Task from the old book. ("Iron" table)
Wandering through the antiquarian ruin for a ridiculous price 35r. bought a book by Hugo Steinhaus "Tasks and Reflections." There I found a good set of mathematical problems suitable for teaching computer science. But I don’t want to share this discovery with Habrasoobshchestvo.
There was also a task for random sampling. The task is classic. I rephrase it somewhat. But the essence does not change.
So ... There are 1000 identical boxes in stock. In the boxes are mixed nails and screws. Screws are 10 times more expensive than nails. How much can you sell a product?
Solution one (Bad) Take the first 10 boxes. Sort, evaluate, extrapolate the result to all boxes. What a bad decision. We do not know the law for which screws and nails are distributed. With the time of filling the warehouse, the number of screws may decrease.
The solution is the second (classical). We take a random sample of 10 boxes, then the algorithm. Than this decision is not perfect. Random numbers are random and may accidentally lie in a handful.) Of course the chance of this randomness is strictly determined mathematically, but still “The Case of the Devil’s Tool.”
The third decision (cool). The book provides a method using "iron" tables. I myself have never met the term itself, but I am not in the subject. So I am waiting for your comments.
Iron tables in an interesting way. All of you (especially designers) could hear about the “golden section.” [Golden is the section where the ratio of the length of a segment to its greater part is equal to its ratio to the smaller one]. If the length of the segment = 1, then the length of the greater part w = 0.618033 ..., Irratic number. Therefore, in the sequence 1 * w, 2 * w, ..., 10000 * w there are no integers, that is, each number consists of an integer and a fractional part. By arranging our sample in ascending order of the fractional part, we get an interesting sample of numbers from 1 to 10,000. (Here I must fully rely on Hugo Steinhaus — I haven’t checked it myself yet.)
1. Numbers are located so little indistinguishable numbers are located far from each other, not too little separated by average distances.
2. This property is preserved if we take not the whole table, but only the numbers lying in a certain interval.
Thus, in order to choose, we have to count ten numbers from an arbitrary place in our tables in the interval from X to X + 1000, where x is an arbitrary number from 0 to 9000 = 10,000-1000.
Then we act according to the algorithm: we open, sort, evaluate.
To be honest, this method made a strong impression on me. Not the sampling method itself with the help of an irrational number, but the depth property of the golden section of such a perfectly distributed number. I felt like an ancient Pythagorean frozen in ecstasy before divine revelation manifested in numbers.
BUT enough lyrics. How I plan to use the revelation given to me.
I plan to give it to the students for analysis. Let him explore it (while purely numerical and statistical methods). We will issue the results in a scientific report, we will bring the children to the conference, and so on. Therefore, I am WAITING for your reaction (advice, suggestions and most importantly, the topic may be trivial, I just don’t know it). In general, I am waiting.
How can you use it? - Well I do not know. Count the nails and screws stored in the closet or something! :)
Shl (in the morning). For those who do not understand what the chip.
Build an iron table to 10.
Build a table number - the fractional part (number * w)
1-0.618
2-0.236
3-0.845
4-0.472
5-0.090
6-0.708
7-0.326
8-0.944
9-0.562
10-0.180
Sorting on the second column will get a row
5,10,2,7,4,9,1,6,3,8.
These are not random numbers. Compare the distance between 10.9.8
or 5,4,3. Therefore, random or root of 2 is worse for sampling - they can give a handful, but this distribution is not.
But to statistically evaluate whether it is better or not - that’s what we will have to do.
At the same time, children will learn something.
There was also a task for random sampling. The task is classic. I rephrase it somewhat. But the essence does not change.
So ... There are 1000 identical boxes in stock. In the boxes are mixed nails and screws. Screws are 10 times more expensive than nails. How much can you sell a product?
Solution one (Bad) Take the first 10 boxes. Sort, evaluate, extrapolate the result to all boxes. What a bad decision. We do not know the law for which screws and nails are distributed. With the time of filling the warehouse, the number of screws may decrease.
The solution is the second (classical). We take a random sample of 10 boxes, then the algorithm. Than this decision is not perfect. Random numbers are random and may accidentally lie in a handful.) Of course the chance of this randomness is strictly determined mathematically, but still “The Case of the Devil’s Tool.”
The third decision (cool). The book provides a method using "iron" tables. I myself have never met the term itself, but I am not in the subject. So I am waiting for your comments.
Iron tables in an interesting way. All of you (especially designers) could hear about the “golden section.” [Golden is the section where the ratio of the length of a segment to its greater part is equal to its ratio to the smaller one]. If the length of the segment = 1, then the length of the greater part w = 0.618033 ..., Irratic number. Therefore, in the sequence 1 * w, 2 * w, ..., 10000 * w there are no integers, that is, each number consists of an integer and a fractional part. By arranging our sample in ascending order of the fractional part, we get an interesting sample of numbers from 1 to 10,000. (Here I must fully rely on Hugo Steinhaus — I haven’t checked it myself yet.)
1. Numbers are located so little indistinguishable numbers are located far from each other, not too little separated by average distances.
2. This property is preserved if we take not the whole table, but only the numbers lying in a certain interval.
Thus, in order to choose, we have to count ten numbers from an arbitrary place in our tables in the interval from X to X + 1000, where x is an arbitrary number from 0 to 9000 = 10,000-1000.
Then we act according to the algorithm: we open, sort, evaluate.
To be honest, this method made a strong impression on me. Not the sampling method itself with the help of an irrational number, but the depth property of the golden section of such a perfectly distributed number. I felt like an ancient Pythagorean frozen in ecstasy before divine revelation manifested in numbers.
BUT enough lyrics. How I plan to use the revelation given to me.
I plan to give it to the students for analysis. Let him explore it (while purely numerical and statistical methods). We will issue the results in a scientific report, we will bring the children to the conference, and so on. Therefore, I am WAITING for your reaction (advice, suggestions and most importantly, the topic may be trivial, I just don’t know it). In general, I am waiting.
How can you use it? - Well I do not know. Count the nails and screws stored in the closet or something! :)
Shl (in the morning). For those who do not understand what the chip.
Build an iron table to 10.
Build a table number - the fractional part (number * w)
1-0.618
2-0.236
3-0.845
4-0.472
5-0.090
6-0.708
7-0.326
8-0.944
9-0.562
10-0.180
Sorting on the second column will get a row
5,10,2,7,4,9,1,6,3,8.
These are not random numbers. Compare the distance between 10.9.8
or 5,4,3. Therefore, random or root of 2 is worse for sampling - they can give a handful, but this distribution is not.
But to statistically evaluate whether it is better or not - that’s what we will have to do.
At the same time, children will learn something.
')
Source: https://habr.com/ru/post/46560/
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