# Raise $ 100 or pass by? Probability Theory in Everyday Work

Surprisingly, we often rely on intuition rather than common sense and calculation. Unfortunately, this applies not only to personal life, but also to work. Remember the old story of whether Bill Gates should pick up a hundred-dollar piece of paper from under his feet? The jokers calculated how much Gates earned per minute and claimed that raising a piece of paper he wastes his time inefficiently.

What do you think, should he raise this money? Do not rush to answer. Let Gates earn 64 thousand dollars a minute. This is a conditional number. Do I need to raise a piece of paper in a hundred dollars? Think about it.

And here we get a trap that was originally laid in the very formulation of the question. Gates does not spend his personal time in order to increase his fortune; money in bank accounts makes it. Therefore, bending down, Bill will receive an additional hundred dollars and this is a winning situation for him. Feel the difference in posing the question? I do not take into account that emotionally, like any person, he will be glad that he found such a bill. And this will be due to the fact that finding a hundred dollars is a rare success and few can boast of it. Did you find one hundred dollars? Just answer honestly. If so, what did you feel? The probability of such an event is extremely small, hence the high emotional coloring.

About the bus and the gorilla on the field, shows on TV and the opening of the door with a race car that you can take home. Probability Theory in Action.

In our work, there are frequent situations when we need to make a decision and we are faced with two types of problems. Lack of information. As well as an incorrect interpretation of the initial conditions, inattention to details. The second type of problem can be corrected by thorough preparation. Let's dwell on such problems.

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

The task sounds like this. You tell your interlocutor to carefully consider, since the test is related to mathematics. And you begin to say that at the final stop of the bus there was nobody in it. Then 5 people got into it. At the next stop, 3 people got off, and 14 came in. Next stop minus 3, plus 11. Then another stop -4, +6. Etc. And again the final stop.

As a rule, they begin to count the number of people, asking you to repeat how many people came out, how many are left. But your question sounds different - "How many stops did the bus drive?" Units answer correctly to this question, since they initially expect a typical action, namely, calculations, since you also mentioned this test for mathematics. 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 stops in any way, 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 people who went out and went in.

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

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

You have a bunch of proposals for new contracts, you are not able to accept each of them. Some look more interesting, some are not so good. The situation of choice in which most of us rely on intuition, but not common sense and calculation, is fully expanded. It will not be difficult for each of us to recall situations of choosing from workdays. But how do we choose? In such situations, I rely on probability theory, which helps to make the final decision. Unfortunately, many higher education institutions do not teach probability theory, or do it so poorly that they discourage all desire to know 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 that has become classic.

In the TV quiz participants must choose one of three doors. There is a car behind one door, there is nothing behind the other two. The participant chooses the door, and the presenter, who knows what is behind each of the doors, opens one of the remaining ones, of course a dummy. Then he tells the participant, “Will you change the door or choose another?” The question that we will consider is whether it is beneficial for the participant to change the door or is it beneficial to leave his choice.

Before moving on, please think and answer this question. Leave the door or change?

In 1990, this issue divided America into two camps. On the one hand, there was Marilyn Vos Savant, who was included in the Guinness Book of Records as the person with the highest level of intelligence equal to 228. On the other hand, mathematicians and readers of the Sunday newspaper, in which Marilyn expressed her point of view on the question of whether or not to change the door. She received several tens of thousands of reviews, of which more than a hundred were written by certified mathematicians, doctors of sciences. 92 percent of those who wrote believed that Marilyn was wrong. Have you made your choice? Write it down honestly on a piece of paper, and then share in the comments what you have chosen. Thanks in advance for your honesty.

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

problem has three doors: one is something valuable, say a race car, the other two are 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:

Machine behind door No. 1

Machine behind door No. 2

Machine behind door No. 3 The

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

Further, according to the scenario, the host, knowingly knowing what is behind each of the doors, opens one door from those not chosen by you, and it turns out that there is a phrase book there. Since, when opening this door, the facilitator used his knowledge of objects behind the doors so as not to reveal the location of the machine, this process cannot be called random in the full sense of the word. There are two options to consider.

First, you initially make the right choice. We will call such a case a "happy guess." The host will randomly open either door 2 or door 3, and if you prefer to change your door, instead of a smart one, with a breeze of travel you will become the owner of the phrasebook. In the case of a “happy guess” it is better, of course, not to be tempted by the offer to change the door, however, the probability of a “happy guess” is only 1 out of 3.

Second, you immediately point to the wrong door. We will call such a case an “erroneous conjecture”. The chances that you will not guess are 2 out of 3, so a “mistaken guess” is twice as likely as a “happy guess”. How is a “mistaken guess” different from a “happy guess”? With a “mistaken guess”, the car is located behind one of those doors that you ignored, and behind the other is a book. In contrast to the “happy guess” in this version, the presenter opens an unselected door not 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 conjecture”, the facilitator intervenes in what until then was called a random process. Thus, the process can no longer be considered random: the leader uses his knowledge to influence the result, and thereby denies the very concept of randomness, guaranteeing that when changing the door, the participant will receive a car. Due to such an intervention, the following occurs: you find yourself in a situation of a “mistaken guess”, and, therefore, win

when changing 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 that you stay with your choice. If you find yourself in a situation of “erroneous conjecture” (probability 2 of 3), then under the influence of the presenter’s actions you win, provided that you change the initial choice. So, your decision comes down to a hunch, what situation will you find yourself in? If you feel that your initial choice is driven by the sixth sense that fate itself guides you, maybe you should not change your mind. But if you are not given to tie the spoons with knots only by the power of thought, then the chances that you will find yourself in a situation of “mistaken guess” are 2 to 1, so it is better to change the door.

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 probability theory, and also begin to apply it in your work. Believe me, this is interesting and exciting, and there is a practical sense. I hope Friday’s reflections on psychology, prerequisites of problems and probability theory, did not make you bored.

PS Monty Hall took the description of the task from Leonard Mlodinov's book “Imperfect Accident”. I recommend reading it, it is a scientific study.

What do you think, should he raise this money? Do not rush to answer. Let Gates earn 64 thousand dollars a minute. This is a conditional number. Do I need to raise a piece of paper in a hundred dollars? Think about it.

And here we get a trap that was originally laid in the very formulation of the question. Gates does not spend his personal time in order to increase his fortune; money in bank accounts makes it. Therefore, bending down, Bill will receive an additional hundred dollars and this is a winning situation for him. Feel the difference in posing the question? I do not take into account that emotionally, like any person, he will be glad that he found such a bill. And this will be due to the fact that finding a hundred dollars is a rare success and few can boast of it. Did you find one hundred dollars? Just answer honestly. If so, what did you feel? The probability of such an event is extremely small, hence the high emotional coloring.

About the bus and the gorilla on the field, shows on TV and the opening of the door with a race car that you can take home. Probability Theory in Action.

In our work, there are frequent situations when we need to make a decision and we are faced with two types of problems. Lack of information. As well as an incorrect interpretation of the initial conditions, inattention to details. The second type of problem can be corrected by thorough preparation. Let's dwell on such problems.

**Problem number 1. Incorrect interpretation of the initial conditions**At the institute, we conducted a mathematical test for the ability to count in the mind. You can practice it, with your friends and acquaintances, it takes literally a few minutes.

The task sounds like this. You tell your interlocutor to carefully consider, since the test is related to mathematics. And you begin to say that at the final stop of the bus there was nobody in it. Then 5 people got into it. At the next stop, 3 people got off, and 14 came in. Next stop minus 3, plus 11. Then another stop -4, +6. Etc. And again the final stop.

As a rule, they begin to count the number of people, asking you to repeat how many people came out, how many are left. But your question sounds different - "How many stops did the bus drive?" Units answer correctly to this question, since they initially expect a typical action, namely, calculations, since you also mentioned this test for mathematics. 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 stops in any way, 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 people who went out and went in.

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

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

**Problem number 2. How to make the right choice**You have a bunch of proposals for new contracts, you are not able to accept each of them. Some look more interesting, some are not so good. The situation of choice in which most of us rely on intuition, but not common sense and calculation, is fully expanded. It will not be difficult for each of us to recall situations of choosing from workdays. But how do we choose? In such situations, I rely on probability theory, which helps to make the final decision. Unfortunately, many higher education institutions do not teach probability theory, or do it so poorly that they discourage all desire to know 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 that has become classic.

**Task Monty Hall**In the TV quiz participants must choose one of three doors. There is a car behind one door, there is nothing behind the other two. The participant chooses the door, and the presenter, who knows what is behind each of the doors, opens one of the remaining ones, of course a dummy. Then he tells the participant, “Will you change the door or choose another?” The question that we will consider is whether it is beneficial for the participant to change the door or is it beneficial to leave his choice.

Before moving on, please think and answer this question. Leave the door or change?

In 1990, this issue divided America into two camps. On the one hand, there was Marilyn Vos Savant, who was included in the Guinness Book of Records as the person with the highest level of intelligence equal to 228. On the other hand, mathematicians and readers of the Sunday newspaper, in which Marilyn expressed her point of view on the question of whether or not to change the door. She received several tens of thousands of reviews, of which more than a hundred were written by certified mathematicians, doctors of sciences. 92 percent of those who wrote believed that Marilyn was wrong. Have you made your choice? Write it down honestly on a piece of paper, and then share in the comments what you have chosen. Thanks in advance for your honesty.

The majority was outraged 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 task The Monty Hall**problem has three doors: one is something valuable, say a race car, the other two are 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:

Machine behind door No. 1

Machine behind door No. 2

Machine behind door No. 3 The

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

Further, according to the scenario, the host, knowingly knowing what is behind each of the doors, opens one door from those not chosen by you, and it turns out that there is a phrase book there. Since, when opening this door, the facilitator used his knowledge of objects behind the doors so as not to reveal the location of the machine, this process cannot be called random in the full sense of the word. There are two options to consider.

First, you initially make the right choice. We will call such a case a "happy guess." The host will randomly open either door 2 or door 3, and if you prefer to change your door, instead of a smart one, with a breeze of travel you will become the owner of the phrasebook. In the case of a “happy guess” it is better, of course, not to be tempted by the offer to change the door, however, the probability of a “happy guess” is only 1 out of 3.

Second, you immediately point to the wrong door. We will call such a case an “erroneous conjecture”. The chances that you will not guess are 2 out of 3, so a “mistaken guess” is twice as likely as a “happy guess”. How is a “mistaken guess” different from a “happy guess”? With a “mistaken guess”, the car is located behind one of those doors that you ignored, and behind the other is a book. In contrast to the “happy guess” in this version, the presenter opens an unselected door not 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 conjecture”, the facilitator intervenes in what until then was called a random process. Thus, the process can no longer be considered random: the leader uses his knowledge to influence the result, and thereby denies the very concept of randomness, guaranteeing that when changing the door, the participant will receive a car. Due to such an intervention, the following occurs: you find yourself in a situation of a “mistaken guess”, and, therefore, win

when changing 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 that you stay with your choice. If you find yourself in a situation of “erroneous conjecture” (probability 2 of 3), then under the influence of the presenter’s actions you win, provided that you change the initial choice. So, your decision comes down to a hunch, what situation will you find yourself in? If you feel that your initial choice is driven by the sixth sense that fate itself guides you, maybe you should not change your mind. But if you are not given to tie the spoons with knots only by the power of thought, then the chances that you will find yourself in a situation of “mistaken guess” are 2 to 1, so it is better to change the door.

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 probability theory, and also begin to apply it in your work. Believe me, this is interesting and exciting, and there is a practical sense. I hope Friday’s reflections on psychology, prerequisites of problems and probability theory, did not make you bored.

PS Monty Hall took the description of the task from Leonard Mlodinov's book “Imperfect Accident”. I recommend reading it, it is a scientific study.