Probability theory is a fairly extensive independent branch of mathematics. In the school course, probability theory is discussed very superficially, but in the Unified State Examination and the State Examination Academy there are problems on this topic. However, solving school course problems is not so difficult (at least as far as arithmetic operations are concerned) - here you do not need to count derivatives, take integrals and solve complex trigonometric transformations- the main thing is to be able to handle simple numbers and fractions.
Probability theory - basic terms
The main terms of probability theory are test, outcome and random event. A test in probability theory is an experiment - tossing a coin, drawing a card, drawing lots - all these are tests. The result of the test, as you may have guessed, is called the outcome.
What is a random event? In probability theory, it is assumed that the test is carried out more than once and there are many outcomes. A random event is a set of outcomes of a trial. For example, if you toss a coin, two random events can happen - heads or tails.
Do not confuse the concepts of outcome and random event. An outcome is one result of one trial. A random event is a set of possible outcomes. By the way, there is such a term as an impossible event. For example, the event “rolling the number 8” on a standard dice is impossible.
How to find probability?
We all roughly understand what probability is, and quite often use this word in our vocabulary. In addition, we can even draw some conclusions regarding the likelihood of a particular event, for example, if there is snow outside the window, we can most likely say that it is not summer. However, how can we express this assumption numerically?
In order to introduce a formula for finding probability, we introduce one more concept - favorable outcome, that is, an outcome that is favorable for a particular event. The definition is quite ambiguous, of course, but according to the conditions of the problem it is always clear which outcome is favorable.
For example: There are 25 people in the class, three of them are Katya. The teacher assigns Olya to duty, and she needs a partner. What is the probability that Katya will become your partner?
In this example, the favorable outcome is partner Katya. We will solve this problem a little later. But first, using an additional definition, we introduce a formula for finding the probability.
- P = A/N, where P is the probability, A is the number of favorable outcomes, N is the total number of outcomes.
All school problems revolve around this one formula, and the main difficulty usually lies in finding the outcomes. Sometimes they are easy to find, sometimes not so much.
How to solve probability problems?
Problem 1
So now let's solve the above problem.
The number of favorable outcomes (the teacher will choose Katya) is three, because there are three Katyas in the class, and the total outcomes are 24 (25-1, because Olya has already been chosen). Then the probability is: P = 3/24=1/8=0.125. Thus, the probability that Olya’s partner will be Katya is 12.5%. Not difficult, right? Let's look at something a little more complicated.
Problem 2
The coin was tossed twice, what is the probability of getting one head and one tail?
So, let's consider the general outcomes. How can coins land - heads/heads, tails/tails, heads/tails, tails/heads? This means that the total number of outcomes is 4. How many favorable outcomes? Two - heads/tails and tails/heads. Thus, the probability of getting a heads/tails combination is:
- P = 2/4 = 0.5 or 50 percent.
Now let's look at this problem. Masha has 6 coins in her pocket: two with a face value of 5 rubles and four with a face value of 10 rubles. Masha moved 3 coins to another pocket. What is the probability that 5-ruble coins will end up in different pockets?
For simplicity, let's designate the coins by numbers - 1,2 - five-ruble coins, 3,4,5,6 - ten-ruble coins. So, how can coins be in your pocket? There are 20 combinations in total:
- 123, 124, 125, 126, 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456.
At first glance, it may seem that some combinations are missing, for example, 231, but in our case, combinations 123, 231 and 321 are equivalent.
Now we count how many favorable outcomes we have. For them we take those combinations that contain either the number 1 or the number 2: 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256. There are 12 of them. Thus, the probability is equal to:
- P = 12/20 = 0.6 or 60%.
The probability problems presented here are quite simple, but do not think that probability is a simple branch of mathematics. If you decide to continue your education at a university (except humanitarian specialties), you will definitely have pairs higher mathematics, where you will be introduced to more complex terms of this theory, and the tasks there will be much more difficult.
Brief theory
To quantitatively compare events according to the degree of possibility of their occurrence, a numerical measure is introduced, which is called the probability of an event. The probability of a random event is a number that expresses the measure of the objective possibility of an event occurring.
The quantities that determine how significant the objective reasons are to expect the occurrence of an event are characterized by the probability of the event. It must be emphasized that probability is an objective quantity that exists independently of the knower and is conditioned by the entire set of conditions that contribute to the occurrence of an event.
The explanations we have given for the concept of probability are not a mathematical definition, since they do not quantify the concept. There are several definitions of the probability of a random event, which are widely used in solving specific problems (classical, axiomatic, statistical, etc.).
Classic definition of event probability reduces this concept to the more elementary concept of equally possible events, which is no longer subject to definition and is assumed to be intuitively clear. For example, if a die is a homogeneous cube, then the loss of any of the faces of this cube will be equally possible events.
Let a reliable event be divided into equally possible cases, the sum of which gives the event. That is, the cases into which it breaks down are called favorable for the event, since the appearance of one of them ensures the occurrence.
The probability of an event will be denoted by the symbol.
The probability of an event is equal to the ratio of the number of cases favorable to it, out of total number of the only possible, equally possible and incompatible cases to the number, i.e.
This is the classic definition of probability. Thus, to find the probability of an event, it is necessary, having considered the various outcomes of the test, to find a set of uniquely possible, equally possible and incompatible cases, calculate their total number n, the number of cases m favorable for a given event, and then perform the calculation using the above formula.
The probability of an event equal to the ratio of the number of experimental outcomes favorable to the event to the total number of experimental outcomes is called classical probability random event.
The following properties of probability follow from the definition:
Property 1. The probability of a reliable event is equal to one.
Property 2. The probability of an impossible event is zero.
Property 3. The probability of a random event is a positive number between zero and one.
Property 4. The probability of the occurrence of events that form a complete group is equal to one.
Property 5. The probability of the occurrence of the opposite event is determined in the same way as the probability of the occurrence of event A.
The number of cases favoring the occurrence of an opposite event. Hence, the probability of the occurrence of the opposite event is equal to the difference between unity and the probability of the occurrence of event A:
An important advantage of the classical definition of the probability of an event is that with its help the probability of an event can be determined without resorting to experience, but based on logical reasoning.
When a set of conditions is met, a reliable event will definitely happen, but an impossible event will definitely not happen. Among the events that may or may not occur when a set of conditions is created, the occurrence of some can be counted on with good reason, and the occurrence of others with less reason. If, for example, there are more white balls in an urn than black balls, then there is more reason to hope for the appearance of a white ball when drawn from the urn at random than for the appearance of a black ball.
Example of problem solution
Example 1
A box contains 8 white, 4 black and 7 red balls. 3 balls are drawn at random. Find the probabilities of the following events: – at least 1 red ball is drawn, – there are at least 2 balls of the same color, – there are at least 1 red and 1 white ball.
The solution of the problem
We find the total number of test outcomes as the number of combinations of 19 (8+4+7) elements of 3:
Let's find the probability of the event– at least 1 red ball is drawn (1,2 or 3 red balls)
Required probability:
Let the event– there are at least 2 balls of the same color (2 or 3 white balls, 2 or 3 black balls and 2 or 3 red balls)
Number of outcomes favorable to the event:
Required probability:
Let the event– there is at least one red and 1 white ball
(1 red, 1 white, 1 black or 1 red, 2 white or 2 red, 1 white)
Number of outcomes favorable to the event:
Required probability:
Answer: P(A)=0.773;P(C)=0.7688; P(D)=0.6068
Example 2
Two dice are thrown. Find the probability that the sum of points is at least 5.
Solution
Let the event be a score of at least 5
Let's use the classic definition of probability:
Total number of possible test outcomes
Number of trials favoring the event of interest
On the dropped side of the first dice, one point, two points..., six points may appear. similarly, six outcomes are possible when rolling the second die. Each of the outcomes of throwing the first die can be combined with each of the outcomes of the second. Thus, the total number of possible elementary test outcomes is equal to the number of placements with repetitions (choice with placements of 2 elements from a set of volume 6):
Let's find the probability of the opposite event - the sum of points is less than 5
The following combinations of dropped points will favor the event:
| № | 1st bone | 2nd bone | 1 | 1 | 1 | 2 | 1 | 2 | 3 | 2 | 1 | 4 | 3 | 1 | 5 | 1 | 3 |
The geometric definition of probability is presented and the solution to the well-known meeting problem is given.
Whether we like it or not, our life is full of all kinds of accidents, both pleasant and not so pleasant. Therefore, it would not hurt each of us to know how to find the probability of a particular event. This will help you take right decisions under any circumstances that involve uncertainty. For example, such knowledge will be very useful when choosing investment options, assessing the possibility of winning a stock or lottery, determining the reality of achieving personal goals, etc., etc.
Probability theory formula
In principle, studying 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 understand 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 a total number of elementary outcomes. For example, we are interested in how to find the probability that there will be even number points. Then A is a roll of m - rolling out 2, 4 or 6 points (three favorable options), and n is all six possible options.
The calculation formula itself is as follows:
With one outcome everything is extremely easy. But how to find the probability if events happen one after another? Consider this example: one card is shown from a card deck (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 a complex event is considered that 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 happens when several occur simultaneously? Then we multiply the results! For example, the probability that when two coins are tossed simultaneously, two heads will appear will be equal to: ½ * ½ = 0.25.
Now let's take even more complex example. Suppose we entered a book lottery in which ten out of thirty tickets are winning. You need to determine:
- The probability that both will be winners.
- 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 will also be lucky. Let's take into account that the events are dependent, since after each pull out the total number of options decreases. We get:
10 / 30 * 9 / 29 = 0,1034.
In the second case, you will need to determine the probability of a losing ticket and take into account that it can be either the first or the second: 10/30 * 20/29 + 20/29 * 10/30 = 0.4598.
Finally, the third case, when you won’t be able to get even one book from the lottery: 20 / 30 * 19 / 29 = 0.4368.
A professional bettor must have a good understanding of the odds, quickly and correctly estimate the probability of an event by coefficient and, if necessary, be able to convert odds from one format to another. In this manual we will talk about what types of coefficients there are, and also use examples to show how you can calculate the probability using a known coefficient and vice versa.
What types of odds are there?
There are three main types of odds that bookmakers offer players: decimal odds, fractional odds(English) and American odds. The most common odds in Europe are decimal. American odds are popular in North America. Fractional odds are the most traditional type; they immediately reflect information about how much you need to bet to get a certain amount.
Decimal odds
Decimal or they are also called European odds is the familiar number format represented by decimal accurate to hundredths, and sometimes even thousandths. An example of a decimal odd is 1.91. Calculating profit in the case of decimal odds is very simple; you just need to multiply the amount of your bet by this odds. For example, in the match “Manchester United” - “Arsenal”, the victory of “Manchester United” is set with a coefficient of 2.05, a draw is estimated with a coefficient of 3.9, and a victory of “Arsenal” is equal to 2.95. Let's say we're confident United will win and we bet $1,000 on them. Then our possible income is calculated as follows:
2.05 * $1000 = $2050;
It’s really not that complicated, is it?! The possible income is calculated in the same way when betting on a draw or victory for Arsenal.
Draw: 3.9 * $1000 = $3900;
Arsenal win: 2.95 * $1000 = $2950;
How to calculate the probability of an event using decimal odds?
Now imagine that we need to determine the probability of an event based on the decimal odds set by the bookmaker. This is also done very simply. To do this, we divide one by this coefficient.
Let's take the existing data 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 suggests fractional coefficient presented ordinary fraction. An example of English odds is 5/2. The numerator of the fraction contains a number that is the potential amount of the net winnings, and the denominator contains a number indicating the amount that must be bet in order to receive this winning. Simply put, we have to bet $2 dollars to win $5. Odds of 3/2 means that in order to get $3 in net winnings, we will have to bet $2.
How to calculate the probability of an event using fractional odds?
It is also not difficult to calculate the probability of an event using fractional odds; 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 much so in North America. Perhaps this type of coefficients is the most complex, but this is only at first glance. In fact, there is nothing complicated in this type of coefficients. Now let's figure it all out in order.
The main feature of American odds is that they can be either positive, so negative. Example of American odds - (+150), (-120). The American odds (+150) means that in order to earn $150 we need to bet $100. In other words, a positive American coefficient reflects the potential net earnings at a bet of $100. A negative American odds reflect the amount of bet that needs to be made in order to get a net win 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 using the American coefficient is calculated using 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); substitute the value (-120) for “M”;
(-(-120)) / ((-(-120)) + 100 = 120 / (120 + 100) = 120 / 220 = 0,545 = 54,5%;
Thus, the probability of an event with American odds (-120) is 54.5%.
For example, we have a coefficient (+150), then the probability is calculated as follows:
100/(P+100); substitute the value (+150) for “P”;
100 / (150 + 100) = 100 / 250 = 0,4 = 40%;
Thus, the probability of an event with American odds (+150) is 40%.
How, knowing the percentage of probability, convert it into a decimal coefficient?
In order to calculate the decimal coefficient based on a known percentage of probability, you need to divide 100 by the probability of the event as a percentage. For example, 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, convert it into a fractional coefficient?
In order to calculate the fractional coefficient based on a known percentage of probability, you need to subtract one from dividing 100 by the probability of an event as a percentage. For example, if 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, convert it into an American coefficient?
If the probability of an event is more than 50%, then the calculation is made using the formula:
- ((V) / (100 - V)) * 100, where V is probability;
For example, if the probability of an event is 80%, 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 using the formula:
((100 - V) / V) * 100, where V is probability;
For example, if we have a percentage probability 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 odds from one format to another. For example, we have a fractional odds of 3/2 and we need to convert it to decimal. To convert a fractional odds to a decimal odds, we first determine the probability of an event with a fractional odds, and then convert this probability into a decimal odds.
The probability of an event with a fractional odds of 3/2 is 40%.
2 / (3+2) = 2 / 5 = 0,4 = 40%;
Now let’s convert the probability of an event into a decimal coefficient; to do this, divide 100 by the probability of the event as a percentage:
100 / 40% = 2.5;
Thus, the fractional odds of 3/2 are equal to the decimal odds of 2.5. In a similar way, for example, American odds are converted to fractional, decimal to American, etc. The most difficult thing in all this is just the calculations.
This is the ratio of the number of those observations in which the event in question occurred to the total number of observations. This interpretation is acceptable in the case of sufficient large quantity 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 will be a woman is 1/2. In other words, an estimate of the probability of an event can be the frequency of its occurrence in a long series of independent repetitions of a random experiment.
Probability in mathematics
In the modern mathematical approach, classical (that is, not quantum) probability is given by the Kolmogorov axiomatics. Probability is a measure P, which is defined on the set X, called probability space. This measure must have the following properties:
From these conditions it follows that the probability measure P also has the property additivity: if sets A 1 and A 2 do not intersect, then . To prove 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 is enough to define it on a sigma algebra, consisting of some subsets of the set X. In this case, random events are defined as measurable subsets of space X, that is, as elements of sigma algebra.
Probability sense
When we find that the reasons for some possible fact actually occurring outweigh the contrary reasons, we consider that fact probable, otherwise - incredible. This preponderance of positive bases over negative ones, and vice versa, can represent an indefinite set of degrees, as a result of which probability(And improbability) It happens more or less .
Complex individual facts do not allow for an exact calculation of the degrees of their probability, but even here it is important to establish some large subdivisions. So, for example, in the legal field, when a personal fact subject to trial is established on the basis of 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 adopted here: probatio plena(where the probability practically turns into reliability), Further - probatio minus plena, then - probatio semiplena major and finally probatio semiplena minor .
In addition to the question of the likelihood of the case, the question may arise, both in the field of law and in the moral field (with a certain ethical point of view), of how likely it is that a given particular fact constitutes a violation common law. This question, which serves as the main motive in the religious jurisprudence of the Talmud, also gave rise to very complex systematic constructions and a huge literature, dogmatic and polemical, in Roman Catholic moral theology (especially from the end of the 16th century) (see Probabilism).
The concept of probability allows for a certain numerical expression when applied only to such facts that are part of certain homogeneous series. Thus (in the simplest example), when someone throws a coin a hundred times in a row, we find here one general or large series (the sum of all falls of the coin), consisting of two particular or smaller ones, in in this case numerically equal rows (falling “heads” and falling “tails”); The probability that this time the coin will land heads, that is, that this new member of the general series will belong to this of the two smaller series, is equal to the fraction expressing the numerical relationship between this small series and the larger one, namely 1/2, that is, the same probability belongs to one or the other of two particular series. In less simple examples, the conclusion cannot be deduced directly from the data of the problem itself, but requires prior induction. So, for example, the question is: what is the probability for a given newborn to live to be 80 years old? Here there must be a general, or large, series of a certain 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 maintain the homogeneity of the series, for for a person, born, for example, in St. Petersburg into a wealthy, cultured family, the entire million-strong population of the city, a significant part of which consists of people from various groups who can die prematurely - soldiers, journalists, workers dangerous professions, - represents a group too heterogeneous for a true probability determination); let this general series consist of ten thousand human lives; it includes smaller series representing the number of people surviving to a particular age; one of these smaller series represents the number of people living to age 80. But it is impossible to determine the number of this smaller series (like all others) a priori; this is done purely inductively, through statistics. Let's say statistical research found that out of 10,000 middle-class St. Petersburg residents, only 45 live to be 80; Thus, this smaller series is related to the larger one as 45 is to 10,000, and the probability for a given person to belong to this smaller series, that is, to live to be 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 - probability theory.
see also
Notes
Literature
- Alfred Renyi. Letters on probability / trans. from Hungarian D. Saas and A. Crumley, eds. B.V. Gnedenko. M.: Mir. 1970
- Gnedenko B.V. Probability theory course. M., 2007. 42 p.
- Kuptsov V.I. Determinism and probability. M., 1976. 256 p.
Wikimedia Foundation.
2010.:Synonyms:
Antonyms
General scientific and philosophical. a category denoting the quantitative degree of possibility of the occurrence of mass random events under fixed observation conditions, characterizing the stability of their relative frequencies. In logic, semantic degree... ... Philosophical Encyclopedia
PROBABILITY, a number in the range from zero to one inclusive, representing the possibility of a given event occurring. 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 similar expressions. under. ed. N. Abramova, M.: Russian Dictionaries, 1999. probability possibility, likelihood, chance, objective possibility, maza, admissibility, risk. Ant. impossibility... ... Synonym dictionary
probability- A measure that an event is likely to occur. Note The mathematical definition of probability is: “a real number between 0 and 1 that relates to random event" The number may reflect the relative frequency in a series of observations... ... Technical Translator's Guide
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-mathematical dictionary
- (probability) The possibility of the occurrence of an event or a certain result. It can be presented in the form of 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... Dictionary of business terms
