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:
- you called to 1st Door
- you called to 2nd Door
- 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 let's take for - the number of favorable outcomes, and for - the total number of 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. Let's say we rang at one of the doors, but a stranger opened it for us. 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.
- 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.
- 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:
- If the experiment is carried out once (once a coin is tossed, the doorbell rings once, etc.), then the events are always independent.
- 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 all possible options:
- eagle eagle
- tails eagle
- tails-eagle
- 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 as a 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.
- What is the probability of drawing a white ball?
- 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:
- Eagle-eagle-eagle
- Eagle-head-tails
- Head-tails-eagle
- Head-tails-tails
- tails-eagle-eagle
- Tails-heads-tails
- Tails-tails-heads
- 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:
- Eagle-eagle-eagle
- Eagle-head-tails
- Head-tails-eagle
- Head-tails-tails
- tails-eagle-eagle
- Tails-heads-tails
- Tails-tails-heads
- 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 up 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.
- What is the probability of drawing clubs in a row (we put the first card drawn back into the deck and shuffle)?
- What is the probability of drawing a black card (spades or clubs)?
- What is the probability of drawing a picture (jack, queen, king or ace)?
- What is the probability of drawing two pictures in a row (we remove the first card drawn from the deck)?
- 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:
- In a deck of cards of each value, it means:
- 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:
- 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):
- What is the probability that the toss of a coin will land on heads? And what is the probability of a tails?
- What is the probability that when a dice is thrown, even number? And with what - odd?
- 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:
- 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: .
- 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. - 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. General rule 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:
- A die is thrown twice. What is the probability that it will come up both times?
- A coin is tossed times. What is the probability of getting heads first and then tails twice?
- The player rolls two dice. What is the probability that the sum of the numbers on them will be equal?
Answers:
- The events are independent, which means that the multiplication rule works: .
- The probability of an eagle is equal. Tails probability too. We multiply:
- 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:
- What is the probability that two coin tosses come up with the same side both times?
- A die is thrown twice. What is the probability that the sum will drop points?
Solutions:
- (Heads up and heads up) or (tails up and tails up): .
- 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! Let's talk 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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Math for Programmers: Probability Theory
Ivan Kamyshan
Some programmers, after working in the development of conventional commercial applications, are thinking about mastering machine learning and becoming a data analyst. Often they do not understand why certain methods work, and most machine learning methods seem like magic. In fact, machine learning is based on mathematical statistics, which, in turn, is based on the theory of probability. Therefore, in this article we will pay attention to the basic concepts of probability theory: we will touch on the definitions of probability, distribution, and analyze a few simple examples.
You may know that probability theory is conditionally divided into 2 parts. Discrete probability theory studies phenomena that can be described by a distribution with a finite (or countable) number of possible behaviors (throws of dice, coins). Continuous probability theory studies phenomena distributed on some dense set, for example, on a segment or in a circle.
You can consider the subject of probability theory on simple example. Imagine yourself as a shooter developer. An integral part of the development of games in this genre is the mechanics of shooting. It is clear that a shooter in which all weapons shoot absolutely accurately will be of little interest to players. Therefore, it is necessary to add spread to the weapon. But simply randomizing weapon hit points will not make it fine tuning, therefore, adjusting the game balance will be difficult. At the same time, using random variables and their distributions, you can analyze how the weapon will work with a given spread, and help make the necessary adjustments.
Space of elementary outcomes
Suppose, from some random experiment that we can repeat many times (for example, tossing a coin), we can extract some formalizable information (heads or tails). This information is called an elementary outcome, and it is advisable to consider the set of all elementary outcomes, often denoted by the letter Ω (Omega).
The structure of this space depends entirely on the nature of the experiment. For example, if we consider shooting at a sufficiently large circular target, the space of elementary outcomes will be a circle, for convenience, placed with the center at zero, and the outcome will be a point in this circle.
In addition, they consider sets of elementary outcomes - events (for example, hitting the "top ten" is a concentric circle of small radius with a target). In the discrete case, everything is quite simple: we can get any event, including or excluding elementary outcomes in a finite time. In the continuous case, however, everything is much more complicated: we need some good enough family of sets to consider, called an algebra, by analogy with simple real numbers that can be added, subtracted, divided and multiplied. Sets in an algebra can be intersected and combined, and the result of the operation will be in the algebra. This is a very important property for the mathematics behind all these concepts. The minimal family consists of only two sets - the empty set and the space of elementary outcomes.
Measure and Probability
Probability is a way of making inferences about the behavior of very complex objects without understanding how they work. Thus, the probability is defined as a function of an event (from that very good family of sets), which returns a number - some characteristic of how often such an event can occur in reality. For definiteness, mathematicians agreed that this number should lie between zero and one. In addition, requirements are imposed on this function: the probability of an impossible event is zero, the probability of the entire set of outcomes is unity, and the probability of combining two independent events (disjoint sets) is equal to the sum of the probabilities. Another name for probability is a probability measure. The most commonly used Lebesgue measure, which generalizes the concepts of length, area, volume to any dimensions (n-dimensional volume), and thus it is applicable to a wide class of sets.
Together, the set of a set of elementary outcomes, a family of sets, and a probability measure is called probability space. Let's look at how we can construct a probability space for the target shooting example.
Consider shooting at a large round target of radius R that cannot be missed. As a set of elementary events, we put a circle centered at the origin of coordinates of radius R . Since we are going to use the area (the Lebesgue measure for two-dimensional sets) to describe the probability of an event, we will use the family of measurable (for which this measure exists) sets.
Note Actually, this is a technical point and in simple tasks the process of determining the measure and the family of sets does not play a special role. But it is necessary to understand that these two objects exist, because in many books on probability theory, theorems begin with the words: “ Let (Ω,Σ,P) be a probability space…».
As mentioned above, the probability of the entire space of elementary outcomes must be equal to one. The area (the two-dimensional Lebesgue measure, which we will denote by λ 2 (A), where A is the event) of the circle, according to the well-known formula from school, is π * R 2. Then we can introduce the probability P(A) = λ 2 (A) / (π *R 2) , and this value will already lie between 0 and 1 for any event A.
If we assume that hitting any point of the target is equally probable, the search for the probability of hitting by the shooter in some area of the target is reduced to finding the area of this set (hence we can conclude that the probability of hitting a specific point is zero, because the area of the point is zero).
For example, we want to know what is the probability that the shooter will hit the "ten" (event A - the shooter hit the right set). In our model, "ten" is represented by a circle centered at zero and with radius r. Then the probability of falling into this circle is P(A) = λ 2 /(A)π *R 2 = π * r 2 /(π R 2)= (r/R) 2 .
This is one of the simplest varieties of "geometric probability" problems - most of these problems require finding an area.
random variables
A random variable is a function that converts elementary outcomes into real numbers. For example, in the considered problem, we can introduce a random variable ρ(ω) - the distance from the point of impact to the center of the target. The simplicity of our model allows us to explicitly specify the space of elementary outcomes: Ω = (ω = (x,y) numbers such that x 2 +y 2 ≤ R 2 ) . Then the random variable ρ(ω) = ρ(x,y) = x 2 +y 2 .
Means of abstraction from the probability space. Distribution function and density
It is good when the structure of space is well known, but in reality this is not always the case. Even if the structure of space is known, it can be complex. To describe random variables, if their expression is unknown, there is the concept of a distribution function, which is denoted by F ξ (x) = P(ξ< x) (нижний индекс ξ здесь означает случайную величину). Т.е. это вероятность множества всех таких элементарных исходов, для которых значение random variableξ on this event is less than the given parameter x .
The distribution function has several properties:
- First, it is between 0 and 1 .
- Second, it does not decrease when its argument x increases.
- Third, when the number -x is very large, the distribution function is close to 0, and when x itself is large, the distribution function is close to 1.
Probably, the meaning of this construction is not very clear on the first reading. One of useful properties– the distribution function allows you to look for the probability that the value takes a value from the interval. So, P (random variable ξ takes values from the interval ) = F ξ (b)-F ξ (a) . Based on this equality, we can investigate how this value changes if the boundaries a and b of the interval are close.
Let d = b-a , then b = a+d . And therefore, F ξ (b)-F ξ (a) = F ξ (a+d) - F ξ (a) . For small values of d, the above difference is also small (if the distribution is continuous). It makes sense to consider the relation p ξ (a,d)= (F ξ (a+d) - F ξ (a))/d . If for sufficiently small values of d this ratio differs little from some constant p ξ (a) , independent of d, then at this point the random variable has a density equal to p ξ (a) .
Note Readers who have previously encountered the concept of a derivative may notice that p ξ (a) is the derivative of the function F ξ (x) at the point a . In any case, you can study the concept of a derivative in an article dedicated to this topic on the Mathprofi website.
Now the meaning of the distribution function can be defined as follows: its derivative (density p ξ , which we defined above) at point a describes how often a random variable will fall into a small interval centered at point a (neighborhood of point a) compared to neighborhoods of other points . In other words, the faster the distribution function grows, the more likely it is that such a value will appear in a random experiment.
Let's go back to the example. We can calculate the distribution function for a random variable, ρ(ω) = ρ(x,y) = x 2 +y 2 , which denotes the distance from the center to the point of a random hit on the target. By definition, F ρ (t) = P(ρ(x,y)< t) . т.е. множество {ρ(x,y) < t)} – состоит из таких точек (x,y) , расстояние от которых до нуля меньше, чем t . Мы уже считали вероятность такого события, когда вычисляли вероятность попадания в «десятку» - она равна t 2 /R 2 . Таким образом, Fρ(t) = P(ρ(x,y) < t) = t 2 /R 2 , для 0 We can find the density p ρ of this random variable. We note right away that it is zero outside the interval, since the distribution function on this interval is unchanged. At the ends of this interval, the density is not determined. Inside the interval, it can be found using a table of derivatives (for example, from the Mathprofi website) and elementary differentiation rules. The derivative of t 2 /R 2 is 2t/R 2 . This means that we found the density on the entire axis of real numbers. Another useful property of density is the probability that a function takes a value from an interval is calculated using the integral of the density over this interval (you can find out what it is in the articles on proper, improper, indefinite integrals on the Mathprofi website). On the first reading, the span integral of the function f(x) can be thought of as the area of a curvilinear trapezoid. Its sides are a fragment of the Ox axis, a gap (of the horizontal coordinate axis), vertical segments connecting the points (a,f(a)), (b,f(b)) on the curve with points (a,0), (b,0 ) on the x-axis. The last side is a fragment of the graph of the function f from (a,f(a)) to (b,f(b)) . We can talk about the integral over the interval (-∞; b] , when for sufficiently large negative values, a, the value of the integral over the interval will change negligibly small compared to the change in the number a. The integral over the intervals is determined in a similar way Topics information technology in general EN probability theory of chance probability calculation … Technical Translator's Handbook Probability theory- there is a part of mathematics that studies the relationships between the probabilities (see Probability and Statistics) of various events. We list the most important theorems related to this science. The probability of occurrence of one of several incompatible events is equal to ... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron PROBABILITY THEORY- mathematical a science that allows, according to the probabilities of some random events (see), to find the probabilities of random events associated with k. l. way with the first. Modern TV based on the axiomatics (see Axiomatic method) of A. N. Kolmogorov. On the… … Russian sociological encyclopedia Probability theory- a branch of mathematics in which, according to the given probabilities of some random events, the probabilities of other events are found, related in some way to the first. Probability theory also studies random variables and random processes. One of the main… … Concepts of modern natural science. Glossary of basic terms probability theory- tikimybių teorija statusas T sritis fizika atitikmenys: engl. probability theory vok. Wahrscheinlichkeitstheorie, f rus. probability theory, f pranc. theorie des probabilités, f … Fizikos terminų žodynas Probability Theory- ... Wikipedia Probability theory- a mathematical discipline that studies the patterns of random phenomena ... Beginnings of modern natural science PROBABILITY THEORY- (probability theory) see Probability ... Big explanatory sociological dictionary Probability theory and its applications- (“Probability Theory and Its Applications”), a scientific journal of the Department of Mathematics of the USSR Academy of Sciences. Publishes original articles and brief communications on probability theory, general problems of mathematical statistics and their applications in natural science and ... ... Great Soviet Encyclopedia INTRODUCTION Many things are incomprehensible to us, not because our concepts are weak; The main goal of studying mathematics in secondary specialized educational institutions is to give students a set of mathematical knowledge and skills necessary for studying other program disciplines that use mathematics to one degree or another, for the ability to perform practical calculations, for the formation and development of logical thinking. In this paper, all the basic concepts of the section of mathematics "Fundamentals of Probability Theory and Mathematical Statistics", provided for by the program and the State Educational Standards of Secondary Vocational Education (Ministry of Education of the Russian Federation. M., 2002), are consistently introduced, the main theorems are formulated, most of which are not proved . The main tasks and methods for their solution and technologies for applying these methods to solving practical problems are considered. The presentation is accompanied by detailed comments and numerous examples. Methodical instructions can be used for initial acquaintance with the studied material, when taking notes of lectures, for preparing for practical exercises, for consolidating the acquired knowledge, skills and abilities. In addition, the manual will be useful for undergraduate students as a reference tool that allows you to quickly restore in memory what was previously studied. At the end of the work, examples and tasks are given that students can perform in self-control mode. Methodical instructions are intended for students of correspondence and full-time forms of education. BASIC CONCEPTS Probability theory studies the objective regularities of mass random events. It is a theoretical basis for mathematical statistics, dealing with the development of methods for collecting, describing and processing the results of observations. Through observations (tests, experiments), i.e. experience in the broad sense of the word, there is a knowledge of the phenomena of the real world. In our practical activities, we often encounter phenomena, the outcome of which cannot be predicted, the result of which depends on chance. A random phenomenon can be characterized by the ratio of the number of its occurrences to the number of trials, in each of which, under the same conditions of all trials, it could occur or not occur. Probability theory is a branch of mathematics in which random phenomena (events) are studied and regularities are revealed during their mass repetition. Mathematical statistics is a branch of mathematics that has as its subject the study of methods for collecting, systematizing, processing and using statistical data to obtain scientifically sound conclusions and make decisions. At the same time, statistical data is understood as a set of numbers that represent the quantitative characteristics of the features of the studied objects that are of interest to us. Statistical data are obtained as a result of specially designed experiments and observations. Statistical data in its essence depend on many random factors, so mathematical statistics is closely related to probability theory, which is its theoretical basis. I. PROBABILITY. THEOREMS OF ADDITION AND PROBABILITY MULTIPLICATION 1.1. Basic concepts of combinatorics In the section of mathematics called combinatorics, some problems are solved related to the consideration of sets and the compilation of various combinations of elements of these sets. For example, if we take 10 different numbers 0, 1, 2, 3,:, 9 and make combinations of them, we will get different numbers, for example 143, 431, 5671, 1207, 43, etc. We see that some of these combinations differ only in the order of the digits (for example, 143 and 431), others in the numbers included in them (for example, 5671 and 1207), and others also differ in the number of digits (for example, 143 and 43). Thus, the obtained combinations satisfy various conditions. Depending on the compilation rules, three types of combinations can be distinguished: permutations, placements, combinations.
Let's first get acquainted with the concept factorial.
The product of all natural numbers from 1 to n inclusive is called n-factorial
and write. Calculate: a) ; b) ; in) . Solution. a) . b) as well as Then we get in) Permutations. A combination of n elements that differ from each other only in the order of the elements is called a permutation. Permutations are denoted by the symbol P n
, where n is the number of elements in each permutation. ( R- the first letter of the French word permutation- permutation). The number of permutations can be calculated using the formula or with factorial: Let's remember that 0!=1 and 1!=1.
Example 2. In how many ways can six different books be arranged on one shelf? Solution. The desired number of ways is equal to the number of permutations of 6 elements, i.e. Accommodations. Placements from m elements in n in each, such compounds are called that differ from each other either by the elements themselves (at least one), or by the order of the location. Locations are denoted by the symbol , where m is the number of all available elements, n is the number of elements in each combination. ( BUT- first letter of the French word arrangement, which means "placement, putting in order"). At the same time, it is assumed that nm. The number of placements can be calculated using the formula those. the number of all possible placements from m elements by n is equal to the product n consecutive integers, of which the greater is m. We write this formula in factorial form: Example 3. How many options for the distribution of three vouchers to a sanatorium of various profiles can be made for five applicants? Solution. The desired number of options is equal to the number of placements of 5 elements by 3 elements, i.e. Combinations. Combinations are all possible combinations of m elements by n, which differ from each other by at least one element (here m and n- natural numbers, and nm). Number of combinations from m elements by n are denoted ( FROM- the first letter of the French word combination- combination). In general, the number of m elements by n equal to the number of placements from m elements by n divided by the number of permutations from n elements: Using factorial formulas for placement and permutation numbers, we get: Example 4. In a team of 25 people, you need to allocate four to work in a certain area. In how many ways can this be done? Solution. Since the order of the chosen four people does not matter, this can be done in ways. We find by the first formula In addition, when solving problems, the following formulas are used that express the main properties of combinations: (by definition, and are assumed); 1.2. Solving combinatorial problems Task 1. 16 subjects are studied at the faculty. On Monday, you need to put 3 subjects in the schedule. In how many ways can this be done? Solution. There are as many ways to schedule three items out of 16 as there are placements of 16 elements of 3 each. Task 2. Out of 15 objects, 10 objects must be selected. In how many ways can this be done? Task 3. Four teams participated in the competition. How many options for the distribution of seats between them are possible? Problem 4. In how many ways can a patrol of three soldiers and one officer be formed if there are 80 soldiers and 3 officers? Solution. Soldier on patrol can be selected ways, and officers ways. Since any officer can go with each team of soldiers, there are only ways. Task 5. Find if it is known that . Since , we get By definition of combination it follows that , . That. . 1.3. The concept of a random event. Event types. Event Probability Any action, phenomenon, observation with several different outcomes, realized under a given set of conditions, will be called test.
The result of this action or observation is called event
. If an event under given conditions can occur or not occur, then it is called random
. In the event that an event must certainly occur, it is called reliable
, and in the case when it certainly cannot happen, - impossible.
The events are called incompatible
if only one of them can appear each time. The events are called joint
if, under the given conditions, the occurrence of one of these events does not exclude the occurrence of the other in the same test. The events are called opposite
, if under the test conditions they, being its only outcomes, are incompatible. Events are usually denoted by capital letters of the Latin alphabet: A, B, C, D, : . A complete system of events A 1 , A 2 , A 3 , : , A n is a set of incompatible events, the occurrence of at least one of which is mandatory for a given test. If a complete system consists of two incompatible events, then such events are called opposite and are denoted by A and . Example. There are 30 numbered balls in a box. Determine which of the following events are impossible, certain, opposite: got a numbered ball (BUT); draw an even numbered ball (AT); drawn a ball with an odd number (FROM); got a ball without a number (D). Which of them form a complete group? Solution . BUT- certain event; D- impossible event; In and FROM- opposite events. The complete group of events is BUT and D, V and FROM. The probability of an event is considered as a measure of the objective possibility of the occurrence of a random event. 1.4. The classical definition of probability The number, which is an expression of the measure of the objective possibility of the occurrence of an event, is called probability
this event and is denoted by the symbol P(A). Definition. Probability of an event BUT is the ratio of the number of outcomes m that favor the occurrence of a given event BUT, to the number n all outcomes (incompatible, unique and equally possible), i.e. . Therefore, in order to find the probability of an event, it is necessary, after considering the various outcomes of the test, to calculate all possible incompatible outcomes n, choose the number of outcomes we are interested in m and calculate the ratio m to n. The following properties follow from this definition: The probability of any trial is a non-negative number not exceeding one. Indeed, the number m of the desired events lies within . Dividing both parts into n, we get 2. The probability of a certain event is equal to one, because . 3. The probability of an impossible event is zero because . Problem 1. There are 200 winners out of 1000 tickets in the lottery. One ticket is drawn at random. What is the probability that this ticket wins? Solution. The total number of different outcomes is n=1000. The number of outcomes favoring the winning is m=200. According to the formula, we get Task 2. In a batch of 18 parts, there are 4 defective ones. 5 pieces are chosen at random. Find the probability that two out of these 5 parts are defective. Solution. Number of all equally possible independent outcomes n is equal to the number of combinations from 18 to 5 i.e. Let's calculate the number m that favor event A. Among the 5 randomly selected parts, there should be 3 high-quality and 2 defective ones. The number of ways to select two defective parts from 4 available defective parts is equal to the number of combinations from 4 to 2: The number of ways to select three quality parts from 14 available quality parts is equal to Any group of quality parts can be combined with any group of defective parts, so the total number of combinations m is The desired probability of the event A is equal to the ratio of the number of outcomes m that favor this event to the number n of all equally possible independent outcomes: The sum of a finite number of events is an event consisting in the occurrence of at least one of them. The sum of two events is denoted by the symbol A + B, and the sum n events symbol A 1 +A 2 + : +A n . The theorem of addition of probabilities. The probability of the sum of two incompatible events is equal to the sum of the probabilities of these events. Corollary 1. If the event А 1 , А 2 , : , А n form a complete system, then the sum of the probabilities of these events is equal to one. Corollary 2. The sum of the probabilities of opposite events and is equal to one. Problem 1. There are 100 lottery tickets. It is known that 5 tickets get a win of 20,000 rubles, 10 - 15,000 rubles, 15 - 10,000 rubles, 25 - 2,000 rubles. and nothing for the rest. Find the probability that the purchased ticket will win at least 10,000 rubles. Solution. Let A, B, and C be events consisting in the fact that a prize equal to 20,000, 15,000 and 10,000 rubles falls on the purchased ticket. since the events A, B and C are incompatible, then Task 2. The correspondence department of the technical school receives tests in mathematics from cities A, B and FROM. The probability of receipt of control work from the city BUT equal to 0.6, from the city AT- 0.1. Find the probability that the next control work will come from the city FROM.
The simplest example of a connection between two events is a causal relationship, when the occurrence of one of the events necessarily leads to the occurrence of the other, or vice versa, when the occurrence of one excludes the possibility of the occurrence of the other. To characterize the dependence of some events on others, the concept is introduced conditional probability.
Definition. Let BUT and AT- two random events of the same test. Then the conditional probability of the event BUT or the probability of event A, provided that event B has occurred, is called the number. Denoting the conditional probability , we obtain the formula Task 1. Calculate the probability that a second boy will be born in a family with one boy child. Solution. Let the event BUT consists in the fact that there are two boys in the family, and the event AT- that one boy. Consider all possible outcomes: boy and boy; boy and girl; girl and boy; girl and girl. Then , and by the formula we find Event BUT called independent
from the event AT if the occurrence of the event AT has no effect on the probability of an event occurring BUT. Probability multiplication theorem The probability of the simultaneous occurrence of two independent events is equal to the product of the probabilities of these events: The probability of the occurrence of several events that are independent in the aggregate is calculated by the formula Problem 2. The first urn contains 6 black and 4 white balls, the second urn contains 5 black and 7 white balls. One ball is drawn from each urn. What is the probability that both balls are white. A and AT there is an event AB. Consequently, b) If the first element works, then an event occurs (the opposite of the event BUT- the failure of this element); if the second element works - event AT. Find the probabilities of events and : Then the event consisting in the fact that both elements will work is, and, therefore,Books
but because these things do not enter the circle of our concepts.
Kozma Prutkov
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