no summer holidays;
b) the sandwich will fall butter side down;
c) the school year will end someday.
6) Homework:
Come up with two reliable, random and impossible events.
Make a drawing for one of them.
The events (phenomena) we observe can be divided into the following three types: reliable, impossible and random.
Reliable they call an event that will definitely occur if a certain set of conditions S is fulfilled. 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 reliable. In this example, the given atmospheric pressure and water temperature constitute the set of conditions S.
Impossible they call an event that certainly will not happen if the set of conditions S is fulfilled. For example, the event “water in the vessel is in a solid state” will certainly not happen if the set of conditions of the previous example is fulfilled.
Random call an event that, when a set of conditions S is fulfilled, can either occur or not occur. For example, if a coin is thrown, it may fall so that there is either a coat of arms or an inscription on top. Therefore, the event “when throwing a coin, the “coat of arms” fell out is random. Each random event, in particular the appearance of a “coat of arms,” is a consequence of the action of many random causes (in our example: the force with which the coin was thrown, the shape of the coin, and many others). It is impossible to take into account the influence of all these reasons on the result, since their number is very large and the laws of their action are unknown. Therefore, probability theory does not set itself the task of predicting whether a single event will occur or not - it simply cannot do this.
The situation is different if we consider random events that can be observed repeatedly when the same conditions S are met, 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, are subject to certain patterns, namely probabilistic patterns. The theory of probability is concerned with establishing these regularities.
Thus, the subject of probability theory is the study of probabilistic patterns of mass homogeneous random events.
Methods of probability theory are widely used in various branches of natural science and technology. Probability theory 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 tossed. 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 inconsistent, then one and only one of these events will appear as a result of the trial. This particular case is of greatest interest to us, since it will be used further.
Example 2. Two cash and clothing lottery tickets were purchased. One and only one of the following events will definitely happen: “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”, “there were no winnings 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 will definitely happen: hit, miss. These two incompatible events form a complete group.
Events are called equally possible, if there is reason to believe that neither of them is more possible than the other.
Example 4. The appearance of a “coat of arms” and the appearance of an inscription when throwing a coin 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 minting does not affect the loss of one side or another of the coin.
I am denoted by capital letters of the Latin alphabet: A, B, C,.. A 1, A 2..
Opposites are two uniquely possible mutine-types that form a complete group. If one of the two is opposite sex. events are designated by A, then another designation is A`.
Example 5. Hit and miss when shooting at a target - opposite field. personal
5th grade. Introduction to Probability (4 hours)
(development of 4 lessons on this topic)
Learning goals : - introduce the definition of a random, reliable and impossible event;
Provide first ideas about solving combinatorial problems: using a tree of options and using the multiplication rule.
Educational goal: development of students' worldview.
Developmental goal : development of spatial imagination, improvement of the skill of working with a ruler.
Reliable, impossible and random events (2 hours)
Combinatorial problems (2 hours)
Reliable, impossible and random events.
First lesson
Lesson equipment: dice, coin, backgammon.
Our life largely consists of accidents. There is such a science as “Probability Theory”. Using its language, you can describe many phenomena and situations.
Even the primitive leader understood that a dozen hunters had a greater “probability” of hitting a bison with a spear than one. That's why they hunted collectively back then.
Such ancient commanders as Alexander the Great or Dmitry Donskoy, preparing for battle, relied not only on the valor and art 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 problem that you solve, everyone gets the same answer - you just need to not make mistakes in the decision.
Real life is not so simple and straightforward. The outcome of many events cannot be predicted in advance. It is impossible, for example, to say for sure which side a coin thrown up 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, chance also has its own laws, which begin to manifest themselves when random phenomena are repeated many times. If you toss a coin 1000 times, it will come up heads approximately half the time, which is not the case with two or even ten tosses. "Approximately" does not mean half. This generally may or may not be the case. The law does not state anything for certain, but it does provide a certain degree of confidence that some random event will occur. Such patterns 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 for sure) both the date of the first snowfall and the number of phone calls.
Probability theory is inextricably linked with our everyday life. This gives us a wonderful opportunity to establish many probabilistic laws experimentally, repeating random experiments many times. 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 is related to games in one way or another. The fact is that the case appears here in its most frequent form. And the first probabilistic tasks were related to assessing the players’ chances of winning.
Modern probability theory has moved away from gambling, but its props still remain the simplest and most reliable source of chance. After practicing with a roulette and a dice, you will learn to calculate the probability of random events in real life situations, which will allow you to evaluate 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 every step you take, because no other area of mathematics contains so many paradoxes. Like probability theory. And perhaps the main explanation for this is its connection with the real world in which we live.
Many games use a die with a different number of dots from 1 to 6 marked on each side. The player throws the dice, looks at how many dots appear (on the side that is located on top), and makes the corresponding 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 occurrence of this or that event and predicting its outcome. What predictions can they make when they roll the dice? First prediction: one of the numbers 1,2,3,4,5, or 6 will appear. Do you think the predicted event will happen or not? Of course, it will definitely come. An event that is sure to occur in a given experience is called a reliable event.
Second prediction : the number 7 will appear. Do you think the predicted event will happen or not? Of course it won’t happen, it’s simply impossible. An event that cannot occur in a given experience is called impossible event.
Third prediction : the number 1 will appear. Do you think the predicted event happened or not? We are not able to answer this question with complete certainty, since the predicted event may or may not occur. An event that may or may not occur in a given experience is called a random event.
Exercise : Describe the events discussed in the tasks below. Like certain, impossible or random.
Let's toss a coin. A coat of arms appeared. (random)
The hunter shot at the wolf and hit it. (random)
The schoolboy goes for a walk every evening. While walking on Monday, he met three acquaintances. (random)
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 spill out. (reliable)
Three shots were fired at the target.” There were five hits" (impossible)
Throw the stone up. The stone remains hanging in the air. (impossible)
We rearrange the letters of the word “antagonism” at random. The result is the word “anachroism.” (impossible)
№959. Petya thought of a natural number. The event is as follows:
a) an even number is intended; (random) b) an odd number is intended; (random)
c) a number is conceived that is neither even nor odd; (impossible)
d) a number is conceived that is even or odd. (reliable)
№ 961. Petya and Tolya compare their birthdays. The event is as follows:
a) their birthdays do not coincide; (random) b) their birthdays are the same; (random)
d) both of their birthdays fall on holidays - New Year (January 1) and Russian Independence Day (June 12). (random)
№ 962. When playing backgammon, two dice are used. The number of moves that a participant in the game makes is determined by adding the numbers on the two sides of the cube that fall out, and if a “double” is rolled (1 + 1.2 + 2.3 + 3.4 + 4.5 + 5.6 + 6), then the number of moves doubles. You roll the dice and figure out how many moves you have to make. The event is as follows:
a) you must make one move; b) you must make 7 moves;
c) you must make 24 moves; d) you must make 13 moves.
a) – impossible (1 move can be made if the combination 1 + 0 is rolled, but there is no number 0 on the dice).
b) – random (if 1 + 6 or 2 + 5 is rolled).
c) – random (if the combination 6 +6 appears).
d) – impossible (there are no combinations of numbers from 1 to 6, the sum of which is 13; this number cannot be obtained even when a “double” is rolled, since it is odd).
Check yourself. (mathematical dictation)
1) Indicate which of the following events are impossible, which are reliable, which are random:
The football match "Spartak" - "Dynamo" will end in a draw. (random)
You will win by participating in a win-win lottery (reliable)
Snow will fall at midnight and the sun will shine 24 hours later. (impossible)
Tomorrow there will be a math test. (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 provides a two-year warranty. Which of the following events are impossible, which are random, which are reliable:
The TV will not break for a year. (random)
The TV will not break for two years. (random)
You won't have to pay for TV repairs for two years. (reliable)
The TV will break in the third year. (random)
3) A bus carrying 15 passengers has to make 10 stops. Which of the following events are impossible, which are random, which are reliable:
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 where no one gets 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)
Homework : p. 53 No. 960, 963, 965 (come up with two reliable, random and impossible events yourself).
Second lesson.
Checking homework. (orally)
a) Explain what certain, random and impossible events are.
b) Indicate which of the following events is reliable, which is impossible, which is random:
There will be no summer holidays. (impossible)
The sandwich will fall butter side down. (random)
The school year will end someday. (reliable)
They'll ask me in class tomorrow. (random)
Today I will meet a black cat. (random)
№ 960. You opened this textbook to any page and chose the first noun that came up. The event is as follows:
a) there is a vowel in the spelling of the selected word. ((reliable)
b) the spelling of the chosen word contains the letter “o”. (random)
c) there are no vowels in the spelling of the selected word. (impossible)
d) there is a soft sign in the spelling of the selected word. (random)
№ 963. You are playing backgammon again. Describe the following event:
a) the player must make no more than two moves. (impossible - with a combination of the smallest numbers 1 + 1 the player makes 4 moves; a combination of 1 + 2 gives 3 moves; all other combinations give more than 3 moves)
b) the player must make more than two moves. (reliable - any combination gives 3 or more moves)
c) the player must make no more than 24 moves. (reliable - the combination of the largest numbers 6 + 6 gives 24 moves, and all others give less than 24 moves)
d) the player must make a double-digit number of moves. (random – for example, the combination 2 + 3 gives a single-digit number of moves: 5, and rolling two fours gives a double-digit number of moves)
2. Problem solving.
№ 964. There are 10 balls in a bag: 3 blue, 3 white and 4 red. Describe the following event:
a) 4 balls were taken from the bag, and they are all blue; (impossible)
b) 4 balls were taken from the bag, and they are all red; (random)
c) 4 balls were taken out of the bag, and they all turned out to be different colors; (impossible)
d) 4 balls were taken out of the bag, and among them there was no black ball. (reliable)
Task 1. The box contains 10 red, 1 green and 2 blue pens. Two objects are drawn at random from the box. Which of the following events are impossible, which are random, which are certain:
a) two red pens are taken out (random)
b) two green handles are taken out; (impossible)
c) two blue pens are taken out; (random)
d) handles of two different colors are taken out; (random)
e) two handles are removed; (reliable)
f) two pencils are taken out. (impossible)
Task 2. Winnie the Pooh, Piglet and everyone - everyone - everyone sits down at the round table to celebrate his birthday. At what number of all - all - all is the event “Winnie the Pooh and Piglet sitting next to each other” reliable, and at what number is it random?
(if there are only 1 of all - all - all of them, then the event is reliable, if there is more than 1, then it is random).
Task 3. Among 100 charity lottery tickets, 20 are winning ones. How many tickets do you need to buy to make the “you won’t win anything” event impossible?
Task 4. There are 10 boys and 20 girls in the class. Which of the following events are impossible for this class, which are random, which are reliable
There are two people in the class who were born in different months. (random)
There are two people in the class who were born in the same month. (reliable)
There are two boys in the class who were born in the same month. (random)
There are two girls in the class who were born in the same month. (reliable)
All boys were born in different months. (reliable)
All girls were born in different months. (random)
There is a boy and a girl born in the same month. (random)
There is a boy and a girl born in different months. (random)
Task 5. There are 3 red, 3 yellow, 3 green balls in the box. We pull out 4 balls at random. Consider the event “Among the drawn balls there will be balls of exactly M colors.” For each M from 1 to 4, determine what kind of event it is - impossible, reliable or random, and fill out the table:
Independent work.
Ioption
a) your friend’s birthday number is less than 32;
c) tomorrow there will be a test in mathematics;
d) Next year the first snow in Moscow will fall on Sunday.
Throwing a dice. Describe the event:
a) the cube, having fallen, will stand on its edge;
b) one of the numbers will appear: 1, 2, 3, 4, 5, 6;
c) the number 6 will appear;
d) a number that is a multiple of 7 will be rolled.
A box contains 3 red, 3 yellow and 3 green balls. Describe the event:
a) all the drawn balls are of the same color;
b) all the drawn balls are of different colors;
c) among the drawn balls there are balls of different colors;
c) among the drawn balls there is a red, yellow and green ball.
IIoption
Describe the event in question as reliable, impossible or accidental:
a) a sandwich that falls off the table will fall face down on the floor;
b) snow will fall in Moscow at midnight, and after 24 hours the sun will shine;
c) you will win by participating in a win-win lottery;
d) next year in May the first thunder of spring will be heard.
All two-digit numbers are written on the cards. One card is chosen at random. Describe the event:
a) there was a zero on the card;
b) there was a number on the card that was a multiple of 5;
c) there was a number on the card that was a multiple of 100;
d) there was a number on the card greater than 9 and less than 100.
The box contains 10 red, 1 green and 2 blue pens. Two objects are drawn at random from the box. Describe the event:
a) two blue pens are taken out;
b) two red pens are taken out;
c) two green handles are taken out;
d) the green and black handles are taken out.
Homework: 1). Come up with two reliable, random and impossible events.
2). Task . There are 3 red, 3 yellow, 3 green balls in the box. We draw N balls at random. Consider the event “among the drawn balls there will be balls of exactly three colors.” For each N from 1 to 9, determine what kind of event it is - impossible, reliable or random, and fill out the table:
Combinatorial problems.
First lesson
Checking homework. (orally)
a) we check the problems that the students came up with.
b) an additional task.
I’m reading an excerpt from V. Levshin’s book “Three Days in Karlikania.”
“At first, to the sounds of a smooth waltz, the numbers formed a group: 1 + 3 + 4 + 2 = 10. Then the young skaters began to change places, forming more and more new groups: 2 + 3 + 4 + 1 = 10
3 + 1 + 2 + 4 = 10
4 + 1 + 3 + 2 = 10
1 + 4 + 2 + 3 = 10, etc.
This continued until the skaters returned to their starting position.”
How many times did they change places?
Today in class we will learn how to solve such problems. They're called combinatorial.
3. Studying new material.
Task 1. How many two-digit numbers can be made from the numbers 1, 2, 3?
Solution: 11, 12, 13
31, 32, 33. 9 numbers in total.
When solving this problem, we searched through all possible options, or, as they usually say in these cases. All possible combinations. Therefore, such problems are called combinatorial. You have to calculate possible (or impossible) options in life quite often, so it’s useful to get acquainted with combinatorial problems.
№ 967. Several countries have decided to use symbols for their national flag in the form of three horizontal stripes of the same width in different colors - white, blue, red. How many countries can use such symbols, provided that each country has its own flag?
Solution. Let's assume that the first stripe is white. Then the second stripe can be blue or red, and the third stripe, respectively, red or blue. We got two options: white, blue, red or white, red, blue.
Let now the first stripe be blue, then again we get two options: white, red, blue or blue, red, white.
Let the first stripe be red, then there are two more options: red, white, blue or red, blue, white.
There were 6 possible options in total. This flag can be used by 6 countries.
So, when solving this problem, we were looking for a way to enumerate possible options. In many cases, it turns out to be useful to construct a picture - a diagram of enumerating options. This, firstly, is clear, and secondly, it allows us to take everything into account and not miss anything.
This diagram is also called a tree of possible options.
Front page
Second stripe
Third lane
The resulting combination
№ 968. How many two-digit numbers can be made from the numbers 1, 2, 4, 6, 8?
Solution. For the two-digit numbers we are interested in, the first place can be any of the given digits, except 0. If we put the number 2 in the first place, then any of the given digits can be in the second place. You will get five two-digit numbers: 2.,22, 24, 26, 28. Likewise, there will be five two-digit numbers with the first digit 4, five two-digit numbers with the first digit 6 and five two-digit numbers with the first digit 8.
Answer: There will be 20 numbers in total.
Let's build a tree of possible options to solve this problem.
Double figures
First digit
Second digit
Received numbers
20, 22, 24, 26, 28, 60, 62, 64, 66, 68,
40, 42, 44, 46, 48, 80, 82, 84, 86, 88.
Solve the following problems by constructing a tree of possible options.
№ 971. The leadership of a certain country decided to make its national flag look like this: on a single-color rectangular background, a circle of a different color is placed in one of the corners. It was decided to choose colors from three possible ones: red, yellow, green. How many variants of this flag?
exists? The figure shows some of the possible options.
Answer: 24 options.
№ 973. a) How many three-digit numbers can be made from the numbers 1,3, 5,? (27 numbers)
b) How many three-digit numbers can be made from the numbers 1,3, 5, provided that the numbers should not be repeated? (6 numbers)
№ 979. Modern pentathletes participate in competitions in five sports over two days: show jumping, fencing, swimming, shooting, and running.
a) How many options are there for the order of completing the types of competition? (120 options)
b) How many options are there for the order of the events of the competition, if it is known that the last event should be running? (24 options)
c) How many options are there for the order of the competition events if it is known that the last event should be running, and the first should be show jumping? (6 options)
№ 981. Two urns contain five balls each in five different colors: white, blue, red, yellow, green. One ball is drawn from each urn at a time.
a) how many different combinations of drawn balls are there (combinations like “white - red” and “red - white” are considered the same)?
(15 combinations)
b) How many combinations are there in which the drawn balls are of the same color?
(5 combinations)
c) how many combinations are there in which the drawn balls are of different colors?
(15 – 5 = 10 combinations)
Homework: p. 54, No. 969, 972, come up with a combinatorial problem yourself.
№ 969. Several countries have decided to use symbols for their national flag in the form of three vertical stripes of the same width in different colors: green, black, yellow. How many countries can use such symbols, provided that each country has its own flag?
№ 972. a) How many two-digit numbers can be made from the numbers 1, 3, 5, 7, 9?
b) How many two-digit numbers can be made from the numbers 1, 3, 5, 7, 9, provided that the numbers should not be repeated?
Second lesson
Checking homework. a) No. 969 and No. 972a) and No. 972b) - build a tree of possible options on the board.
b) we check the completed tasks orally.
Problem solving.
So, before this, we learned how to solve combinatorial problems using a tree of options. Is this a good way? Probably yes, but very cumbersome. Let's try to solve homework problem No. 972 differently. Who can guess how this can be done?
Answer: For each of the five colors of T-shirts there are 4 colors of panties. Total: 4 * 5 = 20 options.
№ 980. The urns contain five balls each in five different colors: white, blue, red, yellow, green. One ball is drawn from each urn at a time. Describe the following event as certain, random, or impossible:
a) taken out balls of different colors; (random)
b) taken out balls of the same color; (random)
c) black and white balls are drawn; (impossible)
d) two balls are drawn, both of which are colored one of the following colors: white, blue, red, yellow, green. (reliable)
№ 982. A group of tourists plans to hike along the route Antonovo - Borisovo - Vlasovo - Gribovo. From Antonovo to Borisovo you can raft on the river or walk. From Borisovo to Vlasovo you can walk or ride bicycles. From Vlasovo to Gribovo you can swim along the river, ride bicycles or walk. How many trekking options can tourists choose from? How many hiking options can tourists choose, provided that they must use bicycles on at least one part of the route?
(12 route options, 8 of them using bicycles)
Independent work.
1 option
a) How many three-digit numbers can be made from the digits: 0, 1, 3, 5, 7?
b) How many three-digit numbers can be made from the digits: 0, 1, 3, 5, 7, provided that the numbers should not be repeated?
Athos, Porthos and Aramis have only a sword, a dagger and a pistol.
a) In how many ways can musketeers be armed?
b) How many weapon options are there if Aramis must wield a sword?
c) How many weapon options are there if Aramis must wield the sword and Porthos must wield the pistol?
Somewhere God sent Raven a piece of cheese, as well as feta cheese, sausage, white and black bread. Having perched on a spruce tree, the crow was just about ready to have breakfast, but she began to think: in how many ways can sandwiches be made from these products?
Option 2
a) How many three-digit numbers can be made from the digits: 0, 2, 4, 6, 8?
b) How many three-digit numbers can be made from the digits: 0, 2, 4, 6, 8, provided that the digits should not be repeated?
Count Monte Cristo decided to give Princess Hayde earrings, a necklace and a bracelet. Each piece of jewelry must contain one of the following types of gemstones: diamonds, rubies or garnets.
a) How many options are there for combining precious stone jewelry?
b) How many jewelry options are there if the earrings should be diamond?
c) How many jewelry options are there if the earrings should be diamond and the bracelet should be garnet?
For breakfast you can choose a bun, sandwich or gingerbread with coffee or kefir. How many breakfast options can you create?
Homework : No. 974, 975. (by compiling a tree of options and using the multiplication rule)
№ 974 . a) How many three-digit numbers can be made from the numbers 0, 2, 4?
b) How many three-digit numbers can be made from the numbers 0, 2, 4, provided that the numbers should not be repeated?
№ 975 . a) How many three-digit numbers can be made from the numbers 1,3, 5,7?
b) How many three-digit numbers can be made from the numbers 1,3, 5,7 under the condition. What numbers should not be repeated?
Problem numbers taken from the textbook
"Mathematics-5", I.I. Zubareva, A.G. Mordkovich, 2004.
