V.Chernobrov, candidate of technical sciences, came to interesting conclusions in the course of studying the properties of time and the possibility of traveling into the past and the future. Thus, in particular, he writes:

“The present is a transition, the transformation of a multi-variant, easily changeable Future into a single-variant and unchanging Past. From this it follows that flights to the Past (with a "negative" density-velocity t/tо) and to the Future will take place in different ways.

To some extent, they can be compared with the movement of an ant along a tree: from any point in the tree (from the Present), only 1 path down (to the Past) and many paths up (to the Future) open for the ant.

However, among all the paths to the Future, there are undoubtedly the most probable options, improbable and almost improbable. The movement into the Future will be the more unstable and energy-intensive, the less probable this variant of the Future turns out to be.

In accordance with this “law of the tree crown”, a return to the Present is possible only if, while staying in the Past, the traveler does not interfere in what is happening around him and does not change the course of past History; otherwise, the time traveler will return to the parallel Present from the Past along another branch of History.

Penetration into the Future from the Present is difficult by choosing the branch of movement, but returning from any version of the Future to the Present is possible under any scenario of behavior. If there is no merger in front of you different options Stories".

Thus, even modern Scientific research confirm the multidimensionality of time and the multivariance of the future, as well as the possibility of moving to its various probabilities.

There is a hypothesis that key points the fate of each person, the so-called "forks" of probabilities, give rise to various "branches" of reality depending on our actions.

All these "branches" exist in the Universe at the same time. But a person can only exist on one such "branch", although sometimes there are cases of spontaneous transition from one "branch" of reality to another.

In favor of the existence of various probabilities of the future (“branches” of the Tree of Life, “grooves” of the Wheel of Time, etc.) is evidenced by the story that happened to Gustav and Johan Schroederman. It began in the spring of 1973, when the Schroedermann family (husband, wife and son) moved from Berlin to a farm near Salzburg.

The youngest of the Schroedermans ran around the neighborhood all summer and once found a rickety house in the forest, bypassing which he almost fell into an overgrown well, but clung to a bush in time. Returning home, he experienced a strange dizziness and at home immediately went to bed. The next morning, there was a knock on the door of the house, and when the boy opened it, he saw himself, wet and covered in mud.

It turned out that the whole past of both boys completely coincides, different probabilities of fate begin after the incident at the well, into which one of them fell, and the other survived.

It is possible that the strong stress and fear of the failed boy, thanks to an altered state of consciousness, moved him to another branch of reality, where he already existed, but did not fall into the well.

It is characteristic that later the parents gave the boys new names and each of them lived his own destiny: one was engaged in the export of beer, the other became an architect.

Rice. 7.2. Payoff matrix taking into account the probabilities of event outcomes

p i is the probability of the i-th variant of the outcome of events.

M j - mat. expectation of the criterion when choosing the j-th variant of action alternatives, determined by the formula:

The two approaches mentioned above make it possible to implement four different decision selection algorithms.

1. Decision based on the rule of maximum probability - maximization of the most probable values ​​of the criterion (profit or income).

2. Decision based on the rule of maximum probability - minimization of the most probable values ​​of the criterion (possible losses or direct losses).

3. Solution based on the maximization rule mathematical expectation(average value) criterion (profit or income).

4. Decision based on the rule of minimization of the mathematical expectation (mean value) of the criterion (losses or losses).

The examples we've looked at so far in this chapter have included a single solution. However, in practice, the result of one decision forces us to make the next one, and so on. This sequence cannot be expressed by a payoff matrix, so some other decision-making process must be used.

scheme decision tree used when you need to make several decisions under conditions of uncertainty, when each decision depends on the outcome of the previous one or the outcome of events.

When compiling a "tree" of decisions, you need to draw a "trunk" and "branches" that reflect the structure of the problem.

· Arranged "trees" from left to right. "Branches" refer to the possible alternative decisions that could be made and the possible outcomes resulting from those decisions.

· "Branches" emerge from nodes. Nodes are of two types.

The square node denotes the place where the decision is made.

The round knot marks the place where the various outcomes appear.

The diagram uses two types of "branches":

The first one is dotted lines coming out of the squares possible solutions, movement on them depends on the decisions made. On the corresponding dotted "branch" all costs caused by the decision are put down.

The second is the solid lines emerging from the circles of possible outcomes. Movement along them is determined by the outcome of events. The solid line indicates the probability of this outcome.

decision node.

branching node of event outcomes.

branches, the movement along which depends on the decision made.

branches, the movement along which depends on the outcome of events.

The search for a solution is divided into three stages.

Stage 1. A "tree" is being built (an example will be discussed in practical exercises). When all decisions and their outcomes are indicated on the "tree", each of the options is calculated, and at the end its monetary income is affixed.

Stage 2. Calculated and put down on the corresponding branches of the probability of each outcome.

Stage 3. At this stage, from right to left, the monetary outcomes of each of the "nodes" are calculated and put down. Any expenses encountered are deducted from expected income.

After the squares of "decisions" have been passed, a "branch" is selected that leads to the highest possible expected income for a given decision (an arrow is placed on this branch).

The other "branch" is crossed out, and the expected return is placed above the decision box.

Thus, at the end of the third stage, a sequence of decisions is formed leading to the maximum income.

In principle, as a criterion, it can act as a maximization of the mat. expectations of income, and minimization of mat. loss expectations.

of a person contains a certain plan with which the soul came here, all variants of the development of events, including. You can go there and see the consequences of important decisions that we make. For example, about changing jobs and lifestyles. This can be done both in independent meditations and in joint master-slave processes. Below is a description of how this was done in a session

Probability Lines

I project three branches:

1) stay in Moscow at an existing job;

2) sell or rent an apartment and go to Asia with friends in order to become a partner in their tourism business;

3) ideal option: I leave work, participate in the business of friends on a project basis, while having my own house, but not in Moscow (either Asia too, but different, or Eastern Europe, or Latin America - a large bright villa in which you can receive guests and conduct retreats), there are a couple - their own partnerships, and they have their own business.

We build all three branches as roads, see if there are branches.

The Moscow branch is a strong thick gray rope, dull and reliable, you won't break away, you won't get lost. Several thinner ropes come from the rope, some of them are brighter and more interesting, but none of them attracts, calls or glows. The feeling is that I still love Moscow, but this topic has become obsolete.

The branch with Asia and friends is very bright and visual, but short and liquid, or something. It lacks the potential to confidently turn around in the future. Not enough resource.

The ideal third picture is divided into several geographical points on the map, each with its own specific flavor. The third branch, inside which there is my own story, is the most attractive, of course, for me. She is not as tangible now as Moscow and not as colorful as the second, But she calls to her. And it glows, filled from the inside. Like a thin living ray, it pulsates and shimmers.

Choosing your path

In this version of events, I freely move around the world at will. My income is lower than in Moscow, but it is enough to not need anything and not deny myself anything, albeit in moderation. I come to projects with friends, they stay with me. I write something and work with people, I do it for pleasure. There is also some kind of secular business project, which is also more or less successful, and gives a stable income.

At the same time, there is a close person with whom we will jointly realize this story, in a pair. In order for it to manifest, not only my intention is needed, and some payment will be required on both sides, of course, as for any choice. As soon as you choose something, you automatically refuse something .. It's always scary and unsafe, besides. Payment as a waiver of existing comfort or freedom. Payment as permission to enter into your life something completely new and unknown, albeit tempting. Pure free will and purity of intentions on both sides. And there too - how it will turn out .. In a different vein (not on a pure will), this topic simply will not take off.

This entire process is currently in development. This branch is in the maturing stage, and if everything goes well, then it can fully manifest itself in my reality. See if there are obstacles or stones on this ideal line for me. I see a fallen tree, right on the road. It is fear and self-doubt. From the series - it's too good for it all to turn out that way, it doesn't happen like that, these are all illusions and fairy tales invented by itself. I'm clearing the road.

The next important step is to make your own final decision - whether it is necessary to throw attention there at all, into this dream branch, since it will not be so easy to “rewind” later. I understand for myself that one way or another I have been energizing it for a long time and activating it internally. And this is not even due to stubbornness or the desire to have it my way.

Much more subtle things and signs that signal that this is fate, no matter how loud it may sound. This branch is gradually becoming more and more tangible. It condenses, slowly and surely. Although, of course, it is still extremely uncertain and can collapse at any moment, but there is a feeling that she herself is coming to me, this thread.

Since it has long been designed and predetermined, ordered, one might say. And I understand where this is leading. And how it develops. And that this is the correct development of events. Even though sometimes I'm afraid to believe it.

And still very much it would not be desirable to cement this branch. Make it rigid and unambiguous .. There is no need to build a rigid binding into it to a certain place or occupation, or to something else. I want it to have a lot of elements: air, water, fire, earth, so that it breathes, so that it is flexible and indestructible - mobile, transformable and reconfigurable. And so that everything that happens in it would be the result of co-creation, not autonomous actions. In any case, this is a paired story, it cannot be born as coercion, the maximum correctness is important here - in no case should you impose or pressure .. Everything is free will. And then - where will he call *

Strengthening the branch with attention

I stretch a ray from my Spark in the direction of this branch, to the point where it aspires, I connect with it with my attention. Thus, the spark begins to work towards the realization of this goal, anchors itself in it. I may not be aware of this, but the work will be carried out: the formation of events in space will take place in such a way that this goal is as close as possible to my reality, to its implementation.

The Spark Beam transforms into a gravitational beam and attracts objects and events from that branch of probabilities to me like a magnet. The goal is getting very close, you can say I'm in it now. Like a teleport, when you do not try to move to a new place with your whole body, but materialize the desired space around you: you tune in to the target and attract it to you. And the closer it is to you, the more your will extends to its implementation. And already Iskra is responsible for shaping those events that will entail the embodiment of this branch in reality, will allow it to play.

I paint my future with the light of my Spark. It’s so cool there, in this line of probabilities there is a very beautiful story where I want to invite everyone to visit .. A large bright room filled with life, sun and air .. I give it fuel, charge it with potential so that it has the opportunity to manifest itself in reality. When you are ready to make a final decision or you need to see some answers on the development of this branch, you can simply remember this state of attraction, soak up the emotional atmosphere and mood of this room, feel the emotion of creativity and partnership. The emotion of creation is always love.

Manifestation and consolidation of the result

To capture that picture that looks so attractive, but unsteady now, you need to let light through it, pour in emotion, charge it with positive. Enter the state of ananda - a joyful upsurge, a loving and beloved being, in love and filled with love, and redirect this internal fuel into an ideal scenario.

Clear the path and remove questions. Align with other branches of reality surrounding me and the players involved, so that all this is synchronized in place and in time. Coincided with intentions, will and freedom of choice. Saturate all this with your own light, warmth and love for the realization of your future. creativity in the way you like. Expose the desired result in such a way that the image is imprinted with light on a sensitive film - the canvas of future events, burns its imprint in it as a light projection. And hold for a little while so that the effect is as bright as possible.

Now you need to process the created dream imprint so that it passes into the layer of material reality. The next step is stabilization. It is necessary to add a little energy of darkness and cold to the picture so that it crystallizes and acquires a more solid outline, passes from the state of a magical mirage into denser layers, consolidates and manifests itself.

Working with a negative print .. The result is literally fixed on a sheet of reality, approximately the same as when we project an image from an analog photographic film onto analog photographic paper, and then pour the developer and fixer in turn so that we can see in detail what we captured with the help of light and intentions and enter there when appropriate and timely.

Because for communication with the world and creative realization The throat chakra answers, I send a ray from the throat chakra to the chosen branch. Behind him asked for a ray from the second chakra, followed by the third. Then the rest of the chakras were connected, it turned out such a ray shower, like from a seven-color flower. I wash and dry everything that has turned out, I fill it with movement, the material energy of the earth, vision, all the qualities of life force and magnetism, I attract the branch of probability into my reality even more, I connect it directly with each of the chakra centers, I prescribe it there in them ..

* a person forgets that the future is multivariate and often adheres to template models (these are usually determined by numerology, astrology, etc.). In fact, each of us is a flow, and the flow needs to flow, not to get hung up on the frames, to easily let go of the old and let in the new, to adapt. Therefore, if you do such practices, in no case "cement" your intention, as the world always offers even more cool options that we ourselves may not even be aware of, especially now.

Reality is multidimensional, opinions about it are multifaceted. Only one or a few faces are shown here. You should not take them as the ultimate truth, because, but for each level of consciousness and. We learn to separate what is ours from what is not ours, or to extract information autonomously)

THEMATIC SECTIONS:
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What is a probability?

Faced with this term for the first time, I would not understand what it is. So I'll try to explain in an understandable way.

Probability is the chance that the desired event will occur.

For example, you decided to visit a friend, remember the entrance and even the floor on which he lives. But I forgot the number and location of the apartment. And now you are standing on the stairwell, and in front of you are the doors to choose from.

What is the chance (probability) that if you ring the first doorbell, your friend will open it for you? Whole apartment, and a friend lives only behind one of them. With equal chance, we can choose any door.

But what is this chance?

Doors, the right door. Probability of guessing by ringing the first door: . That is, one time out of three you will guess for sure.

We want to know by calling once, how often will we guess the door? Let's look at all the options:

1. you called to 1st Door
2. you called to 2nd Door
3. you called to 3rd Door

And now consider all the options where a friend can be:

a. Per 1st door
b. Per 2nd door
in. Per 3rd door

Let's compare all the options in the form of a table. A tick indicates the options when your choice matches the location of a friend, a cross - when it does not match.

How do you see everything Maybe options friend's location and your choice of which door to ring.

BUT favorable outcomes of all . That is, you will guess the times from by ringing the door once, i.e. .

This is the probability - the ratio of a favorable outcome (when your choice coincided with the location of a friend) to the number of possible events.

The definition is the formula. Probability is usually denoted p, so:

It is not very convenient to write such a formula, so we will take for - the number of favorable outcomes, and for - total outcomes.

The probability can be written as a percentage, for this you need to multiply the resulting result by:

Probably, the word “outcomes” caught your eye. Since mathematicians call various actions (for us, such an action is a doorbell) experiments, it is customary to call the result of such experiments an outcome.

Well, the outcomes are favorable and unfavorable.

Let's go back to our example. Suppose we rang at one of the doors, but it was opened to us stranger. We didn't guess. What is the probability that if we ring one of the remaining doors, our friend will open it for us?

If you thought that, then this is a mistake. Let's figure it out.

We have two doors left. So we have possible steps:

1) Call to 1st Door
2) Call 2nd Door

A friend, with all this, is definitely behind one of them (after all, he was not behind the one we called):

a) a friend 1st door
b) a friend for 2nd door

Let's draw the table again:

As you can see, there are all options, of which - favorable. That is, the probability is equal.

Why not?

The situation we have considered is example of dependent events. The first event is the first doorbell, the second event is the second doorbell.

And they are called dependent because they affect the following actions. After all, if a friend opened the door after the first ring, what would be the probability that he was behind one of the other two? Correctly, .

But if there are dependent events, then there must be independent? True, there are.

A textbook example is tossing a coin.

1. We toss a coin. What is the probability that, for example, heads will come up? That's right - because the options for everything (either heads or tails, we will neglect the probability of a coin to stand on edge), but only suits us.
2. But the tails fell out. Okay, let's do it again. What is the probability of coming up heads now? Nothing has changed, everything is the same. How many options? Two. How much are we satisfied with? One.

And let tails fall out at least a thousand times in a row. The probability of falling heads at once will be the same. There are always options, but favorable ones.

Distinguishing dependent events from independent events is easy:

1. If the experiment is carried out once (once a coin is tossed, the doorbell rings once, etc.), then the events are always independent.
2. If the experiment is carried out several times (a coin is tossed once, the doorbell is rung several times), then the first event is always independent. And then, if the number of favorable or the number of all outcomes changes, then the events are dependent, and if not, they are independent.

Let's practice a little to determine the probability.

Example 1

The coin is tossed twice. What is the probability of getting heads up twice in a row?

Solution:

Consider everything possible options:

1. eagle eagle
2. tails eagle
3. tails-eagle
4. Tails-tails

As you can see, all options. Of these, we are satisfied only. That is the probability:

If the condition asks simply to find the probability, then the answer must be given in the form decimal fraction. If it were indicated that the answer must be given as a percentage, then we would multiply by.

Answer:

Example 2

In a box of chocolates, all candies are packed in the same wrapper. However, from sweets - with nuts, cognac, cherries, caramel and nougat.

What is the probability of taking one candy and getting a candy with nuts. Give your answer in percentage.

Solution:

How many possible outcomes are there? .

That is, taking one candy, it will be one of those in the box.

And how many favorable outcomes?

Because the box contains only chocolates with nuts.

Answer:

Example 3

In a box of balls. of which are white and black.

1. What is the probability of drawing a white ball?
2. We added more black balls to the box. What is the probability of drawing a white ball now?

Solution:

a) There are only balls in the box. of which are white.

The probability is:

b) Now there are balls in the box. And there are just as many whites left.

Answer:

Full Probability

The probability of all possible events is ().

For example, in a box of red and green balls. What is the probability of drawing a red ball? Green ball? Red or green ball?

Probability of drawing a red ball

Green ball:

Red or green ball:

As you can see, the sum of all possible events is equal to (). Understanding this point will help you solve many problems.

Example 4

There are felt-tip pens in the box: green, red, blue, yellow, black.

What is the probability of drawing NOT a red marker?

Solution:

Let's count the number favorable outcomes.

NOT a red marker, that means green, blue, yellow, or black.

Probability of all events. And the probability of events that we consider unfavorable (when we pull out a red felt-tip pen) is .

Thus, the probability of drawing NOT a red felt-tip pen is -.

Answer:

The probability that an event will not occur is minus the probability that the event will occur.

Rule for multiplying the probabilities of independent events

You already know what independent events are.

And if you need to find the probability that two (or more) independent events will occur in a row?

Let's say we want to know what is the probability that by tossing a coin once, we will see an eagle twice?

We have already considered - .

What if we toss a coin? What is the probability of seeing an eagle twice in a row?

Total possible options:

1. Eagle-eagle-eagle
2. Eagle-head-tails
3. Head-tails-eagle
4. Head-tails-tails
5. tails-eagle-eagle
6. Tails-heads-tails
7. Tails-tails-heads
8. Tails-tails-tails

I don't know about you, but I made this list wrong once. Wow! And only option (the first) suits us.

For 5 rolls, you can make a list of possible outcomes yourself. But mathematicians are not as industrious as you.

Therefore, they first noticed, and then proved, that the probability of a certain sequence independent events each time decreases by the probability of one event.

In other words,

Consider the example of the same, ill-fated, coin.

Probability of coming up heads in a trial? . Now we are tossing a coin.

What is the probability of getting tails in a row?

This rule does not only work if we are asked to find the probability that the same event will occur several times in a row.

If we wanted to find the TAILS-EAGLE-TAILS sequence on consecutive flips, we would do the same.

The probability of getting tails - , heads - .

The probability of getting the sequence TAILS-EAGLE-TAILS-TAILS:

You can check it yourself by making a table.

The rule for adding the probabilities of incompatible events.

So stop! New definition.

Let's figure it out. Let's take our worn out coin and flip it once.
Possible options:

1. Eagle-eagle-eagle
2. Eagle-head-tails
3. Head-tails-eagle
4. Head-tails-tails
5. tails-eagle-eagle
6. Tails-heads-tails
7. Tails-tails-heads
8. Tails-tails-tails

So here are incompatible events, this is a certain, given sequence of events. are incompatible events.

If we want to determine what is the probability of two (or more) incompatible events, then we add the probabilities of these events.

You need to understand that the loss of an eagle or tails is two independent events.

If we want to determine what is the probability of a sequence) (or any other) falling out, then we use the rule of multiplying probabilities.
What is the probability of getting heads on the first toss and tails on the second and third?

But if we want to know what is the probability of getting one of several sequences, for example, when heads come up exactly once, i.e. options and, then we must add the probabilities of these sequences.

Total options suits us.

We can get the same thing by adding up the probabilities of occurrence of each sequence:

Thus, we add probabilities when we want to determine the probability of some, incompatible, sequences of events.

There is a great rule to help you not get confused when to multiply and when to add:

Let's go back to the example where we tossed a coin times and want to know the probability of seeing heads once.
What is going to happen?

Should drop:
(heads AND tails AND tails) OR (tails AND heads AND tails) OR (tails AND tails AND heads).
And so it turns out:

Let's look at a few examples.

Example 5

There are pencils in the box. red, green, orange and yellow and black. What is the probability of drawing red or green pencils?

Solution:

What is going to happen? We have to pull out (red OR green).

Now it’s clear, we add up the probabilities of these events:

Answer:

Example 6

A die is thrown twice, what is the probability that a total of 8 will come up?

Solution.

How can we get points?

(and) or (and) or (and) or (and) or (and).

The probability of falling out of one (any) face is .

We calculate the probability:

Answer:

Workout.

I think now it has become clear to you when you need to how to count the probabilities, when to add them, and when to multiply them. Is not it? Let's get some exercise.

Tasks:

Let's take a deck of cards in which the cards are spades, hearts, 13 clubs and 13 tambourines. From to Ace of each suit.

1. What is the probability of drawing clubs in a row (we put the first card drawn back into the deck and shuffle)?
2. What is the probability of drawing a black card (spades or clubs)?
3. What is the probability of drawing a picture (jack, queen, king or ace)?
4. What is the probability of drawing two pictures in a row (we remove the first card drawn from the deck)?
5. What is the probability, taking two cards, to collect a combination - (Jack, Queen or King) and Ace The sequence in which the cards will be drawn does not matter.

Answers:

1. In a deck of cards of each value, it means:
2. The events are dependent, since after the first card drawn, the number of cards in the deck has decreased (as well as the number of "pictures"). Total jacks, queens, kings and aces in the deck initially, which means the probability of drawing the “picture” with the first card:

   Since we are removing the first card from the deck, it means that there is already a card left in the deck, of which there are pictures. Probability of drawing a picture with the second card:

   Since we are interested in the situation when we get from the deck: “picture” AND “picture”, then we need to multiply the probabilities:

   Answer:

3. After the first card is drawn, the number of cards in the deck will decrease. Thus, we have two options:
   1) With the first card we take out Ace, the second - jack, queen or king
   2) With the first card we take out a jack, queen or king, the second - an ace. (ace and (jack or queen or king)) or ((jack or queen or king) and ace). Don't forget about reducing the number of cards in the deck!

If you were able to solve all the problems yourself, then you are a great fellow! Now tasks on the theory of probability in the exam you will click like nuts!

PROBABILITY THEORY. AVERAGE LEVEL

Consider an example. Let's say we throw a die. What kind of bone is this, do you know? This is the name of a cube with numbers on the faces. How many faces, so many numbers: from to how many? Before.

So we roll a die and want it to come up with an or. And we fall out.

In probability theory they say what happened favorable event(not to be confused with good).

If it fell out, the event would also be auspicious. In total, only two favorable events can occur.

How many bad ones? Since all possible events, then the unfavorable of them are events (this is if it falls out or).

Definition:

Probability is the ratio of the number of favorable events to the number of all possible events.. That is, the probability shows what proportion of all possible events are favorable.

The probability is denoted by a Latin letter (apparently, from English word probability - probability).

It is customary to measure the probability as a percentage (see topics and). To do this, the probability value must be multiplied by. In the dice example, probability.

And in percentage: .

Examples (decide for yourself):

1. What is the probability that the toss of a coin will land on heads? And what is the probability of a tails?
2. What is the probability that an even number will come up when a dice is thrown? And with what - odd?
3. In a drawer of plain, blue and red pencils. We randomly draw one pencil. What is the probability of pulling out a simple one?

Solutions:

1. How many options are there? Heads and tails - only two. And how many of them are favorable? Only one is an eagle. So the probability

   Same with tails: .

2. Total options: (how many sides a cube has, so many different options). Favorable ones: (these are all even numbers :).
   Probability. With odd, of course, the same thing.
3. Total: . Favorable: . Probability: .

Full Probability

All pencils in the drawer are green. What is the probability of drawing a red pencil? There are no chances: probability (after all, favorable events -).

Such an event is called impossible.

What is the probability of drawing a green pencil? There are exactly as many favorable events as there are total events (all events are favorable). So the probability is or.

Such an event is called certain.

If there are green and red pencils in the box, what is the probability of drawing a green or a red one? Yet again. Note the following thing: the probability of drawing green is equal, and red is .

In sum, these probabilities are exactly equal. That is, the sum of the probabilities of all possible events is equal to or.

Example:

In a box of pencils, among them are blue, red, green, simple, yellow, and the rest are orange. What is the probability of not drawing green?

Solution:

Remember that all probabilities add up. And the probability of drawing green is equal. This means that the probability of not drawing green is equal.

Remember this trick: The probability that an event will not occur is minus the probability that the event will occur.

Independent events and the multiplication rule

You flip a coin twice and you want it to come up heads both times. What is the probability of this?

Let's go through all the possible options and determine how many there are:

Eagle-Eagle, Tails-Eagle, Eagle-Tails, Tails-Tails. What else?

The whole variant. Of these, only one suits us: Eagle-Eagle. So, the probability is equal.

Good. Now let's flip a coin. Count yourself. Happened? (answer).

You may have noticed that with the addition of each next throw, the probability decreases by a factor. The general rule is called multiplication rule:

The probabilities of independent events change.

What are independent events? Everything is logical: these are those that do not depend on each other. For example, when we toss a coin several times, each time a new toss is made, the result of which does not depend on all previous tosses. With the same success, we can throw two different coins at the same time.

More examples:

1. A die is thrown twice. What is the probability that it will come up both times?
2. A coin is tossed times. What is the probability of getting heads first and then tails twice?
3. The player rolls two dice. What is the probability that the sum of the numbers on them will be equal?

Answers:

1. The events are independent, which means that the multiplication rule works: .
2. The probability of an eagle is equal. Tails probability too. We multiply:
3. 12 can only be obtained if two -ki fall out: .

Incompatible events and the addition rule

Incompatible events are events that complement each other to full probability. As the name implies, they cannot happen at the same time. For example, if we toss a coin, either heads or tails can fall out.

Example.

In a box of pencils, among them are blue, red, green, simple, yellow, and the rest are orange. What is the probability of drawing green or red?

Solution .

The probability of drawing a green pencil is equal. Red - .

Auspicious events of all: green + red. So the probability of drawing green or red is equal.

The same probability can be represented in the following form: .

This is the addition rule: the probabilities of incompatible events add up.

Mixed tasks

Example.

The coin is tossed twice. What is the probability that the result of the rolls will be different?

Solution .

This means that if heads come up first, tails should be second, and vice versa. It turns out that there are two pairs of independent events here, and these pairs are incompatible with each other. How not to get confused about where to multiply and where to add.

There is a simple rule for such situations. Try to describe what should happen by connecting the events with the unions "AND" or "OR". For example, in this case:

Must roll (heads and tails) or (tails and heads).

Where there is a union "and", there will be multiplication, and where "or" is addition:

Try it yourself:

1. What is the probability that two coin tosses come up with the same side both times?
2. A die is thrown twice. What is the probability that the sum will drop points?

Solutions:

1. (Heads up and heads up) or (tails up and tails up): .
2. What are the options? and. Then:
   Rolled (and) or (and) or (and): .

Another example:

We toss a coin once. What is the probability that heads will come up at least once?

Solution:

Oh, how I don’t want to sort through the options ... Head-tails-tails, Eagle-heads-tails, ... But you don’t have to! We remember about full probability. Remembered? What is the probability that the eagle will never drop? It's simple: tails fly all the time, that means.

PROBABILITY THEORY. BRIEFLY ABOUT THE MAIN

Probability is the ratio of the number of favorable events to the number of all possible events.

Independent events

Two events are independent if the occurrence of one does not change the probability of the other occurring.

Full Probability

The probability of all possible events is ().

The probability that an event will not occur is minus the probability that the event will occur.

Rule for multiplying the probabilities of independent events

The probability of a certain sequence of independent events is equal to the product of the probabilities of each of the events

Incompatible events

Incompatible events are those events that cannot possibly occur simultaneously as a result of an experiment. A number of incompatible events form a complete group of events.

The probabilities of incompatible events add up.

Having described what should happen, using the unions "AND" or "OR", instead of "AND" we put the sign of multiplication, and instead of "OR" - addition.

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Disputes and hypotheses about the existence of twin planets unknown to us, parallel universes and even galaxies have been going on for many decades. All of them are based on the theory of probability without involving the ideas of modern physics. AT last years to them was added the idea of ​​the existence of a superuniverse, based on proven theories - quantum mechanics and the theory of relativity. Polit.ru publishes an article Max Tegmark"Parallel Universes", which puts forward a hypothesis about the structure of the alleged superuniverse, theoretically including four levels. However, already in the next decade, scientists may have a real opportunity to obtain new data on the properties of outer space and, accordingly, confirm or refute this hypothesis. The article was published in the journal "In the world of science" (2003. No. 8).

Evolution has provided us with an intuition about everyday physics vital to our distant ancestors; therefore, as soon as we go beyond the everyday, we may well expect oddities.

The simplest and most popular cosmological model predicts that we have a twin in a galaxy about 10 to the power of 1028 meters away. The distance is so great that it is beyond the reach of astronomical observation, but this does not make our twin any less real. The assumption is based on the theory of probability without involving the ideas of modern physics. Only the assumption is accepted that space is infinite and filled with matter. There may be many habitable planets, including those where people live with the same appearance, the same names and memories, who have gone through the same life ups and downs as we do.

But we will never be able to see our other lives. The farthest distance we can see is that which light can travel in the 14 billion years since the Big Bang. The distance between the most distant visible objects from us is about 431026 m; it determines the region of the Universe available for observation, called the volume of the Hubble, or the volume of the cosmic horizon, or simply the Universe. The universes of our twins are spheres of the same size centered on their planets. This is the simplest example of parallel universes, each of which is only a small part of the superuniverse.

The very definition of "universe" suggests that it will forever remain in the field of metaphysics. However, the boundary between physics and metaphysics is determined by the possibility of experimental testing of theories, and not by the existence of unobservable objects. The boundaries of physics are constantly expanding, including more and more abstract (and previously metaphysical) ideas, for example, about a spherical Earth, invisible electromagnetic fields, time dilation at high speeds, superposition of quantum states, space curvature and black holes. In recent years, the idea of ​​a superuniverse has been added to this list. It is based on proven theories—quantum mechanics and the theory of relativity—and it meets both of the main criteria of empirical science: it allows predictions and can be refuted. Scientists consider four types of parallel universes. The main question is not whether a superuniverse exists, but how many levels it can have.

Level I

Beyond our cosmic horizon

The parallel universes of our counterparts constitute the first level of the superuniverse. This is the least controversial type. We all recognize the existence of things that we cannot see, but could see by moving to another place or simply by waiting, as we wait for the appearance of a ship from the horizon. Objects beyond our cosmic horizon have a similar status. The size of the observable region of the universe increases by one light year each year as light reaches us from ever more distant regions, behind which lies an infinity that has yet to be seen. We will probably die long before our twins are within sight, but if the expansion of the universe helps, our descendants will be able to see them with sufficiently powerful telescopes.

Level I of the superuniverse seems trivially obvious. How can space not be infinite? Is there a sign somewhere that reads "Watch out! End of space? If there is an end to space, what is beyond it? However, Einstein's theory of gravity called this intuition into question. A space can be finite if it has positive curvature or an unusual topology. A spherical, toroidal, or "pretzel" universe can have a finite volume without boundaries. Background cosmic microwave radiation makes it possible to test the existence of such structures. However, the facts still speak against them. The model of the infinite universe corresponds to the data, and strict restrictions are imposed on all other options.

Another option is this: space is infinite, but matter is concentrated in a limited area around us. In one version of the once popular "island universe" model, it is assumed that on large scales matter is rarefied and has a fractal structure. In both cases, almost all universes in a level I superuniverse must be empty and lifeless. Recent studies of the three-dimensional distribution of galaxies and background (relic) radiation have shown that the distribution of matter tends to be uniform on large scales and does not form structures larger than 1024 m. If this trend continues, then the space outside the observable Universe should be replete with galaxies, stars and planets.

For observers in parallel universes of the first level, the same laws of physics apply as for us, but under different starting conditions. According to modern theories, the processes taking place on early stages The big bang, randomly scattered matter, so there was a possibility of any structures.

Cosmologists accept that our Universe with an almost uniform distribution of matter and initial density fluctuations of the order of 1/105 is quite typical (at least among those in which there are observers). Estimates based on this assumption show that your closest replica is at a distance of 10 to the power of 1028 m. At a distance of 10 to the power of 1092 m there should be a sphere with a radius of 100 light years, identical to the one in the center of which we are located; so that everything that we see in the next century will be seen by our counterparts who are there. At a distance of about 10 to the power of 10118 m from us, there should be a Hubble volume identical to ours. These estimates are derived by counting possible number quantum states that a Hubble volume can have if its temperature does not exceed 108 K. The number of states can be estimated by asking the question: how many protons can a Hubble volume with such a temperature contain? The answer is 10118. However, each proton can either be present or absent, giving 2 to the power of 10118 possible configurations. A "box" containing so many Hubble volumes covers all possibilities. Its size is 10 to the power of 10118 m. Beyond it, the universes, including ours, must repeat themselves. Approximately the same figures can be obtained on the basis of thermodynamic or quantum gravitational estimates of the general information content of the Universe.

However, our closest twin is likely to be closer to us than these estimates give, since the process of planet formation and the evolution of life favor this. Astronomers believe that our Hubble volume contains at least 1020 habitable planets, some of which may be Earth-like.

In modern cosmology, the concept of a Level I superuniverse is widely used to test a theory. Consider how cosmologists use the CMB to reject the model of finite spherical geometry. Hot and cold "spots" on the CMB maps have a characteristic size that depends on the curvature of space. So, the size of the observed spots is too small to be consistent with the spherical geometry. Them the average size varies randomly from one Hubble volume to another, so it is possible that our Universe is spherical, but has anomalously small spots. When cosmologists say that they rule out the spherical model at a 99.9% confidence level, they mean that if the model is correct, then less than one Hubble volume in a thousand will have spots as small as those observed. It follows that the superuniverse theory is verifiable and can be rejected, even though we cannot see other universes. The main thing is to predict what the ensemble of parallel universes is like and find the probability distribution, or what mathematicians call the measure of the ensemble. Our universe must be one of the most probable. If not, if our universe turns out to be unlikely within the framework of the superuniverse theory, then this theory will run into difficulties. As we shall see later, the problem of measure can become quite acute.

Level II

Other post-inflationary domains

If it was difficult for you to imagine a level I superuniverse, then try to imagine an infinite number of such superuniverses, some of which have a different space-time dimension and are characterized by different physical constants. Together they constitute the Level II superuniverse predicted by the theory of chaotic perpetual inflation.

The theory of inflation is a generalization of the Big Bang theory, allowing to eliminate the shortcomings of the latter, for example, the inability to explain why the Universe is so large, homogeneous and flat. The rapid expansion of space in ancient times makes it possible to explain these and many other properties of the Universe. Such stretching is predicted by a wide class of theories elementary particles and all available evidence supports it. The expression "chaotic perpetual" in relation to inflation indicates what is happening on the largest scale. In general, the space is constantly expanding, but in some areas the expansion stops, and individual domains appear, like raisins in rising dough. An infinite number of such domains appear, and each of them serves as the germ of a level I superuniverse, filled with matter, born from the energy of the field that causes inflation.

Neighboring domains are more than infinity away from us, in the sense that they cannot be reached even if we move forever at the speed of light, since the space between our domain and neighboring ones is stretching faster than you can move in it. Our descendants will never see their Level II counterparts. And if the expansion of the universe is accelerating, as observations show, then they will never see their counterparts even at level I.

A level II superuniverse is much more diverse than a level I superuniverse. Domains differ not only in their initial conditions, but also in their fundamental properties. The prevailing opinion among physicists is that the dimension of space-time, the properties of elementary particles, and many so-called physical constants are not built into physical laws, but are the result of processes known as symmetry breaking. It is believed that the space in our universe once had nine equal dimensions. At the beginning space history three of them took part in the expansion and became the three dimensions that characterize today's universe. The remaining six are now undetectable, either because they have remained microscopic, retaining a toroidal topology, or because all matter is concentrated in a three-dimensional surface (membrane, or just a brane) in nine-dimensional space. Thus, the original symmetry of measurements was violated. Quantum fluctuations that cause chaotic inflation could cause various violations symmetry in different caverns. Some could become four-dimensional; others contain only two rather than three generations of quarks; and still others, to have a stronger cosmological constant than our universe.

Another way for the emergence of the level II superuniverse can be represented as a cycle of births and destructions of universes. In the 1930s physicist Richard C. Tolman suggested this idea, and recently Paul J. Steinhardt of Princeton University and Neil Turok of Cambridge University have developed it. Steinhardt and Turok's model envisions a second three-dimensional brane, perfectly parallel to ours and only offset from it in a higher dimension. This parallel universe cannot be considered separate, since it interacts with ours. However, the ensemble of universes—past, present, and future—that these branes form is a superuniverse with a variety that appears to be close to that resulting from chaotic inflation. Another superuniverse hypothesis was proposed by physicist Lee Smolin from the Perimeter Institute in Waterloo (Ontario, Canada). His superuniverse is close to level II in diversity, but it mutates and creates new universes through black holes, not branes.

Although we cannot interact with level II parallel universes, cosmologists judge their existence by circumstantial evidence, since they can be the cause of strange coincidences in our universe. For example, in a hotel you are given room 1967, and you note that you were born in 1967. “What a coincidence,” you say. However, upon reflection, come to the conclusion that this is not so surprising. There are hundreds of rooms in the hotel, and it would not occur to you to think about anything if you were offered a room that meant nothing to you. If you didn't know anything about hotels, then you might assume that there are other rooms in the hotel to explain this coincidence.

As a closer example, consider the mass of the Sun. As you know, the luminosity of a star is determined by its mass. Using the laws of physics, we can calculate that life on Earth can only exist if the mass of the Sun lies in the range: from 1.6x1030 to 2.4x1030 kg. Otherwise, Earth's climate would be colder than Mars or hotter than Venus. Measurements of the mass of the Sun gave a value of 2.0x1030 kg. At first glance, the Sun's mass falling into the range of values ​​that ensures life on Earth is accidental.

The masses of stars occupy the range from 1029 to 1032 kg; if the Sun acquired its mass by chance, then the chance to fall into the optimal interval for our biosphere would be extremely small.

The apparent coincidence can be explained by assuming the existence of an ensemble (in this case, many planetary systems) and a selection factor (our planet must be habitable). Such observer-related selection criteria are called anthropic; and although the mention of them usually causes controversy, yet most physicists agree that these criteria should not be neglected in the selection of fundamental theories.

And what do all these examples have to do with parallel universes? It turns out that a slight change in the physical constants determined by symmetry breaking leads to a qualitatively different universe - one in which we could not exist. If the mass of the proton were only 0.2% larger, the protons would decay to form neutrons, making the atoms unstable. If the forces of electromagnetic interaction were weaker by 4%, there would be no hydrogen and ordinary stars. If the weak force were even weaker, there would be no hydrogen; and if it were stronger, supernovae could not fill interstellar space with heavy elements. If the cosmological constant were noticeably larger, the universe would have ballooned incredibly before galaxies could even form.

The given examples allow us to expect the existence of parallel universes with other values ​​of physical constants. Second-level superuniverse theory predicts that physicists will never be able to deduce the values ​​of these constants from fundamental principles, but can only calculate the probability distribution of various sets of constants in the totality of all universes. In this case, the result must be consistent with our existence in one of them.

Level III

Quantum set of universes

The superuniverses of levels I and II contain parallel universes, extremely remote from us beyond the limits of astronomy. However, the next level of the superuniverse lies right around us. It arises from a famous and highly controversial interpretation of quantum mechanics - the idea that random quantum processes cause the universe to "multiply", forming many copies of itself - one for each possible outcome process.

At the beginning of the twentieth century. quantum mechanics explained the nature nuclear world, which did not obey the laws of classical Newtonian mechanics. Despite the obvious successes, there was a heated debate among physicists about what the true meaning of new theory. It defines the state of the universe not in such terms classical mechanics, as the positions and velocities of all particles, but through a mathematical object called the wave function. According to the Schrödinger equation, this state changes over time in a way that mathematicians define by the term "unitary." It means that the wave function rotates in an abstract infinite-dimensional space called the Hilbert space. Although quantum mechanics is often defined as fundamentally random and indeterminate, the wave function evolves in a quite deterministic way. There is nothing random or uncertain about her.

The hardest part is relating the wave function to what we observe. Many valid wave functions correspond to unnatural situations, such as when a cat is both dead and alive in the so-called superposition. In the 20s. 20th century physicists get around this oddity by postulating that the wave function collapses to some particular classical outcome when one makes an observation. This addition made it possible to explain the results of observations, but turned an elegant unitary theory into a sloppy and not unitary one. Fundamental randomness, usually attributed to quantum mechanics, is a consequence of precisely this postulate.

Over time, physicists abandoned this view in favor of another, proposed in 1957 by Princeton University graduate Hugh Everett III. He showed that it is possible to do without the collapse postulate. Pure quantum theory does not impose any restrictions. Although it predicts that one classical reality will gradually split into a superposition of several such realities, the observer subjectively perceives this splitting as just a slight randomness with a probability distribution exactly the same as that given by the old postulate of collapse. This superposition of the classical universes is the level III superuniverse.

For more than forty years, this interpretation has confused scientists. However, the physical theory is easier to understand by comparing two points of view: external, from the position of a physicist studying mathematical equations (like a bird surveying the landscape from the height of its flight); and internal, from the position of an observer (let's call him a frog) living in a landscape overlooked by a bird.

From the point of view of a bird, the level III superuniverse is simple. There is only one wave function that smoothly evolves in time without splitting and parallelism. Abstract quantum world, described by an evolving wave function, contains a huge number of continuously splitting and merging lines of parallel classical histories, as well as a number of quantum phenomena that cannot be described within the framework of classical concepts. But from the point of view of a frog, one can see only a small part of this reality. She can see the level I universe, but a decoherence process similar to the collapse of the wave function, but with unitarity preserved, prevents her from seeing parallel copies of herself at level III.

When an observer is asked a question that he must quickly answer, a quantum effect in his brain leads to a superposition of decisions like "keep reading the article" and "stop reading the article." From the bird's point of view, the act of making a decision causes a person to multiply into copies, some of which continue to read, while others stop reading. However, from an internal point of view, neither of the doubles is aware of the existence of the others and perceives the split simply as a slight uncertainty, some possibility of continuing or stopping reading.

Strange as it may seem, the exact same situation occurs even in the Level I superuniverse. Obviously, you decided to continue reading, but one of your counterparts in a distant galaxy put the magazine down after the first paragraph. Levels I and III differ only in where your counterparts are located. At level I, they live somewhere far away, in good old three-dimensional space, and at level III, they live on another quantum branch of infinite-dimensional Hilbert space.

The existence of level III is possible only under the condition that the evolution of the wave function in time is unitary. So far, experiments have not revealed its deviations from unitarity. In recent decades, it has been confirmed for all larger systems, including C60 fullerene and optical fibers kilometer long. Theoretically, the proposition about unitarity was reinforced by the discovery of coherence violation. Some theorists working in the field of quantum gravity question it. In particular, it is assumed that evaporating black holes can destroy information, and this is not a unitary process. However, recent advances in string theory suggest that even quantum gravity is unitary.

If so, then black holes do not destroy information, but simply transmit it somewhere. If physics is unitary, the standard picture of the impact of quantum fluctuations in the initial stages of the Big Bang must be changed. These fluctuations do not randomly determine the superposition of all possible initial conditions that coexist simultaneously. In this case, the violation of coherence makes the initial conditions behave in a classical way on different quantum branches. The key point is that the distribution of outcomes in different quantum branches of one Hubble volume (Level III) is identical to the distribution of outcomes in different Hubble volumes of one quantum branch (Level I). This property of quantum fluctuations is known in statistical mechanics as ergodicity.

The same reasoning applies to level II. The process of breaking symmetry does not lead to a single outcome, but to a superposition of all outcomes that quickly diverge into their separate paths. Thus, if the physical constants, the dimension of space-time, etc. may differ in parallel quantum branches at level III, they will also differ in parallel universes at level II.

In other words, the level III superuniverse does not add anything new to what is available at levels I and II, only more copies of the same universes - the same historical lines develop over and over again on different quantum branches. The heated controversy surrounding Everett's theory appears to soon subside as a result of the discovery of equally grandiose but less contentious Levels I and II superuniverses.

The applications of these ideas are profound. For example, such a question: is there an exponential increase in the number of universes over time? The answer is unexpected: no. From the bird's point of view, there is only one quantum universe. And what is the number of separate universes at the moment for the frog? This is the number of markedly different Hubble volumes. The differences may be small: imagine the planets moving in different directions, imagine yourself married to someone else, and so on. At the quantum level, there are 10 to the power of 10118 universes with temperatures no higher than 108 K. The number is gigantic, but finite.

For a frog, the evolution of the wave function corresponds to an infinite movement from one of these 10 states to the power of 10118 to another. You are now in universe A, where you are reading this sentence. And now you are already in universe B, where you are reading the following sentence. In other words, there is an observer in B that is identical to the observer in universe A, with the only difference being that he has extra memories. At every moment there are all possible states, so that the passage of time can occur before the eyes of the observer. This idea was expressed in his 1994 science fiction novel Permutation City by writer Greg Egan and developed by physicist David Deutsch of Oxford University, independent physicist Julian Barbour, and others. we see that the idea of ​​a superuniverse can play a key role in understanding the nature of time.

Level IV

Other mathematical structures

The initial conditions and physical constants in the superuniverse levels I, II, and III may differ, but the fundamental laws of physics are the same. Why did we stop there? Why can't physical laws themselves differ? How about a universe that obeys classical laws without any relativistic effects? How about time moving in discrete steps, like in a computer?

What about the universe as an empty dodecahedron? In the level IV superuniverse, all of these alternatives do exist.

That such a superuniverse is not absurd is evidenced by the correspondence of the world of abstract reasoning to our real world. Equations and other mathematical concepts and structures - numbers, vectors, geometric objects - describe reality with amazing plausibility. Conversely, we perceive mathematical structures as real. Yes, they meet the fundamental criterion of reality: they are the same for everyone who studies them. The theorem will be true regardless of who proved it - a person, a computer or an intelligent dolphin. Other inquisitive civilizations will find the same mathematical structures that we know. Therefore, mathematicians say that they do not create, but discover mathematical objects.

There are two logical, but diametrically opposed paradigms of correlation between mathematics and physics, which arose in ancient times. According to Aristotle's paradigm, physical reality is primary, and mathematical language is only a convenient approximation. Within the framework of Plato's paradigm, it is the mathematical structures that are truly real, and observers perceive them imperfectly. In other words, these paradigms differ in their understanding of what is primary - the frog point of view of the observer (Aristotle's paradigm) or the bird's view from the height of the laws of physics (Plato's point of view).

Aristotle's paradigm is how we perceived the world with early childhood, long before they first heard about mathematics. Plato's point of view is acquired knowledge. Modern theoretical physicists lean towards it, suggesting that mathematics describes the universe well precisely because the universe is mathematical in nature. Then all physics is reduced to the solution mathematical problem, and an infinitely smart mathematician can only calculate the picture of the world on the basis of fundamental laws at the level of a frog, i.e. figure out which observers exist in the universe, what they perceive, and what languages ​​they have invented to convey their perception.

Mathematical structure is an abstraction, an unchanging entity outside of time and space. If the story were a movie, then the mathematical structure would correspond not to one frame, but to the film as a whole. Let's take for example a world consisting of zero-size particles distributed in three-dimensional space. From the bird's point of view, in four-dimensional space-time, particle trajectories are spaghetti. If the frog sees particles moving at constant speeds, then the bird sees a bunch of straight, uncooked spaghetti. If a frog sees two particles orbiting, then a bird sees two "spaghetti" twisted into a double helix. For a frog, the world is described by Newton's laws of motion and gravitation, for a bird - by the geometry of "spaghetti", i.e. mathematical structure. The frog itself for her is a thick ball of them, the complex interweaving of which corresponds to a group of particles that store and process information. Our world is more complicated than this example, and scientists do not know which of the mathematical structures it corresponds to.

Plato's paradigm contains the question: why is our world the way it is? For Aristotle, this is a meaningless question: the world exists, and so it is! But the followers of Plato are interested: could our world be different? If the universe is essentially mathematical, then why is it based on only one of the many mathematical structures? There seems to be a fundamental asymmetry at the very core of nature. To solve the puzzle, I suggested that mathematical symmetry exists: that all mathematical structures are physically realizable, and each of them corresponds to a parallel universe. The elements of this superuniverse are not in the same space, but exist outside of time and space. Most of them probably do not have observers. The hypothesis can be seen as extreme Platonism, stating that the mathematical structures of the Platonic world of ideas, or "mental landscape" of San Jose University mathematician Rudy Rucker, exist in physical sense. This is akin to what cosmologist John D. Barrow of the University of Cambridge called "p in the sky", philosopher Robert Nozick of Harvard University described as the “principle of fertility,” and the philosopher David K. Lewis of Princeton University called it “modal reality.” Level IV closes the hierarchy of superuniverses, since any self-consistent physical theory can be expressed in the form of some mathematical structure.

The Level IV superuniverse hypothesis allows for several verifiable predictions. As at level II, it includes the ensemble (in this case, the totality of all mathematical structures) and selection effects. In classifying mathematical structures, scientists should note that the structure that describes our world is the most general of those that are consistent with observations. Therefore, the results of our future observations should become the most general of those that agree with the data of previous studies, and the data of previous studies the most general of those that are generally compatible with our existence.

Assessing the degree of generality is not an easy task. One of the striking and encouraging features of mathematical structures is that the properties of symmetry and invariance that keep our universe simple and orderly tend to be common. Mathematical structures usually have these properties by default, and getting rid of them requires the introduction of complex axioms.

What did Occam say?

Thus, theories of parallel universes have a four-level hierarchy, where at each next level the universes are less and less reminiscent of ours. They can be characterized by different initial conditions (level I), physical constants and particles (level II), or physical laws (level IV). It's funny that level III has been the most criticized in recent decades as the only one that does not introduce qualitatively new types of universes. In the coming decade, detailed measurements of the CMB and the large-scale distribution of matter in the universe will allow us to more accurately determine the curvature and topology of space and confirm or disprove the existence of level I. The same data will allow us to obtain information about level II by testing the theory of chaotic perpetual inflation. Advances in astrophysics and high-energy particle physics will help refine the degree of fine-tuning of physical constants, strengthening or weakening Level II positions. If efforts to create a quantum computer are successful, there will be an additional argument in favor of the existence of level III, since the parallelism of this level will be used for parallel computing. Experimenters are also looking for evidence of unitarity violation, which will allow us to reject the hypothesis of the existence of level III. Finally, the success or failure of an attempt to solve the main problem of modern physics - to combine general theory relativity with quantum theory fields - will answer the question about level IV. Either a mathematical structure will be found that accurately describes our universe, or we will hit the limit of the incredible efficiency of mathematics and be forced to abandon the Level IV hypothesis.

So, is it possible to believe in parallel universes? The main arguments against their existence boil down to the fact that it is too wasteful and incomprehensible. The first argument is that superuniverse theories are vulnerable to Occam's Razor because they postulate the existence of other universes that we will never see. Why should nature be so wasteful and "amuse" itself by creating an infinite number of different worlds? However, this argument can be reversed in favor of the existence of a superuniverse. What exactly is wasteful nature? Certainly not in space, mass or number of atoms: there are already an infinite number of them at level I, the existence of which is not in doubt, so there is no point in worrying that nature will spend some more of them. The real issue is the apparent reduction in simplicity. Skeptics are worried Additional Information needed to describe invisible worlds.

However, the whole ensemble is often simpler than each of its members. The information volume of a number algorithm is, roughly speaking, the length, expressed in bits, of the shortest computer program that generates this number. Let's take the set of all integers as an example. Which is simpler - the whole set or a single number? At first glance - the second. However, the former can be built with a very simple program, and a single number can be extremely long. Therefore, the whole set turns out to be simpler.

Similarly, the set of all solutions to the Einstein equations for a field is simpler than any particular solution - the first consists of only a few equations, and the second requires a huge amount of initial data to be specified on some hypersurface. Thus, complexity increases when we focus on a single element of the ensemble, losing the symmetry and simplicity inherent in the totality of all elements.

In this sense, the superuniverses are more high levels easier. The transition from our universe to a level I superuniverse eliminates the need to set initial conditions. Further transition to level II eliminates the need to specify physical constants, and at level IV nothing needs to be specified at all. Excessive complexity is only a subjective perception, the point of view of a frog. And from the perspective of a bird, this superuniverse could hardly be any simpler. Complaints about incomprehensibility are of an aesthetic, not scientific, nature and are justified only in the Aristotelian worldview. When we ask a question about the nature of reality, shouldn't we expect an answer that may seem strange?

A common feature of all four levels of the superuniverse is that the simplest and perhaps the most elegant theory includes parallel universes by default. To reject their existence, it is necessary to complicate the theory by adding processes that are not confirmed by experiment and postulates invented for this - about the finiteness of space, the collapse of the wave function and ontological asymmetry. Our choice comes down to what is more wasteful and inelegant - a lot of words or a lot of universes. Perhaps, over time, we will get used to the quirks of our cosmos and find its strangeness charming.