Presentation on the topic of measuring information - a meaningful approach
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Uncertainty of Knowledge and Quantity of Information Another approach to measuring information is called the content approach. In this case, the amount of information is associated with the content (meaning) of the message received by a person. Let us remember that from a “human” point of view, information is the knowledge that we obtain from outside world. The amount of information contained in a message should be greater the more it adds to our knowledge. How is the unit of measurement of information determined from this point of view? You already know that this unit is called a bit. The problem of measuring information is studied in information theory, the founder of which is Claude Shannon. In information theory, a bit is defined as follows:
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SUBSTANTIVE APPROACH TO INFORMATION MEASUREMENT The message that one event out of two equally probable ones has occurred (the uncertainty of knowledge has been halved) carries 1 bit of information. 8 colored balls in a basket - 8 equally probable events The uncertainty of knowing that a red ball can be drawn from the basket is 8. A more strict definition of equiprobability: if you increase the number of coin tosses (100, 1000, 10000, etc.), then the number of heads and the number of tails will be increasingly closer to half the number of coin tosses. Therefore, we can say this: The uncertainty of knowledge about the result of some event (throwing a coin or a die, drawing lots, etc.) is the number of possible results.
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The bookcase has eight shelves. The book can be placed on any of them. How much information does the message contain about where the book is? We ask questions: - Is the book located above the fourth shelf? - No. - Is the book below the third shelf? - Yes. - Is the book on the second shelf? - No. - Well, now everything is clear! The book is on the first shelf! Each answer reduced the uncertainty by half. A total of three questions were asked. This means that 3 bits of information have been typed. And if it were immediately said that the book is on the first shelf, then the same 3 bits of information would be transmitted by this message.
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BINARY SEARCH METHOD You need to guess the intended number from the range of numbers from 1 to 8 8 options for possible events 3 questions 3 bits of information What grade did your friend get in the exam? Four equally probable events. 1 2 3 4 5 6 7 8 5 6 7 8 5 6 Game using the binary search method Game rules: You need to guess the intended number from a given range of integers. The player guessing the number asks questions that can only be answered with “yes” or “no.” If each answer cuts off half of the options (reduces the choice by 2 times), then it carries 1 bit of information. Then the total amount of information (in bits) obtained when guessing a number is equal to the amount questions asked. Question No. Questions yes no 1 Is the number less than 5? 2 Is the number less than 7? 3 Is this number equal to 5?
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Now let's try to get a formula that calculates the amount of information contained in a message that one of many equally probable results of some event took place. Let us denote by the letter N the number of possible results of an event, or, as we also called it, the uncertainty of knowledge. The letter i will denote the amount of information in a message about one of N results. In the coin example: N = 2, i = 1 bit. In the example with estimates: N = 4, i = 2 bits. In the example with a rack: N = 8, i = 3 bits. It is easy to see that the relationship between these quantities is expressed by the following formula: 2i = N. Indeed: 21 = 2; 22 = 4 ; 23 = 8.
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We are already familiar with the resulting formula from the basic computer science course, and we will encounter it more than once. The significance of this formula is so great that we called it the main formula of computer science. If the value N is known, a i is unknown, then this formula becomes the equation for determining i. In mathematics it is called an exponential equation. Let the rack have not 8, but 16 shelves. To answer the question of how much information is contained in the message about the location of the book, you need to solve the equation: 2i = N. Since 16 = 2, then i = 4 bits. The amount of information (i) contained in a message about one of N equally probable outcomes of some events is determined by solving the exponential equation: 2i = N. If the value of N is equal to an integer power of two (4, 8, 16, 32, 64, etc. ), then the exponential equation is easy to solve in your head, since i will be an integer. What, for example, is the amount of information in the message about the result of throwing a die, which has six sides and, therefore, N = 6? You can guess that the solution to the equation 2i = 6. will be fractional number, lying between 2 and 3, since 22 = 4< 6, а 2 = 8 >6. How can you find out this number more accurately?
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EXPONENTIAL EQUATION N i Determination of the amount of information contained in a message that one of N equally probable events has occurred N i Determination of the number of equally probable events N, if it is known how much information a person received in a message that one of these events has occurred. 2 i = N N i N i N i N i 1 0.00000 17 4.08746 33 5.04439 49 5.61471 2 1.00000 18 4.16993 34 5.08746 50 5.64386 3 1.58496 19 4.24793 35 5.12928 51 5.67243 4 2.00000 20 4.32193 36 5.16993 52 5.70044 5 2.32193 21 4.39232 37 5.20945 53 5.72792 6 2.58496 22 4.45943 38 5.24793 54 5.75489 7 2.80735 23 4.52356 39 5.28540 55 5.78136 8 3.00000 24 4.58496 40 5.32 193 56 5.80735 9 3.16993 25 4.64386 41 5.35755 57 5.83289 10 3.32193 26 4.70044 42 5.39232 58 5.85798 11 3.45943 27 4.75489 43 5.42626 59 5.88264 12 3.58496 28 4.80735 44 5.45943 60 5.90689 13 3.70044 29 4.85798 45 5.49185 61 5.93074 14 3.80735 30 4.90689 46 5.52356 62 5 .95420 15 3.90689 31 4.95420 47 5.55459 63 5.97728 16 4.00000 32 5.00000 48 5.58496 64 6.00000
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Text content of presentation slides: Measuring information. Content-based approach. 1st category computer science teacher T.Yu. Khokhlova I know that I know nothing. Socrates Paradox: The more knowledge a person has, the more he feels the lack of knowledge. Content approachFrom a “human” point of view, information is the knowledge that we receive from the outside world. The amount of information contained in a message should be greater, the more it adds to our knowledge. How, from this point of view, is the unit of measurement of information determined? Bit
The problem of measuring information is studied in information theory, the founder of which is Claude Shannon. In information theory, the following definition is given: A message that reduces the uncertainty of knowledge by half carries 1 bit of information. What is uncertainty of knowledge?
Equally probable eventsUncertainty of knowledge about the outcome of some event is the number of possible outcomes. Moreover, none of these outcomes has an advantage over the other. That is, they are equally probable. Possible eventsAn event that has occurred
In the case of a coin, before it is tossed, the uncertainty of knowledge about the result is equal to two. A dice with six sides can fall on any of them with equal probability. This means that the uncertainty of knowledge about the result of throwing is equal to six. Therefore, we can say this: The uncertainty of knowledge about the result of some event (throwing a coin or a die, drawing a lot, etc.) is the number of possible results. Having learned the result of throwing a coin, you received 1 bit of information. A message about one of two equally probable the results of some event carry 1 bit of information.
Task No. 1 A student in an exam can receive one of four grades: “2”, “3”, “4”, “5”. To the question: “Well, what did you get?” - answered: “Four!” How many bits of information are in his response? We will guess the rating by asking questions that can only be answered with “yes” and “no”. First Question - Evaluation above "three"? Yes. There are 4 and 5 left. You have received 1 bit of information. Second question Score 5? No. There are 4 left. You have received 1 more bit of information. Conclusion: a message about one of four equally probable events carries 2 bits of information.4
Problem No. 2 There are eight shelves on a bookshelf. The book is placed on any of them. How much information does the message contain about where the book is located. Half division method A search method in which half of the options are discarded at each step is called the half division method. When guessing the mark, two questions were asked, each of which reduced the uncertainty of the situation by half, and in total possible options there were four. 2×2=4, i.e. 22=4When guessing the location of the book, three questions were asked, each of which reduced the uncertainty of the situation by 2 times, and there were eight possible options in total. 2×2×2=8, i.e. 23 = 8 Based on this, we can derive the formula 2i = N, where i is the amount of information in the message, N is the number of options (events). We use the resulting formula to determine the amount of information when tossing a coin: 21 = 2, i = 1 bit Main formula of computer science Quantity information (i), contained in a message about one of the equally probable results of some event, is determined from the solution of the exponential equation: 2i = NOr i = log2N, where i is the amount of information, N is the number of equally probable events (if the value of N is not equal to powers of two). Ralph Vinton Lyon Hartley (1888-1970) contributed to the foundations of information theory by introducing the logarithmic measure of information in 1928.
Problem: Classes can take place in one of the rooms, numbers from 1 to 16. How much information does the teacher’s message contain that classes will be held in room No. 7? Given: N = 16 options Find: i = ? 1 way: 2i = N2i = 162i = 24i = 4 (bits) Method 2: Question 1: Is the cabinet number less than 9? – Yes (1 bit) 2 question: Is the cabinet number greater than 4? – Yes (1 bit) Question 3: Is the office number even? – No (1 bit)4 question: Room number 5? – No (1 bit) Summing up Equally probable results: no result has an advantage over others. Uncertainty of knowledge – the number of possible results ()message options) – N. The amount of information in a message about one result of an event – i bits. The main formula of computer science 2i = N1 bit – the amount of information in a message about one of two equally probable results of some event. HomeworkParagraph 4, questions 4 and 5 in writing
Attached files
Answer the questions orally:
- What is meant by information?
- What can you do with the information?
- What types of information representation in a computer do you know?
- What message encoding techniques were used in ancient times?
- What is code and information encoding?
- Give examples in various ways encoding information.
- List the advantages and disadvantages of coding used in computers.
- What is the name of the encoding used to represent characters entered from the keyboard?
- Let's think about what can serve as an estimate of the amount of information?
- Is it true that it's worn out
- book, if not in it
- torn pages, carries for
- there are exactly the same number of you
- information, how much is the same
- new?
- Let's think about what can serve as an estimate of the amount of information?
- A block of stone weighing three tons carries as much information for archaeologists as a good photograph of it in an archaeological magazine.
- Is not it?
- Let's think about what can serve as an estimate of the amount of information?
- When a Moscow radio studio broadcasts the latest news, both a resident of the Moscow region and a resident of Novosibirsk receive the same information. But the flow of radio wave energy in Novosibirsk is much less than in Moscow.
- Consequently, the signal power, as well as the weight of the carrier, cannot in any way serve as an estimate of the amount of information carried by the signal.
- How then to measure the amount of information?
- Different approaches to defining and measuring information
- Meaningful
- (probabilistic) approach:
- Amount of information as a measure of uncertainty reduction
- knowledge
- Watch the video
- Let us
- there is a coin
- which we
- throw it on the flat
- surface.
- Possible events
- Event that happened
- One of the following is equally likely to happen
- two possible events - a coin
- will end up in one of two positions:
- "heads or tails".
- Events are equally probable if, with an increasing number of experiments, the numbers of heads and tails gradually become closer.
- Before the throw there is uncertainty of our knowledge ( two events are possible), and after the throw there is complete certainty.
- The uncertainty of our knowledge is reduced by half, since out of two possible equally probable events, one was realized.
- When throwing an equilateral tetrahedral pyramid, there are 4 equally probable events.
- When throwing a six-sided die, there is
- 6 equally probable events.
- A message that reduces the uncertainty of knowledge by half,
- carries 1 bit of information.
- Bit – minimum unit measuring information.
- 1 byte = 23 bits = 8 bits
- 1 KB = 210 bytes = 1024 bytes
- 1 MB = 210 KB = 1024 KB
- 1 GB = 210 MB = 1024 MB
- GB
- kbyte
- MB
- TB
- :1024
- :1024
- :1024
- :1024
- *1024
- *1024
- *1024
- *1024
- The amount i of information contained in the message that one of N equally probable events has occurred is determined by solving the exponential equation
- 2i = N
- Task: In roulette, the total number of holes is 128. How much information will we receive in a visual message about the ball stopping in one of the holes?
- N=128
- i - ?
- Given:
- Solution:
- 2i = N
- 2i = 128
- 27 = 128
- i = 7 bits
- Answer: i = 7 bits
- Number of possible events and amount of information
- Task:
- There are 32 pencils in the box, all pencils are different colors. They pulled out a red one at random. How much information was obtained?
- Solution.
- Since drawing a pencil of any color from the 32 pencils in the box is equally probable, the number of possible events is 32.
- N = 32, i = ?
- N = 2i, 32 = 25, i = 5 bits.
- Answer: 5 bits
- The book has 512 pages. How much information does the message that a bookmark is on a page convey?
- Solving problems in a notebook
- How much information does the message that on a 4x4 square field contain, one of the cells is shaded?
- Solving problems in a notebook
- How much information does the message about rolling a 3 on a six-sided die contain?
- Solving problems in a notebook
- What is the meaning of a content approach to measuring information?
- What formula was studied?
- Name in ascending order what units of measurement of information you know.
- How are units of measurement of information interrelated?
- Fixing the material
- 1. You approached a traffic light when the light was red. After that the yellow light came on. How much information did you receive?
- Solve orally
- 2. You approached a traffic light when the light was yellow. After that the light turned green. How much information did you receive?
- Solve orally
- 3. "Are you getting off at the next stop?" - they asked the man on the bus. “No,” he replied. How much information does the answer contain?
- Solve orally
- 4. How much information is conveyed by the message that the program you need is on one of the eight floppy disks?
- Solve orally
- 1. Analyze the entries in the notebook.
- 2. Solve 2 individual problems on cards.
Class: 10.
The purpose of the lesson: To teach how to measure the information volume of a message through meaningful approach.
Lesson Objectives:
- Educational: teach to measure the information volume of a message through a meaningful approach.
- Developmental: development of thinking, speech, fine motor skills, imaginative perception.
- Educational: grafting careful attitude to information and technology, personal responsibility for work results, accuracy, perseverance, self-discipline.
Lesson type: Explanation of new material with elements of a workshop.
Textbooks:
- “Informatics 10” (basic course), ed. N.V. Makarova, “Peter”, 2003.
- Ugrinovich N.D. Computer science. Basic course Grade 10. - M.: Publishing house "BINOM".
Basic concepts:
- Half division method;
During the classes
I. Organizational moment
Setting the mood for the work environment.
II. New material
In the last lesson we learned to distinguish informative messages from uninformative ones.
We found out that to determine the amount of information in a message about the occurrence of one event out of more than two equally possible ones, the following formulation is necessary: “A message that reduces uncertainty by 2 times contains 1 bit of information.” We analyzed the problem with tossing a coin: “Before tossing the coin, there were two equally probable outcomes. This determines the uncertainty of the situation. In other words, uncertainty is the number of possible events. After receiving a message about the result, there was only one option left. How much has the uncertainty of the situation decreased?”
Now let's solve the problem of determining the amount of information in a message using the method of halves (dichotomy). So that at each search step exactly half of the options can be discarded. We will organize the work in the form of a game “Guess the answer”.
For example, I think that a book is on some shelf, but I don’t tell you about it. You need to determine which of the 8 shelves the book is on. Questions must be asked in such a way that each answer (“yes” or “no”) reduces the uncertainty by exactly half. Therefore, no matter how many questions are asked, so many bits of information carry the message about the guessed object. Fills in during the game table 2, establishing the relationship between the number of events and the amount of information in the message.
Analyzing the solution to previous problems, we introduce symbols and we deduce the formula of R. Hartley. For example, the chain of reasoning could be as follows:
- When guessing the mark, two questions were asked, each of which reduced the uncertainty of the situation by half, and there were four possible options in total. Formalization of the reasoning – 2 · 2 = 4, i.e. 2 2 = 4.
- When guessing the location of the book, three questions were asked, each of which reduced the uncertainty of the situation by half, and there were eight possible options in total. Formalization of the reasoning – 2 · 2 · 2 = 8, i.e. 2 3 = 8.
- Based on this, we can derive the formula 2 i = N, Where i– amount of information in the message, N– number of options (events).
- We use the resulting formula to determine the amount of information when tossing a coin. 2 1 = 2, i= 1 bit.
Number 2 in the formula means reducing the uncertainty by half, in accordance with the definition of the concept of “bit”. Using the formula, we fill out the table of integer powers of two up to 210 = 1024. The table establishes the relationship between the quantities of information in the message ( i) and the number of equally probable events ( N) and is a support for students in solving problems.
Let's create a general diagram:
Let's solve the problem using an example.
Task 1. Classes can take place in one of the rooms, numbered from 1 to 16. How much information does the teacher's message contain that classes will be held in room No. 7?
III. Summarizing
Today we studied:
- Half division method;
- Measuring the amount of information in a message in two ways: using a formula and the halving method,
- Measuring the amount of information in a message over several actions,
- Measuring the number of events if the information volume of the message is known.
IV. Homework
Solve the problem: There are 16 red apples in a bag. How much information does the message that you got a red apple contain?
