Probability Theory and Anthropogenic Factor

Introduction

Among people there is an opinion that a person who entered the Faculty of Mathematics will definitely leave from there a teacher of mathematics. It was not I who invented it, it was by experience, because a rather large number of not very educated people asked where I was going to go to work after graduation from the university. Of course, you can find and much more extensive areas of application of their knowledge. One of them is related to the theory of probability. I do not want to delve into the complex details of the subject, because people who do not have the necessary mathematical base are likely to get confused. But I don’t want to talk about anything at all. Therefore, I want to write about the relationship of man and this very theory of probability, and in simple, understandable to any language. If interested - please under the cat.

general information

I will enter a couple of definitions to formalize a little.
1) If there are several possible random outcomes that are "equally possible" among themselves, then the classical probability is the ratio of the number of "good" random (elementary) events to their total number. For example, if you have 5 balls, 2 of which are white, then the probability of taking exactly the white ball will be 2/5.
2) A random variable is a quantity that as a result of experience takes one of a multitude of values, and the occurrence of one or another value of this quantity before its measurement cannot be accurately predicted. A classic example is dice. Throwing it, you can randomly get one of six possible values.
3) The mathematical expectation of a random variable is the sum of all its possible values ​​multiplied by their probability. In simple terms, this is the “average value” of the random variable being taken. For a die, it is equal to (1 + 2 + 3 + 4 + 5 + 6) * 1/6 = 3.5. What does this give us? The fact that throwing a bone is a lot (for example, 100) times, on average, 3.5 times will fall each time, and in total, approximately 100 * 3.5 = 350 will fall out. With an increase in the number of shots, the relative error of the real result and its expectation multiplied by the number of shots will decrease more and more.

The essence

Now the essence of what I, in fact, wanted to tell: mathematical calculations predict various events quite well, if they are not directly dependent on a person’s choice. If the anthropogenic factor interferes, then it is necessary to make some plans, relying only on the theory of probability, with caution. I will give a couple of simple examples. Perhaps they are a bit flimsy, but simple and understandable.

Coin

Occasion times

During a couple at the university (class at work, working day) you got bored and you offered your neighbor (colleague) to play the following game: throw a coin; if an eagle has fallen - your friend pays you 5 rubles, if he has a tail, then you pay 5 rubles. Out of boredom, a person can agree. You will play like this all day, and in the end both will remain with practically the same money that they were originally. The probability of either side of the coin losing 1/2 and, as a result, the expectation of your winnings is zero. So the average win / loss will be in the region of plus or minus 10 rubles. Well, maybe a little more. In any case, for the budget is not critical.
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Case two

The situation is the same, but you offered to pay for the loss not for 5, but for 1000 rubles each. Most likely your friend / colleague will refuse. For you do not want to just lose a significant amount of money.

What has changed? The expected payoff is still zero. From the point of view of mathematics, everything is practically the same. And here the human factor has already intervened, and your plan to pass the boring day has failed.

Lottery

You have decided to organize a lottery. They made tickets at the price of 10 rubles with a fifty percent chance of winning 15. The mathematical expectation of winning is 15 * 0.5 = 7.5 rubles, but since the ticket costs 10, it turns out -2.5 rubles. Yes, the client is not very profitable, but you are not going to work at a loss, right? However, such a lottery is unlikely to be popular. Because it is proposed to spend 10 rubles with a dubious chance to win 15. The difference is small.

You change the conditions and make the lottery almost charitable. Now win 25 rubles. The mathematical expectation of winning minus the cost of the ticket - 2.5 rubles! You will even be at a loss! But the majority of people still will not favor your lottery, because winning is a little more than the ticket price. Only schoolchildren who do not have enough stuff for ice cream will play the lottery.

At the same time, your enterprising neighbor also arranges his lottery. Only he takes the ticket 50 rubles, and the prize is a car worth 500,000 rubles. The probability of winning is 0.001%. The expectation of winning is 5 rubles. Minus the ticket price, we get -45 rubles. Yes, the neighbor's lottery is just looting! Selling a sufficiently large number of tickets, even playing the car at the same time, he still notably rich. But people can easily buy tickets, because what is 50 rubles before the prospect of getting a good car for free?

The reader may decide that the matter is simply in the amount of the winnings. But this is far from necessary. I will give one more rather far-fetched, but illustrative example:

Very large lottery

They offer you a gift of unheard of generosity. "Super Lottery". One of two, to choose from. You can play it only once . In the first "lottery" you are guaranteed to pay a million dollars. And in the second with a 50% chance you will get 2 million, with a 40% chance a million and with a 10% chance you leave with nothing. The expectation of winning the first "lottery" is 1 million. In the second - 1.4 million. But what will you choose? Maybe someone will choose the second option, but conducting a survey among a certain number of people will show that most probably will choose the first option. After all, as they say, a tit in hands is better ... Especially, if a tit is a million, and in the second “lottery” there is a chance not to get anything. And a hypothetical 2 million does not solve anything.

Last example

You have written a good and high-quality application for your phone. Spent a lot of manpower and money. You expose it to the store for $ 9.99. For such a quality product, this is not very much, it seems. Yes, and you need to pay off and earn extra money. But nobody buys your application. People thought it was expensive. Downloads are minimal. You are desperate to reduce the price to $ 0.99. Furor, people only download your program this way, but there is not enough money from them. Then you raise the price again, but now to $ 4.99. Yes, the download stream is reduced relative to the lowest price, but still it is higher than at the beginning. And lo and behold, you get quite a good profit from your product. From the point of view of primitive calculations, the number of people who want to have this program has always been the same. However, you have reduced the price relative to the original, and profits have increased. Again, purely human factor.

So what is the result?

As a result, on the one hand, mathematical calculations can give results that are not quite obvious from the point of view of mathematics. A person can choose from almost the same conditions strictly one, and among several offers take the most unfavorable for himself. Why? That's the way a person is. The benefit of one particular person may not always be just calculated.
On the other hand, if you look from the point of view of various firms, corporations, etc., then having many clients, you can get good money, even if from the point of view of mathematics, the offer to the client is not the most profitable. That is why there are banks, lotteries, insurance companies. And people take loans at wild interest rates, buy questionable lottery tickets and insure things that are likely to be all right.
So, trying to use some kind of “stupid” calculations for people, thinking like a robot, most likely, nothing useful and useful will come of it. But if you act wisely, imagine yourself in the place of other people, then you can move mountains and earn billions with the help of mathematics.

In general, think like people, but do not forget about mathematics too.

PS If I somewhere wrote some nonsense (I took examples from my head), do not kick much, tell me. I am interested in the opinions of other people.