"Randomness is not accidental"... It sounds like a philosopher said, but in fact, the study of accidents is the destiny of the great science of mathematics. In mathematics, chance is the theory of probability. Formulas and examples of tasks, as well as the main definitions of this science will be presented in the article.
What is Probability Theory?
Probability theory is one of the mathematical disciplines that studies random events.
To make it a little clearer, let's give a small example: if you toss a coin up, it can fall heads or tails. As long as the coin is in the air, both of these possibilities are possible. That is, the probability of possible consequences correlates 1:1. If one is drawn from a deck with 36 cards, then the probability will be indicated as 1:36. It would seem that there is nothing to explore and predict, especially with the help of mathematical formulas. Nevertheless, if you repeat a certain action many times, then you can identify a certain pattern and, on its basis, predict the outcome of events in other conditions.
To summarize all of the above, the theory of probability in the classical sense studies the possibility of the occurrence of one of the possible events in a numerical sense.
From the pages of history
The theory of probability, formulas and examples of the first tasks appeared in the distant Middle Ages, when attempts to predict the outcome of card games first arose.
Initially, the theory of probability had nothing to do with mathematics. It was justified by empirical facts or properties of an event that could be reproduced in practice. The first works in this area as a mathematical discipline appeared in the 17th century. The founders were Blaise Pascal and Pierre Fermat. For a long time they studied gambling and saw certain patterns, which they decided to tell the public about.
The same technique was invented by Christian Huygens, although he was not familiar with the results of the research of Pascal and Fermat. The concept of "probability theory", formulas and examples, which are considered the first in the history of the discipline, were introduced by him.
Of no small importance are the works of Jacob Bernoulli, Laplace's and Poisson's theorems. They made probability theory more like a mathematical discipline. Probability theory, formulas and examples of basic tasks got their present form thanks to Kolmogorov's axioms. As a result of all the changes, the theory of probability has become one of the mathematical branches.
Basic concepts of probability theory. Developments
The main concept of this discipline is "event". Events are of three types:
- Reliable. Those that will happen anyway (the coin will fall).
- Impossible. Events that will not happen in any scenario (the coin will remain hanging in the air).
- Random. Those that will or will not happen. They can be influenced by various factors that are very difficult to predict. If we talk about a coin, then random factors that can affect the result: the physical characteristics of the coin, its shape, initial position, throw force, etc.
All events in the examples are denoted by capital Latin letters, with the exception of R, which has a different role. For example:
- A = "students came to the lecture."
- Ā = "students didn't come to the lecture".
In practical tasks, events are usually recorded in words.
One of the most important characteristics of events is their equal possibility. That is, if you toss a coin, all variants of the initial fall are possible until it falls. But events are also not equally probable. This happens when someone deliberately influences the outcome. For example, "marked" playing cards or dice, in which the center of gravity is shifted.
Events are also compatible and incompatible. Compatible events do not exclude the occurrence of each other. For example:
- A = "the student came to the lecture."
- B = "the student came to the lecture."
These events are independent of each other, and the appearance of one of them does not affect the appearance of the other. Incompatible events are defined by the fact that the occurrence of one precludes the occurrence of the other. If we talk about the same coin, then the loss of "tails" makes it impossible for the appearance of "heads" in the same experiment.
Actions on events
Events can be multiplied and added, respectively, logical connectives "AND" and "OR" are introduced in the discipline.
The amount is determined by the fact that either event A, or B, or both can occur at the same time. In the case when they are incompatible, the last option is impossible, either A or B will drop out.
The multiplication of events consists in the appearance of A and B at the same time.
Now you can give a few examples to better remember the basics, probability theory and formulas. Examples of problem solving below.
Exercise 1: The firm is bidding for contracts for three types of work. Possible events that may occur:
- A = "the firm will receive the first contract."
- A 1 = "the firm will not receive the first contract."
- B = "the firm will receive a second contract."
- B 1 = "the firm will not receive a second contract"
- C = "the firm will receive a third contract."
- C 1 = "the firm will not receive a third contract."
Let's try to express the following situations using actions on events:
- K = "the firm will receive all contracts."
In mathematical form, the equation will look like this: K = ABC.
- M = "the firm will not receive a single contract."
M \u003d A 1 B 1 C 1.
We complicate the task: H = "the firm will receive one contract." Since it is not known which contract the firm will receive (the first, second or third), it is necessary to record the entire range of possible events:
H \u003d A 1 BC 1 υ AB 1 C 1 υ A 1 B 1 C.
And 1 BC 1 is a series of events where the firm does not receive the first and third contract, but receives the second one. Other possible events are also recorded by the corresponding method. The symbol υ in the discipline denotes a bunch of "OR". If we translate the above example into human language, then the company will receive either the third contract, or the second, or the first. Similarly, you can write other conditions in the discipline "Probability Theory". The formulas and examples of solving problems presented above will help you do it yourself.
Actually, the probability
Perhaps, in this mathematical discipline, the probability of an event is a central concept. There are 3 definitions of probability:
- classical;
- statistical;
- geometric.
Each has its place in the study of probabilities. Probability theory, formulas and examples (Grade 9) mostly use the classic definition, which sounds like this:
- The probability of situation A is equal to the ratio of the number of outcomes that favor its occurrence to the number of all possible outcomes.
The formula looks like this: P (A) \u003d m / n.
And, actually, an event. If the opposite of A occurs, it can be written as Ā or A 1 .
m is the number of possible favorable cases.
n - all events that can happen.
For example, A \u003d "pull out a heart suit card." There are 36 cards in a standard deck, 9 of them are of hearts. Accordingly, the formula for solving the problem will look like:
P(A)=9/36=0.25.
As a result, the probability that a heart-suited card will be drawn from the deck will be 0.25.
to higher mathematics
Now it has become a little known what the theory of probability is, formulas and examples of solving tasks that come across in the school curriculum. However, the theory of probability is also found in higher mathematics, which is taught in universities. Most often, they operate with geometric and statistical definitions of the theory and complex formulas.
The theory of probability is very interesting. Formulas and examples (higher mathematics) are better to start learning from a small one - from a statistical (or frequency) definition of probability.
The statistical approach does not contradict the classical approach, but slightly expands it. If in the first case it was necessary to determine with what degree of probability an event will occur, then in this method it is necessary to indicate how often it will occur. Here a new concept of “relative frequency” is introduced, which can be denoted by W n (A). The formula is no different from the classic:
If the classical formula is calculated for forecasting, then the statistical one is calculated according to the results of the experiment. Take, for example, a small task.
The department of technological control checks products for quality. Among 100 products, 3 were found to be of poor quality. How to find the frequency probability of a quality product?
A = "the appearance of a quality product."
W n (A)=97/100=0.97
Thus, the frequency of a quality product is 0.97. Where did you get 97 from? Of the 100 products that were checked, 3 turned out to be of poor quality. We subtract 3 from 100, we get 97, this is the quantity of a quality product.
A bit about combinatorics
Another method of probability theory is called combinatorics. Its basic principle is that if a certain choice A can be made in m different ways, and a choice B in n different ways, then the choice of A and B can be made by multiplying.
For example, there are 5 roads from city A to city B. There are 4 routes from city B to city C. How many ways are there to get from city A to city C?
It's simple: 5x4 = 20, that is, there are twenty different ways to get from point A to point C.
Let's make the task harder. How many ways are there to play cards in solitaire? In a deck of 36 cards, this is the starting point. To find out the number of ways, you need to “subtract” one card from the starting point and multiply.
That is, 36x35x34x33x32…x2x1= the result does not fit on the calculator screen, so it can simply be denoted as 36!. Sign "!" next to the number indicates that the entire series of numbers is multiplied among themselves.
In combinatorics, there are such concepts as permutation, placement and combination. Each of them has its own formula.
An ordered set of set elements is called a layout. Placements can be repetitive, meaning one element can be used multiple times. And without repetition, when the elements are not repeated. n is all elements, m is the elements that participate in the placement. The formula for placement without repetitions will look like:
A n m =n!/(n-m)!
Connections of n elements that differ only in the order of placement are called permutations. In mathematics, this looks like: P n = n!
Combinations of n elements by m are such compounds in which it is important which elements they were and what is their total number. The formula will look like:
A n m =n!/m!(n-m)!
Bernoulli formula
In the theory of probability, as well as in every discipline, there are works of outstanding researchers in their field who have taken it to a new level. One of these works is the Bernoulli formula, which allows you to determine the probability of a certain event occurring under independent conditions. This suggests that the appearance of A in an experiment does not depend on the appearance or non-occurrence of the same event in previous or subsequent tests.
Bernoulli equation:
P n (m) = C n m ×p m ×q n-m .
The probability (p) of the occurrence of the event (A) is unchanged for each trial. The probability that the situation will happen exactly m times in n number of experiments will be calculated by the formula that is presented above. Accordingly, the question arises of how to find out the number q.
If event A occurs p number of times, accordingly, it may not occur. A unit is a number that is used to designate all outcomes of a situation in a discipline. Therefore, q is a number that indicates the possibility of the event not occurring.
Now you know the Bernoulli formula (probability theory). Examples of problem solving (the first level) will be considered below.
Task 2: A store visitor will make a purchase with a probability of 0.2. 6 visitors entered the store independently. What is the probability that a visitor will make a purchase?
Solution: Since it is not known how many visitors should make a purchase, one or all six, it is necessary to calculate all possible probabilities using the Bernoulli formula.
A = "the visitor will make a purchase."
In this case: p = 0.2 (as indicated in the task). Accordingly, q=1-0.2 = 0.8.
n = 6 (because there are 6 customers in the store). The number m will change from 0 (no customer will make a purchase) to 6 (all store visitors will purchase something). As a result, we get the solution:
P 6 (0) \u003d C 0 6 × p 0 × q 6 \u003d q 6 \u003d (0.8) 6 \u003d 0.2621.
None of the buyers will make a purchase with a probability of 0.2621.
How else is the Bernoulli formula (probability theory) used? Examples of problem solving (second level) below.
After the above example, questions arise about where C and p have gone. With respect to p, a number to the power of 0 will be equal to one. As for C, it can be found by the formula:
C n m = n! /m!(n-m)!
Since in the first example m = 0, respectively, C=1, which in principle does not affect the result. Using the new formula, let's try to find out what is the probability of buying goods by two visitors.
P 6 (2) = C 6 2 ×p 2 ×q 4 = (6×5×4×3×2×1) / (2×1×4×3×2×1) × (0.2) 2 × (0.8) 4 = 15 × 0.04 × 0.4096 = 0.246.
The theory of probability is not so complicated. The Bernoulli formula, examples of which are presented above, is a direct proof of this.
Poisson formula
The Poisson equation is used to calculate unlikely random situations.
Basic formula:
P n (m)=λ m /m! × e (-λ) .
In this case, λ = n x p. Here is such a simple Poisson formula (probability theory). Examples of problem solving will be considered below.
Task 3 A: The factory produced 100,000 parts. The appearance of a defective part = 0.0001. What is the probability that there will be 5 defective parts in a batch?
As you can see, marriage is an unlikely event, and therefore the Poisson formula (probability theory) is used for calculation. Examples of solving problems of this kind are no different from other tasks of the discipline, we substitute the necessary data into the above formula:
A = "a randomly selected part will be defective."
p = 0.0001 (according to the assignment condition).
n = 100000 (number of parts).
m = 5 (defective parts). We substitute the data in the formula and get:
R 100000 (5) = 10 5 / 5! X e -10 = 0.0375.
Just like the Bernoulli formula (probability theory), examples of solutions using which are written above, the Poisson equation has an unknown e. In essence, it can be found by the formula:
e -λ = lim n ->∞ (1-λ/n) n .
However, there are special tables that contain almost all the values of e.
De Moivre-Laplace theorem
If in the Bernoulli scheme the number of trials is sufficiently large, and the probability of the occurrence of event A in all schemes is the same, then the probability of the occurrence of event A a certain number of times in a series of trials can be found by the Laplace formula:
Р n (m)= 1/√npq x ϕ(X m).
Xm = m-np/√npq.
To better remember the Laplace formula (probability theory), examples of tasks to help below.
First we find X m , we substitute the data (they are all indicated above) into the formula and get 0.025. Using tables, we find the number ϕ (0.025), the value of which is 0.3988. Now you can substitute all the data in the formula:
P 800 (267) \u003d 1 / √ (800 x 1/3 x 2/3) x 0.3988 \u003d 3/40 x 0.3988 \u003d 0.03.
So the probability that the flyer will hit exactly 267 times is 0.03.
Bayes formula
The Bayes formula (probability theory), examples of solving tasks with the help of which will be given below, is an equation that describes the probability of an event, based on the circumstances that could be associated with it. The main formula is as follows:
P (A|B) = P (B|A) x P (A) / P (B).
A and B are definite events.
P(A|B) - conditional probability, that is, event A can occur, provided that event B is true.
Р (В|А) - conditional probability of event В.
So, the final part of the short course "Theory of Probability" is the Bayes formula, examples of solving problems with which are below.
Task 5: Phones from three companies were brought to the warehouse. At the same time, part of the phones that are manufactured at the first plant is 25%, at the second - 60%, at the third - 15%. It is also known that the average percentage of defective products at the first factory is 2%, at the second - 4%, and at the third - 1%. It is necessary to find the probability that a randomly selected phone will be defective.
A = "randomly taken phone."
B 1 - the phone that the first factory made. Accordingly, introductory B 2 and B 3 will appear (for the second and third factories).
As a result, we get:
P (B 1) \u003d 25% / 100% \u003d 0.25; P (B 2) \u003d 0.6; P (B 3) \u003d 0.15 - so we found the probability of each option.
Now you need to find the conditional probabilities of the desired event, that is, the probability of defective products in firms:
P (A / B 1) \u003d 2% / 100% \u003d 0.02;
P (A / B 2) \u003d 0.04;
P (A / B 3) \u003d 0.01.
Now we substitute the data into the Bayes formula and get:
P (A) \u003d 0.25 x 0.2 + 0.6 x 0.4 + 0.15 x 0.01 \u003d 0.0305.
The article presents the theory of probability, formulas and examples of problem solving, but this is only the tip of the iceberg of a vast discipline. And after all that has been written, it will be logical to ask the question of whether the theory of probability is needed in life. It is difficult for a simple person to answer, it is better to ask someone who has hit the jackpot more than once with her help.
When a coin is tossed, it can be said that it will land heads up, or probability of this is 1/2. Of course, this does not mean that if a coin is tossed 10 times, it will necessarily land on heads 5 times. If the coin is "fair" and if it is tossed many times, then heads will come up very close half the time. Thus, there are two kinds of probabilities: experimental and theoretical .
Experimental and theoretical probability
If we toss a coin a large number of times - say 1000 - and count how many times it comes up heads, we can determine the probability that it will come up heads. If heads come up 503 times, we can calculate the probability of it coming up:
503/1000, or 0.503.
it experimental definition of probability. This definition of probability stems from observation and study of data and is quite common and very useful. For example, here are some probabilities that were determined experimentally:
1. The chance of a woman developing breast cancer is 1/11.
2. If you kiss someone who has a cold, then the probability that you will also get a cold is 0.07.
3. A person who has just been released from prison has an 80% chance of going back to prison.
If we consider the toss of a coin and taking into account that it is equally likely to come up heads or tails, we can calculate the probability of coming up heads: 1 / 2. This is the theoretical definition of probability. Here are some other probabilities that have been theoretically determined using mathematics:
1. If there are 30 people in a room, the probability that two of them have the same birthday (excluding the year) is 0.706.
2. During a trip, you meet someone and during the course of the conversation you discover that you have a mutual acquaintance. Typical reaction: "That can't be!" In fact, this phrase does not fit, because the probability of such an event is quite high - just over 22%.
Therefore, the experimental probability is determined by observation and data collection. Theoretical probabilities are determined by mathematical reasoning. Examples of experimental and theoretical probabilities, such as those discussed above, and especially those that we do not expect, lead us to the importance of studying probability. You may ask, "What is true probability?" Actually, there is none. It is experimentally possible to determine the probabilities within certain limits. They may or may not coincide with the probabilities that we obtain theoretically. There are situations in which it is much easier to define one type of probability than another. For example, it would be sufficient to find the probability of catching a cold using theoretical probability.
Calculation of experimental probabilities
Consider first the experimental definition of probability. The basic principle we use to calculate such probabilities is as follows.
Principle P (experimental)
If in an experiment in which n observations are made, the situation or event E occurs m times in n observations, then the experimental probability of the event is said to be P (E) = m/n.
Example 1 Sociological survey. An experimental study was conducted to determine the number of left-handers, right-handers and people in whom both hands are equally developed. The results are shown in the graph.
a) Determine the probability that the person is right-handed.
b) Determine the probability that the person is left-handed.
c) Determine the probability that the person is equally fluent in both hands.
d) Most PBA tournaments have 120 players. Based on this experiment, how many players can be left-handed?
Solution
a) The number of people who are right-handed is 82, the number of left-handers is 17, and the number of those who are equally fluent in both hands is 1. The total number of observations is 100. Thus, the probability that a person is right-handed is P
P = 82/100, or 0.82, or 82%.
b) The probability that a person is left-handed is P, where
P = 17/100 or 0.17 or 17%.
c) The probability that a person is equally fluent with both hands is P, where
P = 1/100 or 0.01 or 1%.
d) 120 bowlers and from (b) we can expect 17% to be left handed. From here
17% of 120 = 0.17.120 = 20.4,
that is, we can expect about 20 players to be left-handed.
Example 2 Quality control
. It is very important for a manufacturer to keep the quality of their products at a high level. In fact, companies hire quality control inspectors to ensure this process. The goal is to release the minimum possible number of defective products. But since the company produces thousands of items every day, it cannot afford to inspect each item to determine if it is defective or not. To find out what percentage of products are defective, the company tests far fewer products.
The USDA requires that 80% of the seeds that growers sell germinate. To determine the quality of the seeds that the agricultural company produces, 500 seeds are planted from those that have been produced. After that, it was calculated that 417 seeds germinated.
a) What is the probability that the seed will germinate?
b) Do the seeds meet government standards?
Solution a) We know that out of 500 seeds that were planted, 417 sprouted. The probability of seed germination P, and
P = 417/500 = 0.834, or 83.4%.
b) Since the percentage of germinated seeds exceeded 80% on demand, the seeds meet the state standards.
Example 3 TV ratings. According to statistics, there are 105,500,000 TV households in the United States. Every week, information about viewing programs is collected and processed. Within one week, 7,815,000 households were tuned in to CBS' hit comedy series Everybody Loves Raymond and 8,302,000 households were tuned in to NBC's hit Law & Order (Source: Nielsen Media Research). What is the probability that one home's TV is tuned to "Everybody Loves Raymond" during a given week? to "Law & Order"?
Solution The probability that the TV in one household is set to "Everybody Loves Raymond" is P, and
P = 7.815.000/105.500.000 ≈ 0.074 ≈ 7.4%.
The possibility that the household TV was set to "Law & Order" is P, and
P = 8.302.000/105.500.000 ≈ 0.079 ≈ 7.9%.
These percentages are called ratings.
theoretical probability
Suppose we are doing an experiment, such as tossing a coin or dart, drawing a card from a deck, or testing items on an assembly line. Each possible outcome of such an experiment is called Exodus . The set of all possible outcomes is called outcome space . Event it is a set of outcomes, that is, a subset of the space of outcomes.
Example 4 Throwing darts. Suppose that in the "throwing darts" experiment, the dart hits the target. Find each of the following:
b) Outcome space
Solution
a) Outcomes are: hitting black (H), hitting red (K) and hitting white (B).
b) There is an outcome space (hit black, hit red, hit white), which can be written simply as (B, R, B).
Example 5 Throwing dice.
A die is a cube with six sides, each of which has one to six dots. 
Suppose we are throwing a die. Find
a) Outcomes
b) Outcome space
Solution
a) Outcomes: 1, 2, 3, 4, 5, 6.
b) Outcome space (1, 2, 3, 4, 5, 6).
We denote the probability that an event E occurs as P(E). For example, "the coin will land on tails" can be denoted by H. Then P(H) is the probability that the coin will land on tails. When all outcomes of an experiment have the same probability of occurring, they are said to be equally likely. To see the difference between events that are equally likely and events that are not equally likely, consider the target shown below. 
For target A, black, red, and white hit events are equally likely, since black, red, and white sectors are the same. However, for target B, the zones with these colors are not the same, that is, hitting them is not equally likely.
Principle P (Theoretical)
If an event E can happen in m ways out of n possible equiprobable outcomes from the outcome space S, then theoretical probability
event, P(E) is
P(E) = m/n.
Example 6 What is the probability of rolling a 3 by rolling a die?
Solution There are 6 equally likely outcomes on the die and there is only one possibility of throwing the number 3. Then the probability P will be P(3) = 1/6.
Example 7 What is the probability of rolling an even number on the die?
Solution The event is the throwing of an even number. This can happen in 3 ways (if you roll 2, 4 or 6). The number of equiprobable outcomes is 6. Then the probability P(even) = 3/6, or 1/2.
We will be using a number of examples related to a standard 52-card deck. Such a deck consists of the cards shown in the figure below. 
Example 8 What is the probability of drawing an ace from a well-shuffled deck of cards?
Solution There are 52 outcomes (the number of cards in the deck), they are equally likely (if the deck is well mixed), and there are 4 ways to draw an ace, so according to the P principle, the probability
P(drawing an ace) = 4/52, or 1/13.
Example 9 Suppose we choose without looking one marble from a bag of 3 red marbles and 4 green marbles. What is the probability of choosing a red ball?
Solution There are 7 equally likely outcomes to get any ball, and since the number of ways to draw a red ball is 3, we get
P(choosing a red ball) = 3/7.
The following statements are results from the P principle.
Probability Properties
a) If the event E cannot happen, then P(E) = 0.
b) If the event E is bound to happen then P(E) = 1.
c) The probability that event E will occur is a number between 0 and 1: 0 ≤ P(E) ≤ 1.
For example, in tossing a coin, the event that the coin lands on its edge has zero probability. The probability that a coin is either heads or tails has a probability of 1.
Example 10 Suppose that 2 cards are drawn from a deck with 52 cards. What is the probability that both of them are spades?
Solution The number of ways n of drawing 2 cards from a well-shuffled 52-card deck is 52 C 2 . Since 13 of the 52 cards are spades, the number m of ways to draw 2 spades is 13 C 2 . Then,
P(stretching 2 peaks) \u003d m / n \u003d 13 C 2 / 52 C 2 \u003d 78/1326 \u003d 1/17.
Example 11 Suppose 3 people are randomly selected from a group of 6 men and 4 women. What is the probability that 1 man and 2 women will be chosen?
Solution Number of ways to choose three people from a group of 10 people 10 C 3 . One man can be chosen in 6 C 1 ways and 2 women can be chosen in 4 C 2 ways. According to the fundamental principle of counting, the number of ways to choose the 1st man and 2 women is 6 C 1 . 4C2. Then, the probability that 1 man and 2 women will be chosen is
P = 6 C 1 . 4 C 2 / 10 C 3 \u003d 3/10.
Example 12 Throwing dice. What is the probability of throwing a total of 8 on two dice?
Solution There are 6 possible outcomes on each dice. The outcomes are doubled, that is, there are 6.6 or 36 possible ways in which the numbers on two dice can fall. (It's better if the cubes are different, say one is red and the other is blue - this will help visualize the result.) 
Pairs of numbers that add up to 8 are shown in the figure below. There are 5 possible ways to get the sum equal to 8, hence the probability is 5/36.
Knowing how to estimate the probability of an event based on odds is essential to choosing the right bet. If you don't understand how to translate betting odds into odds, you'll never be able to determine how betting odds compare to the actual odds that an event will take place. It should be understood that if the probability of an event according to the bookmakers is lower than the probability of the same event according to your own version, a bet on this event will be valuable. You can compare odds for different events on the Odds.ru website.
1.1. Coefficient types
Bookmakers usually offer three types of odds – decimal, fractional and American. Let's take a look at each of the varieties.
1.2. Decimal Odds
Decimal odds, when multiplied by the size of the bet, allow you to calculate the entire amount that you will receive in your hand if you win. For example, if you bet $1 at odds of 1.80, if you win, you will receive $1.80 ($1 is the returned amount of the bet, $0.80 is the winnings on the bet, which is also your net profit).
That is, the probability of an outcome, according to the bookmakers, is 55%.
1.3. Fractional Odds
Fractional odds are the most traditional kind of odds. The numerator shows the potential amount of net winnings. The denominator is the amount of bet that needs to be made in order to get this same win. For example, odds of 7/2 means that in order to get a net win of $7, you need to wager $2.
In order to calculate the probability of an event based on a decimal coefficient, a simple calculation should be made - the denominator is divided by the sum of the numerator and denominator. For the above coefficient 7/2, the calculation will be as follows:
2 / (7+2) = 2 / 9 = 0,22
That is, the probability of an outcome, according to the bookmakers, is 22%.
1.4. American odds
This type of odds is popular in North America. At first glance, they seem rather complicated and incomprehensible, but do not be afraid. Understanding American odds can be useful, for example, when playing in American casinos, to understand quotes shown in North American sports broadcasts. Let's figure out how to evaluate the probability of an outcome based on American odds.
First of all, you need to understand that American odds are positive and negative. Negative American odds are always in the format, for example, "-150". This means that in order to receive $100 in net profit (winning), you need to wager $150.
A positive American coefficient is calculated in reverse. For example, we have a coefficient of "+120". This means that in order to receive $120 net profit (winning), you need to wager $100.
The probability calculation based on negative American odds is done using the following formula:
(-(negative US odds)) / ((-(negative US odds)) + 100)
(-(-150)) / ((-(-150)) + 100) = 150 / (150 + 100) = 150 / 250 = 0,6
That is, the probability of an event for which a negative American coefficient of “-150” is given is 60%.
Now consider similar calculations for a positive American coefficient. The probability in this case is calculated using the following formula:
100 / (positive US odds + 100)
100 / (120 + 100) = 100 / 220 = 0.45
That is, the probability of an event for which a positive American coefficient of “+120” is given is 45%.
1.5. How to convert coefficients from one format to another?
The ability to translate odds from one format to another can serve you well later on. Oddly enough, there are still bookmakers in which odds are not converted and are shown in only one format, which is unusual for us. Let's look at examples of how to do this. But first, we need to learn how to calculate the probability of an outcome based on the coefficient given to us.
1.6. How to calculate a decimal coefficient based on probability?
Everything is very simple here. It is necessary to divide 100 by the probability of the event as a percentage. That is, if the estimated probability of an event is 60%, you need to:
With an estimated probability of an event of 60%, the decimal odds would be 1.66.
1.7. How to calculate a fractional coefficient based on probability?
In this case, it is necessary to divide 100 by the probability of an event and subtract one from the result obtained. For example, the probability of an event is 40%:
(100 / 40) — 1 = 2,5 — 1 = 1,5
That is, we get a fractional coefficient of 1.5/1 or, for the convenience of counting, - 3/2.
1.8. How to calculate the American coefficient based on the probable outcome?
Here, much will depend on the probability of the event - whether it will be more than 50% or less. If the probability of an event is more than 50%, then the calculation will be made according to the following formula:
- ((probability) / (100 - probability)) * 100
For example, if the probability of an event is 80%, then:
— (80 / (100 — 80)) * 100 = — (80 / 20) * 100 = -4 * 100 = (-400)
With an estimated probability of an event of 80%, we got a negative American coefficient of "-400".
If the probability of an event is less than 50 percent, then the formula will be as follows:
((100 - probability) / probability) * 100
For example, if the probability of an event is 40%, then:
((100-40) / 40) * 100 = (60 / 40) * 100 = 1,5 * 100 = 150
With an estimated probability of an event of 40%, we got a positive American coefficient of "+150".
These calculations will help you better understand the concept of bets and odds, learn how to evaluate the true value of a particular bet.
From a practical point of view, event probability is the ratio of the number of those observations in which the event in question occurred to the total number of observations. Such an interpretation is admissible in the case of a sufficiently large number of observations or experiments. For example, if about half of the people you meet on the street are women, then you can say that the probability that the person you meet on the street is a woman is 1/2. In other words, the frequency of its occurrence in a long series of independent repetitions of a random experiment can serve as an estimate of the probability of an event.
Probability in mathematics
In the modern mathematical approach, the classical (that is, not quantum) probability is given by Kolmogorov's axiomatics. Probability is a measure P, which is set on the set X, called the probability space. This measure must have the following properties:
It follows from these conditions that the probability measure P also has the property additivity: if sets A 1 and A 2 do not intersect, then . To prove it, you need to put everything A 3 , A 4 , … equal to the empty set and apply the property of countable additivity.
The probability measure may not be defined for all subsets of the set X. It suffices to define it on the sigma-algebra consisting of some subsets of the set X. In this case, random events are defined as measurable subsets of the space X, that is, as elements of the sigma algebra.
Probability sense
When we find that the reasons for some possible fact to actually occur outweigh the opposite reasons, we consider this fact probable, otherwise - incredible. This predominance of positive bases over negative ones, and vice versa, can represent an indefinite set of degrees, as a result of which probability(and improbability) happens more or less .
Complicated single facts do not allow an exact calculation of their degrees of probability, but even here it is important to establish some large subdivisions. So, for example, in the field of law, when a personal fact subject to trial is established on the basis of witness testimony, it always remains, strictly speaking, only probable, and it is necessary to know how significant this probability is; in Roman law, a quadruple division was accepted here: probatio plena(where the probability practically turns into authenticity), Further - probatio minus plena, then - probatio semiplena major and finally probatio semiplena minor .
In addition to the question of the probability of the case, there may arise, both in the field of law and in the field of morality (with a certain ethical point of view), the question of how likely it is that a given particular fact constitutes a violation of the general law. This question, which serves as the main motive in the religious jurisprudence of the Talmud, gave rise in Roman Catholic moral theology (especially from the end of the 16th century) to very complex systematic constructions and an enormous literature, dogmatic and polemical (see Probabilism).
The concept of probability admits of a definite numerical expression in its application only to such facts which are part of certain homogeneous series. So (in the simplest example), when someone throws a coin a hundred times in a row, we find here one common or large series (the sum of all falls of a coin), which is composed of two private or smaller, in this case numerically equal, series (falls " eagle" and falling "tails"); The probability that this time the coin will fall tails, that is, that this new member of the general series will belong to this of the two smaller series, is equal to a fraction expressing the numerical ratio between this small series and the large one, namely 1/2, that is, the same probability belongs to one or the other of the two private series. In less simple examples, the conclusion cannot be drawn directly from the data of the problem itself, but requires prior induction. So, for example, it is asked: what is the probability for a given newborn to live up to 80 years? Here there must be a general or large series of a known number of people born in similar conditions and dying at different ages (this number must be large enough to eliminate random deviations, and small enough to preserve the homogeneity of the series, because for a person, born, for example, in St. Petersburg in a well-to-do cultural family, the entire million-strong population of the city, a significant part of which consists of people from various groups that can die prematurely - soldiers, journalists, workers in dangerous professions - represents a group too heterogeneous for a real definition of probability) ; let this general series consist of ten thousand human lives; it includes smaller rows representing the number of those who live to this or that age; one of these smaller rows represents the number of those living to 80 years of age. But it is impossible to determine the size of this smaller series (as well as all others). a priori; this is done in a purely inductive way, through statistics. Suppose statistical studies have established that out of 10,000 Petersburgers of the middle class, only 45 survive to the age of 80; thus, this smaller row is related to the larger one as 45 to 10,000, and the probability for a given person to belong to this smaller row, that is, to live to 80 years old, is expressed as a fraction of 0.0045. The study of probability from a mathematical point of view constitutes a special discipline, the theory of probability.
see also
Notes
Literature
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General scientific and philosophical. a category denoting the quantitative degree of the possibility of the appearance of mass random events under fixed observation conditions, characterizing the stability of their relative frequencies. In logic, the semantic degree ... ... Philosophical Encyclopedia
PROBABILITY, a number in the range from zero to one, inclusive, representing the possibility of this event happening. The probability of an event is defined as the ratio of the number of chances that an event can occur to the total number of possible ... ... Scientific and technical encyclopedic dictionary
In all likelihood .. Dictionary of Russian synonyms and expressions similar in meaning. under. ed. N. Abramova, M.: Russian dictionaries, 1999. probability, possibility, probability, chance, objective possibility, maza, admissibility, risk. Ant. impossibility... ... Synonym dictionary
probability- A measure that an event can occur. Note The mathematical definition of probability is "a real number between 0 and 1 related to a random event." The number may reflect the relative frequency in a series of observations ... ... Technical Translator's Handbook
Probability- "a mathematical, numerical characteristic of the degree of possibility of the occurrence of any event in certain specific conditions that can be repeated an unlimited number of times." Based on this classic… … Economic and Mathematical Dictionary
- (probability) The possibility of the occurrence of an event or a certain result. It can be represented as a scale with divisions from 0 to 1. If the probability of an event is zero, its occurrence is impossible. With a probability equal to 1, the onset of ... Glossary of business terms
Choosing the right bet depends not only on intuition, sports knowledge, betting odds, but also on the odds ratio of the event. The ability to calculate such an indicator in betting is the key to success in predicting the upcoming event on which the bet is supposed to be made.
In bookmakers, there are three types of odds (for more details, see the article), the variety of which determines how to calculate the probability of an event for a player.
Decimal Odds
The calculation of the probability of an event in this case occurs according to the formula: 1/coefficient of event. = v.i, where the coefficient of sob. is the coefficient of the event, and c.i is the probability of the outcome. For example, we take an event odds of 1.80 at a bet of one dollar, performing a mathematical action according to the formula, the player gets that the probability of an event outcome according to the bookmaker is 0.55 percent.
Fractional Odds
When using fractional odds, the probability calculation formula will be different. So with a coefficient of 7/2, where the first digit means the possible amount of net profit, and the second is the size of the required rate, to obtain this profit, the equation will look like this: . Here zn.coef is the denominator of the coefficient, chs.coef is the numerator of the coefficient, s.i is the probability of the outcome. Thus, for a fractional odds of 7/2, the equation looks like 2 / (7+2) = 2 / 9 = 0.22, therefore, 0.22 percent of the probability of the outcome of the event according to the bookmaker.
American odds
American odds are not very popular among bettors and are usually used exclusively in the USA, having a complex and intricate structure. To answer the question: “How to calculate the probability of an event in this way?”, You need to know that such coefficients can be negative and positive.
A coefficient with a “-” sign, such as -150, indicates that a player needs to wager $150 to make a net profit of $100. The probability of an event is calculated based on the formula where you need to divide the negative coefficient by the sum of the negative coefficient and 100. This looks like the example of a bet of -150, so (-(-150)) / ((-(-150)) + 100) = 150 / (150 + 100) = 150 / 250 = 0.6, where 0.6 is multiplied by 100 and the outcome of the event is 60 percent. The same formula applies to positive American odds.
