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Choice correct rate depends not only on intuition, sports knowledge, betting odds, but also on the event probability coefficient. The ability to calculate a similar indicator in betting is the key to success in forecasting 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.
A professional better should be well versed in odds, quickly and correctly evaluate the probability of an event by a coefficient and, if necessary, be able convert odds from one format to another. In this manual, we will talk about what types of coefficients are, as well as using examples, we will analyze how you can calculate the probability from a known coefficient and vice versa.
What are the types of coefficients?
There are three main types of odds offered by bookmakers: decimal odds, fractional odds(English) and american odds. The most common odds in Europe are decimal. IN North America American odds are popular. Fractional odds - most traditional look, they immediately reflect information about how much you need to bet in order to get a certain amount.
Decimal Odds
Decimals or else they are called European odds is the usual number format represented by decimal accurate to hundredths, and sometimes even to thousandths. An example of a decimal odd is 1.91. Calculating profit in case of decimal odds is very simple, just multiply your bet amount by this odd. For example, in the match "Manchester United" - "Arsenal", the victory of "MU" is set with a coefficient - 2.05, a draw is estimated with a coefficient - 3.9, and the victory of "Arsenal" is equal to - 2.95. Let's say we're confident United will win and bet $1,000 on them. Then our possible income calculated as follows:
2.05 * $1000 = $2050;
Isn't it really that difficult? In the same way, the possible income is calculated when betting on a draw and the victory of Arsenal.
Draw: 3.9 * $1000 = $3900;
Arsenal win: 2.95 * $1000 = $2950;
How to calculate the probability of an event by decimal odds?
Imagine now that we need to determine the probability of an event by the decimal odds set by the bookmaker. This is also very easy to do. To do this, we divide the unit by this coefficient.
Let's take the data we already have and calculate the probability of each event:
Manchester United win: 1 / 2.05 = 0,487 = 48,7%;
Draw: 1 / 3.9 = 0,256 = 25,6%;
Arsenal win: 1 / 2.95 = 0,338 = 33,8%;
Fractional Odds (English)
As the name implies fractional coefficient presented common fraction. An example of an English odd is 5/2. The numerator of the fraction contains a number that is the potential amount of net winnings, and the denominator contains a number indicating the amount that must be wagered in order to receive this winnings. Simply put, we have to wager $2 dollars to win $5. Odds of 3/2 means that in order to get $3 of net winnings, we will have to bet $2.
How to calculate the probability of an event by fractional odds?
It is also not difficult to calculate the probability of an event by fractional coefficients, you just need to divide the denominator by the sum of the numerator and denominator.
For the fraction 5/2, we calculate the probability: 2 / (5+2) = 2 / 7 = 0,28 = 28%;
For the fraction 3/2, we calculate the probability:
American odds
American odds unpopular in Europe, but very unpopular in North America. Perhaps, this species coefficients is the most difficult, but it is only at first glance. In fact, there is nothing complicated in this type of coefficients. Now let's take a look at everything in order.
The main feature of American odds is that they can be either positive, and negative. An example of American odds is (+150), (-120). The American odds (+150) means that in order to earn $150 we need to bet $100. In other words, the positive US coefficient reflects the potential net income with a bet of $100. The negative American coefficient reflects the amount of bet that must be made in order to receive a net winning of $100. For example, the coefficient (- 120) tells us that by betting $120 we will win $100.
How to calculate the probability of an event using American odds?
The probability of an event according to the American odds is calculated according to the following formulas:
(-(M)) / ((-(M)) + 100), where M is a negative American coefficient;
100/(P+100), where P is a positive American coefficient;
For example, we have a coefficient (-120), then the probability is calculated as follows:
(-(M)) / ((-(M)) + 100); we substitute the value (-120) instead of "M";
(-(-120)) / ((-(-120)) + 100 = 120 / (120 + 100) = 120 / 220 = 0,545 = 54,5%;
Thus, the probability of an event with an American coefficient (-120) is 54.5%.
For example, we have a coefficient (+150), then the probability is calculated as follows:
100/(P+100); we substitute the value (+150) instead of "P";
100 / (150 + 100) = 100 / 250 = 0,4 = 40%;
Thus, the probability of an event with an American coefficient (+150) is 40%.
How, knowing the percentage of probability, translate it into a decimal coefficient?
In order to calculate the decimal coefficient for a known percentage of probability, you need to divide 100 by the probability of an event in percent. For example, if the probability of an event is 55%, then the decimal coefficient of this probability will be equal to 1.81.
100 / 55% = 1,81
How, knowing the percentage of probability, translate it into a fractional coefficient?
In order to calculate a fractional coefficient from a known percentage of probability, you need to subtract one from dividing 100 by the probability of an event in percent. For example, we have a probability percentage of 40%, then the fractional coefficient of this probability will be equal to 3/2.
(100 / 40%) - 1 = 2,5 - 1 = 1,5;
The fractional coefficient is 1.5/1 or 3/2.
How, knowing the percentage of probability, translate it into an American coefficient?
If the probability of an event is more than 50%, then the calculation is made according to the formula:
- ((V) / (100 - V)) * 100, where V is the probability;
For example, we have an 80% probability of an event, then the American coefficient of this probability will be equal to (-400).
- (80 / (100 - 80)) * 100 = - (80 / 20) * 100 = - 4 * 100 = (-400);
If the probability of an event is less than 50%, then the calculation is made according to the formula:
((100 - V) / V) * 100, where V is the probability;
For example, if we have a probability percentage of an event of 20%, then the American coefficient of this probability will be equal to (+400).
((100 - 20) / 20) * 100 = (80 / 20) * 100 = 4 * 100 = 400;
How to convert the coefficient to another format?
There are times when it is necessary to convert coefficients from one format to another. For example, we have a fractional coefficient 3/2 and we need to convert it to decimal. To convert a fractional to decimal odds, we first determine the probability of an event with a fractional odds, and then convert that probability to a decimal odds.
The probability of an event with a fractional coefficient of 3/2 is 40%.
2 / (3+2) = 2 / 5 = 0,4 = 40%;
Now we translate the probability of an event into a decimal coefficient, for this we divide 100 by the probability of an event as a percentage:
100 / 40% = 2.5;
Thus, a fractional odd of 3/2 is equal to a decimal odd of 2.5. In a similar way, for example, American odds are converted to fractional, decimal to American, etc. The hardest part of all this is just the calculations.
"Accidents are not accidental" ... It sounds like a philosopher said, but in fact, studying accidents is the lot great science 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 possible consequences the ratio is 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. 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. Events
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. The ones that will or won't 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 indicated by capital letters. with Latin letters, with the exception of P, 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 events - their equivalence. 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, "labeled" 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 have next view: 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 this example into human language, then the firm 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 problems that come across in 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 one, 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 main principle is that if a certain choice A can be made m different ways, and the choice of B - n different ways, then the choice of A and B can be done by multiplication.
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 their total. 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 eminent researchers in their field who have brought it to 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 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 large enough, and the probability of occurrence of event A in all schemes is the same, then the probability of occurrence of event A a certain amount of times in a series of tests can be found by Laplace's 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 to find 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. To the common man difficult to answer, it is better to ask someone who has hit the jackpot more than once with it.
I understand that everyone wants to know in advance how a sporting event will end, who will win and who will lose. With this information, you can bet on sports events. But is it possible at all, and if so, how to calculate the probability of an event?
Probability is a relative value, therefore it cannot speak with accuracy about any event. This value allows you to analyze and evaluate the need to place a bet on a particular competition. The definition of probabilities is a whole science that requires careful study and understanding.
Probability coefficient in probability theory
In sports betting, there are several options for the outcome of the competition:
- victory of the first team;
- victory of the second team;
- draw;
- total
Each outcome of the competition has its own probability and frequency with which this event will occur, provided that the initial characteristics are preserved. As mentioned earlier, it is impossible to accurately calculate the probability of any event - it may or may not coincide. Thus, your bet can either win or lose.
There can be no exact 100% prediction of the results of the competition, since many factors influence the outcome of the match. Naturally, bookmakers do not know the outcome of the match in advance and only assume the result, making a decision on their analysis system and offer certain coefficients for rates.
How to calculate the probability of an event?
Let's say that the odds of the bookmaker is 2.1/2 - we get 50%. It turns out that the coefficient 2 is equal to the probability of 50%. By the same principle, you can get a break-even probability ratio - 1 / probability.
Many players think that after several repeated losses, a win will definitely happen - this is an erroneous opinion. The probability of winning a bet does not depend on the number of losses. Even if you throw several heads in a row in a coin game, the probability of throwing tails remains the same - 50%.
Whether we like it or not, our life is full of all kinds of accidents, both pleasant and not very. Therefore, each of us would do well to know how to find the probability of an event. This will help you make the right decisions under any circumstances that are associated with uncertainty. For example, such knowledge will be very useful when choosing investment options, evaluating the possibility of winning a stock or lottery, determining the reality of achieving personal goals, etc., etc.
Probability Formula
In principle, the study of this topic does not take too much time. In order to get an answer to the question: "How to find the probability of a phenomenon?", you need to deal with key concepts and remember the basic principles on which the calculation is based. So, according to statistics, the events under study are denoted by A1, A2,..., An. Each of them has both favorable outcomes (m) and the total number of elementary outcomes. For example, we are interested in how to find the probability that an even number of points will be on the top face of the cube. Then A is roll m - rolling 2, 4, or 6 (three favorable choices), and n is all six possible choices.
The calculation formula itself is as follows:
With one outcome, everything is extremely easy. But how to find the probability if the events go one after the other? Consider this example: one card is shown from a deck of cards (36 pieces), then it is hidden back into the deck, and after shuffling, the next one is pulled out. How to find the probability that at least in one case the Queen of Spades was drawn? Exists next rule: if considered complex event, which can be divided into several incompatible simple events, then you can first calculate the result for each of them, and then add them together. In our case, it will look like this: 1/36 + 1/36 = 1/18. But what about when several occur at the same time? Then we multiply the results! For example, the probability that when two coins are tossed at the same time, two tails will fall out will be equal to: ½ * ½ = 0.25.
Now let's take even more complex example. Suppose we enter a book lottery in which ten out of thirty tickets are winning. It is required to determine:
- The probability that both will win.
- At least one of them will bring a prize.
- Both will be losers.
So let's consider the first case. It can be broken down into two events: the first ticket will be lucky, and the second one will also be lucky. Let's take into account that the events are dependent, since after each pulling out the total number of options decreases. We get:
10 / 30 * 9 / 29 = 0,1034.
In the second case, you need to determine the probability of a losing ticket and take into account that it can be both the first in a row and the second one: 10 / 30 * 20 / 29 + 20 / 29 * 10 / 30 = 0.4598.
Finally, the third case, when even one book cannot be obtained from the lottery: 20 / 30 * 19 / 29 = 0.4368.
