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Raise 100 dollars or pass by? Probability theory in everyday work

It's amazing, but we often rely on intuition than on common sense and calculation. Unfortunately, this concerns not only personal life, but also work. Remember the old story about whether Bill Gates is worth picking up a hundred dollar paper from under his feet? The jokers counted on how much Gates earned per minute and claimed that picking up a piece of paper he was wasting his time inefficiently.

Do you think it cost him to raise this money? Do not rush to answer. Let Gates earns 64 thousand dollars per minute. This is a conventional number. Do I need to raise a piece of paper in a hundred dollars? Think about it.

And here we get the trap that was originally laid in the very formulation of the question. Gates does not spend his personal time in order to multiply the state, it makes money in bank accounts. Therefore, bending over, Bill will receive an extra one hundred dollars and this is a winning situation for him. Feel the difference in the formulation of the question? I do not take into consideration that emotionally, like any other person, he will be glad that he has found such a bill. And this will be due to the fact that finding a hundred dollars is a rare success and few people can boast of it. Did you find a hundred dollars? Just answer honestly. If so, what did you feel? The probability of such an event is extremely small, hence the high emotional color.
On the bus and gorilla on the field, the show on TV and the opening of the door with a racing car that can be taken home. Probability theory in action.

In our work, there are frequent situations when it is necessary to make a decision and we face two types of problems. Lack of information. As well as misinterpretation of the initial conditions, inattention to details. The second type of problems can be corrected by careful preparation. Let's dwell on such problems.

Problem number 1. Incorrect interpretation of baseline

At the institute, we conducted a mathematical test of the ability to count in the mind. You can practice in it, with your friends and acquaintances, it takes, literally, a few minutes.

The challenge sounds like this. You tell your interlocutor to read carefully, because the test is related to mathematics. And you begin to say that at the last stop of the bus there was nobody in it. Then 5 people sat in it. At the next stop, 3 people got off, and 14. entered. The next stop is minus 3, plus 11. Then another stop is -4, +6. And so on. And again the final stop.

As a rule, they start counting the number of people, asking you to repeat how many people left, how many are left. But your question sounds different - “How many stops did the bus pass?”. The units answer this question correctly, as they initially expect a typical action, namely, calculations, since the math test was also mentioned. This is a typical test showing that a person does not specify the initial conditions, does not pay attention to details, and acts according to his own understanding of the test. Which, as we see, turns out to be wrong.

When you conduct the test, do not call a stop at all, this facilitates the subsequent calculation, and also spoils the test. The number of stops should be quite large (more than 10), and you should also consider not to be mistaken with the number of those who went out and stopped.

Another version of the test has already become a classic of the genre, it is a gorilla on the basketball field. The subjects are asked to calculate how many passes the players make, in the middle of the game a man in a gorilla costume passes through the players. About half of those who considered passes simply did not notice him. They focused on another task. And this is a feature of our psychology. Below is an example video from a classic study.

As a conclusion, I can say the following: it is very important to correctly and carefully evaluate the initial conditions. What to do, and most importantly why. And then act, but then we turn to the assessment of probabilities or item number 2.

Problem number 2. How to make the right choice

You have a lot of proposals for the conclusion of new contracts, you are not able to accept each of them. Some look more interesting, some are not so good. A situation of choice arises in full growth in which most of us rely on intuition, but not common sense and calculation. Recall the situation of choosing from working days for each of us is not difficult. But how do we choose? In such situations, I rely on probability theory, which helps to make the final decision. Unfortunately, in many higher educational institutions they do not teach probability theory, or they do it so badly that they discourage anyone from knowing about this subject. However, probability theory works and helps to make decisions. Let me interest you in this theory and encourage you to read more, with just one example, which has become a classic.

Monty Hall's challenge

In the game show, participants must select one of the three doors. Behind one door is a car, behind the other two there is nothing. The participant chooses the door, and the presenter, who knows what is behind each door, opens one of the remaining ones, of course, a pacifier. Then he says to the participant, “Will you change the door or choose another one?” The question that we consider is whether it is profitable for a participant to change the door or it is advantageous to leave his choice.

Before going further, please think and answer this question. Leave the door or change?

In 1990, this issue divided America into two camps. On the one hand, Marilyn Vos Savant was included in the Guinness Book of Records as a person with the highest level of intelligence equal to 228. On the other hand, mathematics and readers of the Sunday newspaper, in which Marilyn expressed her point of view on the question whether to change the door. She received tens of thousands of reviews, of which more than a hundred were written by graduated mathematicians and doctors of science. 92 percent of those who wrote thought Marilyn was wrong. Made your choice? Honestly write it on a piece of paper, and then share in the comments that you have chosen. Thanks in advance for your honesty.

The indignation of the majority was caused by the strategy proposed by Marilyn. She offered to change the door. Do not leave, namely, change, as this increases the chances of winning.

The answer to the Monty Hall problem
In the task of Monty Hall, there are three doors: one for something valuable, say a racing car, for two others - something much less interesting, for example, a Russian-Russian phrasebook. You have chosen door number 1. In this case, the space of elementary events is represented by the following three possible outcomes:

The car behind the door number 1
The car behind the door number 2
The car behind the door number 3

The probability of each outcome is 1 out of 3. Since it is assumed that the majority will choose a car, we will consider the first outcome to be a winning one, and the chances of guessing are 1 out of 3.

Further according to the scenario, the presenter, knowing that he is behind each of the doors, opens one door from those you have not chosen, and it turns out that there is a phrase book there. Since, while opening this door, the presenter used his knowledge of the objects behind the doors so as not to reveal the location of the car, this process cannot be called random in the full sense of the word. There are two options worth considering.

The first is that you initially make the right choice. Let's call such a case a “happy guess.” The presenter will randomly open either door 2 or door 3, and if you prefer to change your door instead of a smart one, you will become the owner of a phrasebook with the breeze of the trip. In the case of a “happy guess,” it is better, of course, not to be tempted by the proposal to change the door, but the probability of getting a “happy guess” is only 1 out of 3.

Second, you immediately point out the wrong door. Let's call such a case an “erroneous guess.” The chances of you not guessing are 2 out of 3, so an “erroneous guess” is twice as likely as a “lucky guess”. How is an “erroneous guess” different from a “happy guess”? With an “erroneous guess,” the car is behind one of those doors that you went around with your attention, and after another - a book. In contrast to the “lucky guess” in this version, the presenter opens the door that is not selected at random. Since he is not going to open the door with the car, he chooses the very door for which there is no car. In other words, in the “erroneous guess” the leader intervenes in what was called a random process until then. Thus, the process can no longer be considered random: the presenter uses his knowledge to influence the result, and thus denies the very notion of randomness, ensuring that the participant will receive a car when the door is changed. Because of this kind of intervention, the following happens: you find yourself in a situation of “erroneous guess”, and therefore you win
when you change the door and lose, if you refuse to change it.

The result is: if you find yourself in a situation of “happy guess” (the probability of which is 1 out of 3), you win, provided you remain with your choice. If you find yourself in an “erroneous guess” situation (probability 2 out of 3), then you win under the influence of the lead’s actions, provided you change the initial choice. So, your decision comes down to a guess, what situation will you find yourself in? If you feel that your initial choice is led by the sixth sense that destiny is guiding you, maybe you should not change your decision. But if you are not given to tie spoons with knots only by the power of thought, then surely the chances that you have fallen into the situation of an “erroneous guess” are 2 to 1, so it is better to change the door.

The TV show statistics confirm that those who changed their choice won twice as often. Voila

I hope that this example will make you think about how to quickly pick up a book on the theory of probability, as well as begin to apply it in your work. Believe me, this is interesting and fascinating, and there is a practical sense. I hope Friday's thoughts about psychology, the premises of problems and probability theory did not make you bored.

PS Description of the problem Monty Hall took from the book "Imperfect Accident" Leonard Mlodinov. I recommend it to read it, it is a scientific one.

Source: https://habr.com/ru/post/101695/

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