Probability Theory
Randomness
What is randomness?
Apparently, this is the opposite of the pattern.
I got "two" in physics class because I didn't know
answering the teacher's question because he didn't do his homework.
This, of course, is a rule.
On vacation in another country, I meet a classmate,
about which I have not heard anything for 20 years.
This is, of course, a coincidence.
However, your classmate was not there by chance.
He planned a trip
did the necessary actions for this, etc.
However, meeting with you will also be an accident for him, since he, in turn,
also did not have any information about you.
This means that, generally speaking, randomness does not exist.
And the term "Randomness" means that for the subject (in this example, for me),
the circumstances of some event or process are completely unknown
and / or they cannot be estimated.
Therefore, for us, the result looks completely random.
For example, after rolling the die, the number 5 came up.
Why?
Because during the flight and landing of the cube
many factors interplayed
(throw force, throw direction, air resistance, material properties,
from which the cube is made, the force of gravity of the Earth ...),
as a result of which the cube stops with the number 5 on top.
However, we are absolutely unable to "calculate" all this.
That's why:
for the subject (me) - this is a random result,
Probability
What is "Probability"?
"Probability theory is essentially
nothing but common sense reduced to calculations."
P. Laplace.
This word is widely used in everyday life and people
believe that its meaning is clear to everyone.
If we do not have an absolutely accurate, unambiguous,
clear answer about any event, phenomenon,
then the concept of "Probability" is used, which,
in some approximation, evaluates:
how far (close) is the possible answer from the truth.
However, as usual, not everything is so simple.
There are two approaches to determine the probability:
subjective and empirical.
In the first case, we determine the probability based on observations,
experience, analysis of the situation, and so on.
The evaluation takes into account as many possible factors as possible.
However, the result is highly dependent on the individual(s),
this is clearly reflected in the name of the method - subjective.
As a rule, this method is used in everyday life.
Example:
"How does our neighbor feel about Russia's war against Ukraine?"
"Hmm... I guess at least 90% that he supports that bastard."
In the second case, the term "Probability" is defined as follows:
Probability is the numerical value of the possibility of an event occurring.
It is this definition that the branch of mathematics
called "Probability Theory" uses.
This definition contains one of the key concepts - "Event".
Events must be independent of each other.
At the first toss of a coin (test), an "eagle" fell out.
At the second - "tails".
The result of the second trial is not affected by the first roll.
That is, getting "tails" lies entirely on the conscience of the second throw.
Probability theory studies the probabilistic
patterns of massive homogeneous random independent events.
In ordinary life, we constantly make probabilistic estimates and, perhaps,
We make decisions based on them.
Wrong application of the theory of probability naturally leads to wrong results.
Example 1
You leave the house tomorrow morning and go to work.
Question.
What is the probability that you will meet an elephant on the street?
Answer: 50%. Why? Because there are only two variants.
Either we meet or we don't.
Therefore, one second, that is, 50%.
If you read the above text, you can see where the error is.
It was necessary to conduct "mass" experiments, for example,
50 days to check whether I had shot an elephant.
Let's say I met an elephant once in those 50 days.
Then the probability will be equal to 2% (1:50=0.02).
Example 2
The lover says to his beloved:
"Listen, it's really by Providence that we met you."
"Why?" asks the happy girl?
“Look, what is the probability that I met you then at my friend’s party?
It was necessary that his wife left him (therefore, in order to
distract himself, he decided to be among people as much as possible).
Moreover, you could not have been born at all! And with a high probability.
To do this, it was necessary for your mom and dad to meet,
which was very unlikely, but let's say your mom could also not be born,
and also with a high probability ....
You see, my dear, it must have been a hand from above that connected us."
"That's the way it is" - answers his happiness,
but you incorrectly apply the calculations from the theory of probability.
Events should not only be random,
but also independent.
In other words, there must be no causal relationship between events.
Like flipping a coin, for example.
The result of the experience (throwing) N5
does not depend on the experience of N6 or N47.
But my birth depends on the meeting of my parents,
and their meeting depends on several other events, and so on.
You know, I'll probably look for someone smarter than you."
Example 3
This example is known as the "turkey principle".
The man caught the turkey, brought him into the yard,
and went into the house himself.
The turkey thinks:
everything, apparently the end has come to me, he probably went for an ax.
But the man returns and in his hands instead of an ax is a cup of food.
The turkey is surprised - today it has passed, tomorrow it will definitely kill.
However, the man again brings him food, and this continues for quite some time.
One day, the turkey, as usual,
joyfully awaits the appearance of a person, as he is sure
in accordance with the theory of probability,
that he will receive quite tasty food.
But the man turns out to have an ax in his hands and
he cuts off the head of the turkey.
The error in the reasoning of the turkey was based on the fact
that he applied the theory of probability thinking
that he had complete information about the ongoing process.
Man's goal was to fatten the turkey and then eat it.