Science,English,and math Ⅱ

【The theorem of vector makes us fly in the sky】

《A plane flies in the sky by the composition of lots of power from vector》

If using the theorem and the way of thinking on the composition of vector,we’ll understand at ease why the plane flies in the sky or why the ball which the pitcher throws is a curve or screw ball.

The body of the airplane and the section of the wing is what is called a streamline 流線型. The flow of the air around the section of the wing and body is shown in the illustration of the book.

As for the flow of the air,the upper part of the wing is faster than the lower part,according to the illustration,so the pressure in the upper part on the wing is less than the lower part,according to the theorem of Bernoulli,so the wing receives the lift 揚力. The lift is the one which raises the plane in the air against the gravitation.

When trying to go straight in the air there is resistance against the air,but propelling force like a propeller or jet engine can overcome it.

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【The theorem of vector make us fly in the sky】

《The plane flies in the sky by the composition of lots of power from vector》

A helicopter only looks like a plane. In general,the principle of the helicopter is the same one of a propeller-shaped flying toy made of bamboo,there seem to be some people who think so,but if trying to express on the principle of the helicopter,it’s not enough. In fact the way of thinking from the vector is made use for the helicopter here and there.

Oh! I can understand the relation between the vector and the plane somehow. The lift is equivalent to vertical arrow and the propelling force is the horizontal one,when the two powers are composed well,the plane can fly in the air,roughlying speaking,I’m wondering.

By the way I hate to say but I can’t understand on the matrix very much,so if trying to express it,I have little confidence on my expression,I’m afraid,but it’s useful for me to broaden my lookout,so I’m going to go on. If trying to study math…

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【The matrix also plays an active part in the economy very much】

《We can predict the market share in an automobile with the what is called Marcov chain》

An automobile has spread among us lately,its salespersons would rather place great importance on a user who tries to trade a old car for a new one than other user who doesn’t have any car. When repeating of trading the old one for the new one,how does its market share change? It’s the matter of life and death to not only the makers but to the salespersons.

It’s the technique of Marcov chain which analyzes the market share and shows us how the market share changes. The Marcov chain is application from the matrix. Its basic concept is a process in which almost the same thing is repeated. The probability of each phenomenon is determined by condition alone just before.

“What are you talking about?” I’m afraid almost all of you think like that,to tell the truth,neither I can understand at all,so I’m going to express it concretely.

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【The matrix also plays an active part in the economy】

《We can predict the market share in an automobile with what we call Marcos chain》

Let’s suppose there were cars belonged to N company and T company so as to be simplified the story. The owners of the car belonged to the N company think that the car has such the little trouble and good design that when trading an old one for a new one,90 % of the owners buy the one belong to the N company again,and other 10 % the T company.

On the other hand,let’s suppose that other owners of the car which belonged to the T company thought that the car was costly a little,but its function is good,the owners were content with the cars,so when trading,60 % of the owners bought the same car again,other 40% the N company.Let’s suppose that the time when trading the car was every five year in order to be simplified the story.

In this way,using the probability that the users trade a new brand,and predict the market share in the future is the Marcos
..

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【The matrix also plays an active part in the economy】

《We can predict the market share on an automobile with what we call the Marcov chain》

Thus with the use of the probability that the time when the users trade for a new brand,the way of predicting the market share in the future is called the Marcos chain.

The share approaches a balance before long,and to my surprise,it’s nothing to do with the present share at all. Its changing matrix decides the share. Even if the present share is 50 to 50,or 0 to 100,it’s not related with the share in the future. Even if it’s superior to other at present,the situation is reversed if the company neglects originality and invention.The Marcov chain shows us its fear theoretically.

By the way,we shouldn’t think we rely on a rumor. For example,when telling a news that some person isn’t a criminal,let’s suppose that the wrong news that it’s the criminal was told to the 10 % of the people. Even if it’s not the criminal at all,finally half of the...

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【Winning a game of tennis with the theory of the game】

《We can win all of the races like sports and economy with the theory of the game》

Even if it’s not the criminal at all,half of the whole of people think it’s the criminal because of the wrong information.

Speaking of Japanese word,what is called ゲーム,it reminds of us a video game,go,syogi,or chess,but the game which I’m going to express from now on has been developed not only on them but all the races with which we compete like politics,economy,society,or sports as basic science. We call it theory of the game.

Its origin dates back to a book of which title is the theory and economic activity on the game written by a economist,John von Neumann and a mathematician,Morgenstern. The book has formalized all the subjects on the races and established the theory on them.

After the theory was established,the practical use of the theory has spread more and more,and the theory has influenced on social science widely.

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【Winning a game of tennis with the theory of the game】

《We can win all of the races like sports and even politics with the theory of the game》

In general,when playing a game,we can’t foresee what the opponent will do. Though being uncertain,the player have to do something,what is an effective action then? It searches for a strategy which makes its advantage bigger as probability in a mathematical point of view.

By the way,I’ve tried to express the theory,but I can’t show its equation with little knowledge on both the math and PC. To my sorrow,I can’t understand the essential part. I need to study math carefulully,but I hate it and I’m afraid I can’t understand it at all,so I’m going to do the next chapter.

【In short,finding the value of an area】

《Integration 積分,which has started from flooding at the Nile in Egypt》

As we usually say differential calculus 微分,integral calculus,we use them as a pair. When studying math in a textbook,we learn the differential calculus and master....

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【In short,finding the value of an area】

《It’s started from floodings at the Nile in Ancient Egypt》

In general,we learn the differential calculus at first in a textbook and master it,after that we learn the integral calculus when going to high school. Its order is like that. When looking at the completed compendium 体系 of two of them,its order seems to be logical,but it’s reverse in a historic point of view.

While the idea of differential calculus wasn’t shown to us until two geniuses Newton and Leibnits appeared in the 17th century,the integral calculus has started from the flooding at the Nile in Ancient Egypt.

As it’s said Egypt is a gift from the Nile,the Egyptian civilization has started from the Nile,and floodings in the Nile accelerated the development of geometry and technique of measurement on the land. As I expressed before,the word of geometry originates from the language in Ancient Egypt,geo 土地 and merry 測量.

Finding the value for an area of a figure which was ....

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【In short,finding the value of an area】

《It’s started from flooding at the Nile in Ancient Egypt》

Finding the value for the area of a figure which was surrounded by a complicated curve,or other value for a cubic volume of a quadrangular pyramid 四角錐. Those kinds of things used to be the lesson for the Ancient Egypt then. Those knowledge which was got there came into flower by Archimedes in Greece.

The way of Archimedes was named taking all the things. It’s the basic idea of modern integral calculus 積分. The way of thinking is dividing the complicated figure into fine ones so as to find its value. Archimedes closely resembled 近似 each of the fine ones with the other which is easy to understand.

Archimedes calculated the pi with the way of thinking,and has found the approximate value,3.14 which we use at present. His way is theoretical. His way of thinking is like the next.

There is a circle of which diameter is 1. He put two kinds of regular hexagon around the circle. The one..

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【In short,finding the value of an area】

《It’s started from floodings at the Nile in Ancient Egypt》

The one regular hexagon was inscribed 内接 to the circle,and the other one was circumscribed 外接 to the circle. The length of the circumference 周囲 of each of the regular hexagon was decided by the Pythagorean theorem.

Archimedes increased the number of the angle of the figure with the way more and more like a regular dodecagon 正12角形,or a regular 96 gon 96角形,and he calculated the value less and less minutely.

By the way there is an episode on Archimedes like the next.

When he was absorbed in the calculation of pi,sitting on the road,a Roman soldier tried to kill him. Then it is said he told the soldier to wait until he finished computing the calculation.

【He tried to divide it into a minimum one as minute as possible】

《The idea of Archimedes was an approach to the the differential and integral calculus》

Archimedes tried to resembled the circle closely to the regular 96 gon....

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【He tried to divide it into a minimum one as minute as possible】

《The way of his idea was an approach to the differential and integral calculus》

Archimedes tried to resemble the circle closely to the 96 gon so as to find the value of pi. The bigger the number of 96,the closer,it approaches the exact value more and more,but strictly speaking,even if the number of the regular polygon is 10 billion,its just that an approximate value. To be exact it’s not precise. It’s just the halfway to the precise value.

Even if Archimedes was an excellent mathematician,he couldn’t tried to find the precise value forever,so a concept of the limit appeared. We can almost master of the idea of the differential and integral calculus as long as we can understand the idea of the concept of the limit,the book says so.

What is the limit? If showing it with a sequence of numbers,we can grasp it very much. For example,2,2,2,2,2,... it’s a sequence of numbers on 2 alone,needless to say,it moves toward 2.

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【He tried to divide it into the minimum one as minute as possible】

《The idea of his way was an approach to the differential and integral calculus》

In short,its limit is 2. For example,let’s suppose there was other a sequence of numbers. One,half,one-third,one-fourth,one-fifth....one-nth. When bigger and bigger the n becomes,the one-nth becomes smaller limitlessly,so the limit of the sequence of numbers is 0.

Then,let’s think of pi as an instance. The number of which Archimedes thought was 96,and I’ll try to make it bigger and bigger. Then we can understand the length of the circumference of the regular polygon which is inscribed or circumscribed to the circle approaches the certain value. We call the value the limit. The limit is the pi.

Even if we go on increasing the number of the angles on the regular polygon more and more,if stopping halfway,it will be useless. It has to be done forever. Then the value at which we aim is the limit,and we show it with the sign as the pi.

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【He tried to divide it the minimum one as minute as possible】

《The idea of his way is an approach to the differential and integral calculus》

There is one thing of which we should be careful. It doesn’t always mean that we the human calculate it one after another actually. If being calculated in one’s mind forever,the value of which calculation approaches will be the goal.it’s the pi.

The space of the circle is also the same. We closely resemble the space of a regular polygon to the space of the circle,,and increase the number of angles of the regular polygon more and more,then we can get its limit.

Thus we’ll be able to find the length of a circumference 円周 or the space of a circle if we divide it into the minimum one less and less. We can find them as their limit,it’s the way of thinking on the integral calculus.

【The idea of the integral calculus】

《Let’s find the value of the area on a curveilinear figure with the thought of an idea on the limit》

We closely resembled the...

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【The idea of the integral calculus】

《Let’s find the area of curvilinear figure with an idea on limit》

We closely resembled the figures which we can recognize its area both from outsid and inside to the circle so as to find the area of the circle,and both of the limits agree.so we could find the area of the circle, in fact as to the general function 関数,the idea of finding the area as the limit of the approximate value is the same,and the integral calculus means finding the value.

When showing it with an equation,it’s R ≦ S ≦ T. It seems that the R is the area of the regular polygon which is inscribed to the circle.and there is a small n beside the R,it’s on the lower right. It seems the n is the number of the angles of the regular polygon. The S is the areaof the circle.

The T is the other area of other regular polygon which is circumscribed to the circle.and there is also the small n beside the T,the n is equivalent to the one beside the R.

Bigger and bigger the both n becomes..

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【The idea of integral calculus】

《Let’s find the area of curvilinear future with an idea of limit》

Even if bigger and bigger the both n become limitlessly,both limit of the values exist and the both values agree. Mathematicians have already demonstrated it strictly. Please believe me! The author said like the that.

I’m going to express the idea of the integral calculus. We closely resemble a given function to the one which is easy to understand for the time being,using the idea of limit,and finding the ultimate precise value. The approximate value is a temporary one,so we can get the real one with the limit.

Even if it’s the approximate value or limit,it’s the talk in the world of microorganism. I don’t mind it at all,though if saying so,my thread will be of no use.

【Finding an instant speed】

《If finding the speed which changes unstably,the differential calculus alone catch it》

To tell the truth,we’ve already experienced the differential calculus. For instance,we …

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【Finding an instant speed】

《If waning to find an instant speed which changes unstably,the differential calculus alone catches it.》

For instance we differentiate the speed from the mileage. In short a speed meter is an instrument which differentiates.

There is a graph on the book which is shown a relation between the mileage and time. Its vertical direction is the mileage and horizontal one is the time.Its speed isn’t always fixed,so the graph isn’t always shown with a straight line. Sometime the speed is slow and other time fast,so it’s a curve rather than the straight line.

If the line is inclined steeply on the graph,it means that its speed is fast,and its shown gently,it means its speed is slow. Let’s suppose that the curve on the graph was a slope. Then the speed of the car is shown with the inclination of the slope.

When the curve on the graph is a straight line,it means that the car runs with a fixed speed,for the mileage is in proportion to the time. The speed is the one…

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【Finding an instant speed】

《If wanting to find an instant speed which changes unstably,the differential calculus alone catches it》

The speed is the next one that the mileage is divided with the time,so it's just the inclination of a straight line. Then how should we think on the speed which changes unstably ? If finding the mileage of an hour the car runs so as to know its speed per hour,it seems to be far from an instant speed.

So let's see how far the car runs per a minute. If it runs 1km per a minute,it means it goes 60 k.p.h.,but strictly speaking it doesn't mean that it always runs with a fixed speed,so if wanting to find the speed per a second…

Thus if going on thinking on an instant speed one after another,it's endless. It won't get anywhere,so the one which overcomes the difficult point is the concept of the limit.

The average speed is the one which the mileage is divided by the time,so when we try to make the time smaller and smaller,its value approaches other value…

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【Finding an instant speed】

《If finding an instant speed which changes unstably,the differential》

The average speed is the one which the mileage is divided by the time,so when we try to make the time smaller and smaller,its value approaches other value closer and closer. Then there is a value which we want to find. We define an instant speed as its value.

Differentiate the mileage means the instant speed. If thinking it of on the graph,it means finding the inclination of the curve on the graph. To be concrete,it resembles a thing which we magnify something bigger and bigger with a magnifying glass or a microscope. Its limitation is the differential calculus.

When the speed is unstable,we can't find the instant speed from average one. Average life expectancy or average salary are also the same. The average one is frequently far from our reality.

【Once again,and once more again…】

《A biggest clue which we pursue a function is a derived function 導関数》

The speed of a car is found with…

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【Once again,and once more again…】

《A biggest clue which we pursue a function is a derived function》

The speed of the car is the mileage from which is differentiated. The speed changes every moment,so when paying attention to the speed,we can find the speed itself is the function for the time. To be concrete,the record of the value on the speedometer which changes every time. In short when looking at the entire one which is differentiated,we can find a new function.

In general,when there is a function,and if the function is differentiated,it becomes the other one which is differentiated. We call the new one a derived function from the first one. Even if it's the derived function,it's just an ordinary function,so we can differentiate between the derived function. We call it the quadratic derived function.

We differentiate the mileage of the car by the time,and we find the derived function. Then we differentiated the derived function and find the quadratic derived function,which is…

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【Once again,and once more again…】

《A biggest clue which we pursue the curve on the function is the derived function.》

Then we differentiate the derived function and find the quadratic derived function 二時導関数,which is acceleration. The one which we recognize on the body is the acceleration. Even if an airplane is flying high up in the sky with a marvelous speed,we don't really feel so,for then the acceleration is almost zero.

There is a strap つり革 in a train. The strap is a barometer of acceleration. When the train’s speed is fixed,the strap is turned downward. If adding acceleration to the strap,when starting,it shakes opposite direction from the train moves. When stopping,the strap shakes to the direction where the train has come all together.

By the way,I've expressed on the differential calculus and integral calculus,so I can understand the conception of them,but when looking at their equation,I can't understand them a bit. To my sorrow,I find it that math is hard to understand.

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【The pair is as if it were a happy married couple】

《The differential calculus and integral calculus suit perfectly with a wave of a magic wand》

As we the Japanese always say the differential calculus and integral calculus in Japanese,whenever we call it,we do it with a pair. While the differential calculus is the one in which we find the inclination of a tangent line 接線,the integral calculus is the one in which we find a space.

The two of them are seemingly unrelated,but they cling together with a wave of a magic wand. The magic wand is the eccence of the differential calculus and integral calculus. We call it the basic theorem of the differential calculus and integral calculus.

If saying in other words,the differential calculus and integral calculus are opposite calculations each other. It resembles relations that subtraction is the opposite calculation for addition,and division is the opposition calculation for multiplication. There are detailed descriptions on the relation…

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【The pair is as if it were a happy married couple】

《The differential calculus and integral calculus suit perfectly with a wave of a magic wand》

There are detailed description on the relation between the differential calculus and integral calculus,which means their equations. Though I've said it's the eccence,to my sorrow I can't understand what it's the important things. I should study math again from its base,but I don't feel like doing at all.

If each time we try to find the value of a space or a cubic volume,we divid them into tiny ones and find their sum,it'll be hard and complicated. When handling a complicated figure,using the way was very tiresome and had a limit. The basic theorem of the differential calculus and integral calculus has overcome the drawback.

By the way,we can seemingly differentiate all the functions,but it doesn't always mean that all the functions are able to be differentiated. All the functions aren't shown with equations. The author said like that.

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【If taking advantage of the differential calculus and integral calculus…】

《If looking around us,there are lots of things which we make use of the differential calculus and integral calculus》

Which field is the widest application in math? No one is wider than the differential calculus and integral calculus in math. If looking around us,we'll find lots of them at once. The electricity which we've made use of everyday is the first to say. Not only the source of the electricity like a generator,transformer,or power transmission,but electronic appliances like TV,stereo,or washing machine are based on the principle of the differential calculus and integral calculus.

For example,as for the generator,it works,being based on the principle of electromagnetic induction by Faraday. The principle is like the next.

If an electric wire moves between the north pole and sout pole of a magnet,voltage generates on the electric wire,its strength is in proportion to the ratio of flux. What is the flux?

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【If making use of the differential calculus and integral calculus…】

《If looking around us,there are plenty things which we take advantage of the differential calculus and integral,calculus》

I want to express the word of 磁束 in English,but the dictionary doesn't have it,and to my sorrow,when reading the expression on the electromagnetic induction,I can't understand it very much,so if you are interested in it,please research it for yourself.

The idea of the differential calculus has been adopted the linear motor car which is super high speed one,a principle of maintaining the body of a train in the air.

Condenser or coil have been used for apparatus of audio like TV,stereo,and so on,they act of the differential calculus and integral calculus. Computer graphic has been paid attention among the computer science lately,the idea of the differential calculus and integral calculus plays an important role.

For instance,a drawing for a smooth curve or smooth curved surface is indispensable…

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【If making use of the differential calculus and integral calculus…】

《When looking around us there are plenty of things which we apply the differential calculus and integral calculus.》

For example,drawing for a smooth curve or a smooth curved surface are indispensable so as to design a body of a car,then a technique of a spline curve is used for it. When inputting some points into the display,the computer used the differential calculus and connects them with smooth curves.

As for an automatic pilot on a jumbo jet,the function of gyro is used and a stable horizontal plate which is always horizontal against the surface of the earth and turns to a standard direction is provided. A pilot measures acceleration of front,back,left,right,upward,or downward one by one on the plate and integrates two times. The pilot finds its position though it continues to change every moment.

By the way,what is the gyro? It originates from the Greek. It means a ring or circle,but this time it means the…

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【What is the gyro? And what I've thought】

This time the gyro means what's called the function of gyro. It means searching for angular velocity 角速度 with a computer. In short it's a sensor which detects its inclination. It's used for a smartphone as a function of a compass. When using the compass,where are the users of the smartphone going? I'm wondering.

While I can understand the concept of the differential calculus and integral calculus,when looking at the equations,I cant grasp it at all.

There are lots of electronic appliances around us. Without knowing the information on each electronic appliance,we can use it at will,and when leading our everyday life we don't have to be worried about it. I won't be distressed by lacking of the knowledge without announcing,“I want to express everything in English”

Art is long and life is short,somebody important seemed to say like that long long ago,it's right to the point,I'm sure. When looking around me,there are lots of strange things…

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【Forseeing the future】

Science has developed so much at present,but trying to foresee things in the future is hard to do. If we can predict completely how stock prices change beforehand,no one will be troubled,but can't we foresee the things in the future at all? We can to some extent,and its keyword is a place of vector and equation of differential calculus.

Foe example,let's suppose that we observed wind speed at several places in some area. Its speed is shown with a vector,and it's shown on a map. Thus,a thing in which the vector is indicated each place is called the place of vector. Let's suppose that the wind speed didn't change at all so as to the story is simple.

If we let a balloon fly in the air then,what kind of curve will the balloon show in the air? Its answer is shown with the curve which the tangent is in perfect harmony with the vector of the place of the vector. What is the tangent? While I can express it in English,but to tell the truth,I'm not sure of it clearly.

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【Forseeing the future】

If you're interested in the tangent,please research it for yourself.

The differential calculus is finding the inclination of the tangent. In a word,the place of the vector is finding the value of the differential calculus each place.

On the other hand,can we find original locus 軌跡 from the place of the vector? It means that it's solving the equation of the differential calculus.

【Achilles can catch up with it】

There is a paradox which an Ancient Greek thought up,Achilles and a tortoise. Achilles is the one who appears in the Greek Myth and can run at high speed,but he can't catch up with a tortoise which walk at a slow pace,it's a strange story.

When starting at the same time,let's suppose that Achilles was away 100 meters from the tortoise. Achilles is at the speed of 10 meters per second,and the tortoise is 1 meter per second.

After starting Achilles could catch up with the place where the tortoise was in 10 seconds,but then the tortoise would advance…

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【Achilles can catch up with it】

But then the tortoise would advance ahead of him by 1 meter. When Achilles reaches the place again where the tortoise did,the tortoise advanced ahead of him a little then. In short he can't catch up with the tortoise forever?

The reason which led to the irrational conclusion is the next one. The time when he catch up with the tortoise is the extremity,and the story is limited to the process in the middle of it. When taking the extremity and after the extremity into account,Achilles can catch up with the tortoise and overtake it.

In short it's just that it limited the time as a person who prepared the question is pleased,and saying,“Oh,Achilles can't catch up with it.”

【A way in which we get along with Lady Luck】

《The theory of probability in which we observe an accidental thing scientifically was born from a gamble》

There are plenty of phenomena which we find an accidental thing in the world. When throwing a dice,and it was 1,we think it …

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【A way in which we get along with Lady Luck 運命の女神】

《The theory of probability in which we observe an accidental thing scientifically was born from a gamblie》

When throwing a dice,it was 1. We think it happened to do. Both of winning in a public lottery and an ivory ball rolled and stopped at some number in a roulette are the same,but considering carefully,the phenomena in which we think it happened to do have some cause,so the results occurred.

The way of throwing the dice caused the 1. The instant when the dice was throwing,if we can know all the data like its speed or rolling,and its surrounding conditions like the flow of the air and situation of the place where the dice was thrown completely,the 1 didn't happen to do. It was done inevitably.

However the way in which we can discover its inevitability is beyond the human’s ability,so we are forced to think it happened to do. Instead,the idea of probability has been born as the one which makes up for being lack of the human’s…

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【A way in which we get along with Lady Luck】

《A theory of probability in which we think an accidental thing scientifically was born from a gamble》

The idea of probability has been born as the one which makes up for being lacking of wisdom of us the human. The idea is the next one. Even if we can't recognize each phenomenon clearly,when observing them repeatedly totally,we can find a fixed law there.

It is said that an Italian mathematician,Cardano is the first one who researched the probability systematically. He wrote the first book on the probability and calculated the probability in which throwing the dice twice and three times precisely. This book became a handbook for gamblers.

The theory of probability which was born from the study of gambling was established its base by Pascal,and Fermat and the theory has developed into an accurate science. By the way,Pascal is said to love the gamble very much.

We handle with the statistics as if it were a brother for the theory of…

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【A way in which we get along with Lady Luck】

《A theory of probability in which we think an accidental thing scientifically was born from a gamble》

We handle the statistics as if it were a brother for the theory of probability,and both of them are indispensable tools for our everyday life in this society. Two of them are related to various fields like economy,social science,the manifucture industry,politics,psychology,biology,and insurance.

Whether or not we're conscious of it,it means that the world where we live has a relation to the probability. When doing business,when thinking,“What kind of way does lead us to a big chance for a success?”both of the study of the theory of the probability and statistics are strong weapons for us.

【When throwing 6 times,at least once…?】

《The base of the probability is a law of large numbers. We should be careful of a wrong usage》

When watching a baseball game on a TV program,a commentator often says like the next.“The batter retired three…”

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【WHen throwing 6 times,at least once…】

《The base of the probability is a law of numbers. We should be careful of its usage.》

When watching a baseball game on a TV program,a commentator often says like the next.,A“The batter has retreated three times continuously today,but his batting average is over 30 percent,it is about time he was likely to got a hit.” His comment looks plausible.

There is a similrar thing which we ofte seem to hear like the next.“Though having thrown a dice 5 times,the 1 didn't appear at all. The probability of the 1 is one-sixths,so the probability of the I in the next time is high.” We say like that,but both of them are the one which we interpret the theory of the probability by mistake.

As for the probability,instead of talking each phenomenon,when attempting something plenty of time and looking at them entirely,we inquire into the proportion in which something happens. It doesn't mean that the theory of the probability shows us something happens,for the…

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【When throwing 6 times,at least once…】

《The base of Probability is a law of large numbers. We should be careful of it.》

…for the dice doesn't memorize anything at all.

We need to bear this theory of the probability in our mind firmly. A batter whose batting average is about 30 percent has always about 30 percent of the probability even if he repeated to retire many times,and the probability of the 1 of the dice is one-sixth.

However,when playing baseball,if being at the bat four times,then as the batter may be get used to the ball which the pitcher throws a little,he may get a hit,but it's other factor from the probability.

The probability that the 1 is shown is one-sixth is fixed from the definition of the probability which is shown by Laplace like the next.

“A dice has six kinds of number,and if its cube is made precisely,each number which is shown will be as same degree as its number of the surface. When throwing the dice,there are cases in which 6 kinds of numbers are…”

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【When throwing 6 times,at least once…】

《The base of probability is a law of large numbers.We should be careful of it.》

When throwing the dice,there are cases in which 6 kinds of number is shown as same degree as the number of the surface,so it's one-sixth.

When making an experiment actually,we can understand the theory,though it's not useful in a few dozens of trials,but if repeating to do it hundredth times or thousandth times,it has tendency that the probability approaches one-sixth. We call the property a law of large number on the probability,and it indicates essential nature on the probability.

Though it seems to be persistent,I find it important,so I'm going to say it repeatedly,the author said like that,so I will.

The probability that the 1 is shown is indicated when throwing the dice doesn't guarantee the next thing. When throwing the dice,the 1 is shown once.

By the way,without knowing the theory of the probability,we can enjoy a lottery ticket,but once having known…oh！

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【The way of thinking on a permutation 順列 and combination.】

《In probability,it's important for us to count a number of a case.》

I'm going to prepare for tools which we can figure the probability in this chapter,the author said like that.

When trying to grasp a phenomenon as probability,we should be careful about the next things. When looking around the whole thing,we can think what kind of case,and we should pay attention to which case we should handle as a question.

When the probability on each case is considered to be equal,Laplace,a mathematician,made use of the numbers in every case and defined the probability,so in probability counting the number of cases is base. There are concepts of permutation and combination as tools for counting numbers. I'm going to express on the permutation at first.

The permutation is number which show us kinds of row,as the name indicates. When there are three people,A,B.C,let's think of how many there are kinds of row on the three.

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【The way of thinking on permutation and combination.】

《In probability, it's important for us to count numbers of a case.》

When choosing the first one,anyone is all right,so there are three kinds of permutations ,A,B,C. When the first one is set up,the second one is either of the rest two. When the second one is set,the rest one is the last one.

As a result, in the way of three kinds of row there are three permutations on the head. As to the permutation of the second one,each of the first one has two. If the first one and the second one are set up,the last one will also be done inevitably. I'm sure I should show you with an illustration concretely,but to my sorrow,I can't,so please imagine it. If showing it with an equation,it's 3×2×1=6

When choosing three permutations from six things,its equation is 6×5×4＝120. As you perceive,we choose what kinds of permutations from other many things,we can understand it at once even if they increase so much.

In the next,I'm going to express…

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【The way of thinking on permutation and combination.】

《In probability,it is important for us to count the number of a case.》

I'm going to express on the combination this time.

In permutation,A,B.C,and A,C,B,are different,but the two members which composed the permutation are the same. In this way,when they are the same as member,without distinguishing them we count their number. It's the combination.

For example,the combination which we choose three from other five of A,B,C,D,and E is 10. Its equation is shown in the book,but to my sorrow,I can't express why it is. Without having basic knowledge on math,I find it hard.

When there are some numbers and we express them with N,its permutation is described with the next equitation,N!,which is 1×2×3…×N. We call it N factorial,nの階乗.

By the way when trying to express some equations,small size of numbers are added to a larger number either on the upper left,or right,and lower left or right. I find it complicated.

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【It is said that even a poor shot will hit the mark if he tries often enough,but is it true?】

《As for an issue on “at least”,it becomes easy if thinking complementary things 余事象.》

The season when some students who try to take an exam to enter of their preference have a hard time comes every year. A student who takes an entrance exam,but its parents are worried about.

There are various. Weak-willed ones,strong-willed ones,prudent ones and capricious ones. Some of them focus on a single school which they want to enter and pay no attention to other schools at all. Others take the entrance exam for several schools in case they fail to be admitted to the school of their choice,for they are very careful.

Let's think of safe ways of passing the exam,how and what kind of school they should choose. Some people say like the next as a joke.“The more the ones try to take the entrance exam,the better we go with more students who are worse than us on academy ability.”,though it's out of question

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【It is said that a poor shoot will hit the mark if he tries ofte enough,but is it true?】

《As to an issue on at least,it becomes easier if thinking complementary things. 》

Let's think of more positive way as possibility. In short it's the more they take the entrance exam for schools,the better,as the proverb says,“A poorly shoot will hit the mark if he tries often enough.” If saying it's natural,it is,but considering it as a point of view of possibility,we find that its effect is enormous unexpectedly.

Let's suppose that the percentage of passing an entrance exam for 5 schools are 0.3,0,4,0.5,0,6,and 0,6,judging from the competition to enter the schools or deviation values 偏差値. Then let's think of how much possibility of passing the entrance exam on at least one school there is.

At first we should find the each possibility of being unable to pass the entrance exam for 5 schools. It's a complementary thing,余事象. When thinking of a single phenomenon,there are other phenomena,we call…

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【It is said that a poor hit will shoot the marke if he tries often enough,but is it true?】

《As to an issue on at least,it becomes easy if thinking complementary events.》

When thinking of a single phenonenon,there are other phenomena,and we call the other ones complementary events. The probability of being unable to pass the entrance exam for each school is 1ーthe percentage of being able to pass the entrance exam.

When it's 0.7, 0.6, 0.5, 0.4, and 0.4,so the probability of being unable to pass the entrance exam for all the schools is 0.7×0.6×0.5×0.4×0.4=0.0336. When passing the entrance exam for at least one school,it's all right as long as the one doesn't pass all the entrance exam for 5 schools,1ー0.0336=0.9664,it's the probability for being able to pass the entrance exam.

Its almost close to 1,so I'm sure it is certain the one can pass the entrance exam at least 5 of 1. We come across the situation in which we say at least in various affairs.

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『Supplementation』

A jet plane has several pairs of engine. As long as at least either of one of the engines functions,without falling down on the ground the jet plane can manage to go on flying. It is designed like that. That is why the more engine,the safer the jet plane,though it doesn't always mean that it doesn't drop on the ground,I'm sure.

【An untrustworthy intuition.】

《There is a class composed of 40 students. The probability that a student’s birthday is same with other one is 89%.》

When thinking over the probability on an issue,an expectation with an intuition is sometimes far different from other answer with a study of theory on the probability. It may be that then the significance of the theory on the probability is large. It is a question on a birthday that we often adopt its instance.

Let's suppose there was a class composed of 40 students,and we bet on whether or not a student’s birthday is same with other one. I'm going to study the question with the probability.

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【An untrustworthy intuition.】

《There is a class composed of 40 students. The probability that a student’s birthday is same with other one is no less than 89%.》

The question is whether or not at least the birthday of the two students are same,we should think over the complementary event 余事象,first of all. The complementary event,in short,in this situation,it means all the students’ birthday is different. Then I'm going to get rid of 29th February in order to make the things simple.

I'm going to think over from the case of less students in turn. As for the two students, if a student’s birthday is fixed, the birthday of the other is among the other rest days in a year,so the probability that the two students’ birthday is different is 364-365ths.

What about if there are 3 students? If the two students’ birthday is different from the third one,the rest days are 363. As a result,the probability that the birthday of the first of two students are different from the third one is 363-365ths.

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【Untrustworthy intuition.】

《There is a class composed of 40 students. The probability that a student’s birthday is same with other one is no less than 89%.》

Each day when the three students were born are irrelevant each other,so the probability that the birthday of the three students are different is 364-365ths×363-365ths.

Even if the number of the students increases by 40,we can calculate its probability. At least the two students’ birthday is the same is complementary event,if deducting its probability from 1,we can find the value we want.

If calculating the probability actually,the probability that the students who have the same birthday is no less than 89%.

By the way,my intuition prevents me from thinking it over mathematically, it can't be! To my sorrow,I can't express it theoretically. The ones who have the wrong impression like me may have prevented the society from developing. Some of them may have insisted on the Ptolemaic theory 天動説 even after the Copernican theory.

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【When drawing lots,the earlier,the better?】

《We can see it well if drawing the illustration of possibility.》

Almost all of the shopping districts do a lottery for a big sale at the end of a year. When winning a first prize,a bell is rung and the names of prize winner are posted. The number of the first prize are fixed,so the names who won all the first prizes are shown,no one can win the first prize,so everyone may want to draw lots earlier.

However others may think the latter,the better.for it is said that taking the last helping will bring you luck 残り物には福がある. When drawing lots,is there advantageous or disadvantageous according to order of drawing lots?

For example,let's suppose that there were ten lots and there were three prizes among the ten. Then two people draw the lots in turn. Let's compare the probability of winning the prize,which is better,the former or the latter?

The probability that the former draws the prize is clearly three-tenths,on the other hand the latter is …

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【When drawing lots,the faster the better?】

《We can see it well if drawing the illustration of probability.》

The probability that the former draws the prize is three-tenths clearly,on the other hand the other probability that the latter draws the prize are two cases,for its situation changes,according to whether or not the former draws the prize.

When the former draws the prize,there are two prizes among the rest of nine lots,so the probability that the latter draws the prize is two-ninths.Multiplying three-tenths by two-ninths is equal to one-fifteenths,so the probability that the latter draws the prize then one-fifteenths. It's the situation is A.

When the former doesn't draw the prize,there are three prizes among the rest nine of the lots,then the probability that the latter draws the prize is three-ninths. The probability that the former doesn't draw the prize is seven-tenths. Multiplying three-tenths by three-ninths is equal to seven-thirtieths. It's the situation B.

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昔お世話になった先生ですか？

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To a person who sent its message in 147.

147にメッセージを送ってくれた方に。

Thank you for your response,and how do you do?

レスありがとう。😊 初めまして、かな？

A person whose handle name was 中学生 used to come to my former thread before.

ハンドルネームが中学生の人が自分のスレに来た事があったけど…🤔

I'm so selfish and I can't understand what other one’s feeling at all,so all of the people are apt away from me.

自分は我がままで人の気持ちが分からないので、みんな自分から去って行く傾向にあります

It doesn't always mean that the person whose handle name was 中学生 under the care of mine.

中学生というハンドルネームの人を世話した、というほどの事はないけど…💦

Are you the same one?

同じ人ですか？😁

• << 151 そうだと思います！！ 当時も先生とお呼びしていました。

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【When drawing lots,the faster the better?】

《We can see it well if drawing the illustration of the probability.》

Both of the two situation never happens at the same time. When the one happens,the other one never does. If adding the two probability,it's three-tenths. As a result the probability that the latter draws the prize is three-tenths,though to my sorrow I'm not sure why it is. I can understand until its halfway.

In short the probability that both of the former and the latter draw the prize is three-tenths. It seems that when drawing the lots,the faster the better,but it turns out to be superstition. In general the theory can stand up in every situation,but there is one thing to which we paid attention.

When calculating the probability of the lots,it's equal as long as we look over the whole lots. After some people drew the lots and thinking over the probability later,the probability changes from the former one.

As for the former instance,after the former draws the prize…

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【When drawing lots,the faster the better?】

《We can see it well if drawing its illustration.》

As for the former instance,after the former drew the prizes,the probability that the latter draws the prize is two-ninths,so the latter is less advantageous than the former.

On the other hand,after the former failed to draw the prize,the probability that the latter draws the prize is three-ninths,so the latter is more advantageous.

When assuming that the former can succeed in drawing the prize,a pessimist thinks it disadvantageous if drawing the lots later. Contrary to it,an optimist assumes that the former fails to draw the prize,so it thinks it advantageous if drawing later. There seems to be the tendency like that.

The probability of human activity isn't as same as the one of a dice nor the lots,for it depends on psychological factors.

I said that we can see it well if drawing its illustration,but I can't draw it,so the of the response ended up being different from the subtitle.

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