Initially, being just a collection of information and empirical observations behind the game of dice, the theory of probability became a thorough science. The first to give it a mathematical framework were Fermat and Pascal.
From thinking about the eternal to the theory of probability
The two individuals to whom probability theory owes many of its fundamental formulas, Blaise Pascal and Thomas Bayes, are known as deeply religious people, the latter being a Presbyterian minister. Apparently, the desire of these two scientists to prove the fallacy of the opinion about a certain Fortune giving good luck to her favorites gave impetus to research in this area. After all, in fact, any gambling with its wins and losses, it is just a symphony of mathematical principles.
Thanks to the passion of the gentleman de Mere, who equally being a gambler and a person not indifferent to science, Pascal was forced to find a way to calculate probability. De Mere was interested in the following question: “How many times do you need to throw two dice in pairs so that the probability of getting 12 points exceeds 50%?” The second question, which was of great interest to the gentleman: “How to divide the bet between the participants in the unfinished game?” Of course, Pascal successfully answered both questions of de Mere, who became the unwitting initiator of the development of probability theory. It is interesting that the person of de Mere remained known in this area, and not in literature.
Previously, no mathematician had ever attempted to calculate the probabilities of events, since it was believed that this was only a guessing solution. Blaise Pascal gave the first definition of the probability of an event and showed that it is a specific figure that can be justified mathematically. Probability theory has become the basis for statistics and is widely used in modern science.
What is randomness
Considering a test that can be repeated infinite number times, then we can define a random event. This is one of the likely outcomes of the experiment.
Experience is the implementation concrete actions under constant conditions.
To be able to work with the results of the experiment, events are usually designated by the letters A, B, C, D, E...
Probability of a random event
In order to begin the mathematical part of probability, it is necessary to define all its components.
The probability of an event is expressed in numerical form a measure of the possibility of some event (A or B) occurring as a result of an experience. The probability is denoted as P(A) or P(B).
In probability theory they distinguish:
- reliable the event is guaranteed to occur as a result of the experience P(Ω) = 1;
- impossible the event can never happen P(Ø) = 0;
- random an event lies between certain and impossible, that is, the probability of its occurrence is possible, but not guaranteed (probability random event always within 0≤Р(А)≤ 1).
Relationships between events
Both one and the sum of events A+B are considered, when the event is counted when at least one of the components, A or B, or both, A and B, is fulfilled.
In relation to each other, events can be:
- Equally possible.
- Compatible.
- Incompatible.
- Opposite (mutually exclusive).
- Dependent.
If two events can happen with equal probability, then they equally possible.
If the occurrence of event A does not reduce to zero the probability of the occurrence of event B, then they compatible.
If events A and B never occur simultaneously in the same experience, then they are called incompatible. Coin toss - good example: the appearance of heads is automatically the non-appearance of heads.
The probability for the sum of such incompatible events consists of the sum of the probabilities of each of the events:
P(A+B)=P(A)+P(B)
If the occurrence of one event makes the occurrence of another impossible, then they are called opposite. Then one of them is designated as A, and the other - Ā (read as “not A”). The occurrence of event A means that Ā did not occur. These two events form a complete group with a sum of probabilities equal to 1.
Dependent events have mutual influence, decreasing or increasing the probability of each other.
Relationships between events. Examples
Using examples it is much easier to understand the principles of probability theory and combinations of events.
The experiment that will be carried out consists of taking balls out of a box, and the result of each experiment is an elementary outcome.
An event is one of the possible outcomes of an experiment - a red ball, a blue ball, a ball with number six, etc.
Test No. 1. There are 6 balls involved, three of which are blue with odd numbers on them, and the other three are red with even numbers.
Test No. 2. 6 balls involved of blue color with numbers from one to six.
Based on this example, we can name combinations:
- Reliable event. In Spanish No. 2 the event “get the blue ball” is reliable, since the probability of its occurrence is equal to 1, since all the balls are blue and there can be no miss. Whereas the event “get the ball with the number 1” is random.
- Impossible event. In Spanish No. 1 with blue and red balls, the event “getting the purple ball” is impossible, since the probability of its occurrence is 0.
- Equally possible events. In Spanish No. 1, the events “get the ball with the number 2” and “get the ball with the number 3” are equally possible, and the events “get the ball with an even number” and “get the ball with the number 2” have different probabilities.
- Compatible Events. Getting a six twice in a row while throwing a die is a compatible event.
- Incompatible events. In the same Spanish No. 1, the events “get a red ball” and “get a ball with an odd number” cannot be combined in the same experience.
- Opposite events. Most shining example This is coin tossing, where drawing heads is equivalent to not drawing tails, and the sum of their probabilities is always 1 (full group).
- Dependent Events. So, in Spanish No. 1, you can set the goal of drawing the red ball twice in a row. Whether or not it is retrieved the first time affects the likelihood of being retrieved the second time.
It can be seen that the first event significantly affects the probability of the second (40% and 60%).
Event probability formula
The transition from fortune-telling to precise data occurs through the translation of the topic into a mathematical plane. That is, judgments about a random event such as “high probability” or “minimal probability” can be translated into specific numerical data. It is already permissible to evaluate, compare and enter such material into more complex calculations.
From the point of view of calculation, determining the probability of an event is the ratio of the number of elementary positive outcomes to the number of all possible outcomes of the experience relative to certain event. Probability is denoted by P(A), where P stands for the word “probabilite”, which is translated from French as “probability”.
So, the formula for the probability of an event is:
Where m is the number of favorable outcomes for event A, n is the sum of all outcomes possible for this experience. In this case, the probability of an event always lies between 0 and 1:
0 ≤ P(A)≤ 1.
Calculation of the probability of an event. Example
Let's take Spanish. No. 1 with balls, which was described earlier: 3 blue balls with the numbers 1/3/5 and 3 red balls with the numbers 2/4/6.
Based on this test, several different problems can be considered:
- A - red ball falling out. There are 3 red balls, and there are 6 options in total. This is simplest example, in which the probability of the event is equal to P(A)=3/6=0.5.
- B - rolling an even number. There are 3 even numbers (2,4,6), and the total number of possible numerical options is 6. The probability of this event is P(B)=3/6=0.5.
- C - the occurrence of a number greater than 2. There are 4 such options (3,4,5,6) out of a total number of possible outcomes of 6. The probability of event C is equal to P(C)=4/6=0.67.
As can be seen from the calculations, event C has high probability, since the number of probable positive outcomes is higher than in A and B.
Incompatible events
Such events cannot appear simultaneously in the same experience. As in Spanish No. 1 it is impossible to get a blue and a red ball at the same time. That is, you can get either a blue or a red ball. In the same way, an even and an odd number cannot appear in a dice at the same time.
The probability of two events is considered as the probability of their sum or product. The sum of such events A+B is considered to be an event that consists of the occurrence of event A or B, and the product of them AB is the occurrence of both. For example, the appearance of two sixes at once on the faces of two dice in one throw.
The sum of several events is an event that presupposes the occurrence of at least one of them. The production of several events is the joint occurrence of them all.
In probability theory, as a rule, the use of the conjunction “and” denotes a sum, and the conjunction “or” - multiplication. Formulas with examples will help you understand the logic of addition and multiplication in probability theory.
Probability of the sum of incompatible events
If probability is considered Not joint events, then the probability of the sum of events is equal to the addition of their probabilities:
P(A+B)=P(A)+P(B)
For example: let's calculate the probability that in Spanish. No. 1 with blue and red balls, a number between 1 and 4 will appear. We will calculate not in one action, but by the sum of the probabilities of the elementary components. So, in such an experiment there are only 6 balls or 6 of all possible outcomes. The numbers that satisfy the condition are 2 and 3. The probability of getting the number 2 is 1/6, the probability of getting the number 3 is also 1/6. The probability of getting a number between 1 and 4 is:
The probability of the sum of incompatible events of a complete group is 1.
So, if in an experiment with a cube we add up the probabilities of all numbers appearing, the result will be one.
This is also true for opposite events, for example in the experiment with a coin, where one side is the event A, and the other is the opposite event Ā, as is known,
P(A) + P(Ā) = 1
Probability of incompatible events occurring
Probability multiplication is used when considering the occurrence of two or more incompatible events in one observation. The probability that events A and B will appear in it simultaneously is equal to the product of their probabilities, or:
P(A*B)=P(A)*P(B)
For example, the probability that in Spanish No. 1, as a result of two attempts, a blue ball will appear twice, equal to
That is, the probability of an event occurring when, as a result of two attempts to extract balls, only blue balls are extracted is 25%. It is very easy to do practical experiments on this problem and see if this is actually the case.
Joint events
Events are considered joint when the occurrence of one of them can coincide with the occurrence of another. Despite the fact that they are joint, the probability is considered Not dependent events. For example, throwing two dice can give a result when the number 6 appears on both of them. Although the events coincided and appeared at the same time, they are independent of each other - only one six could fall out, the second die has no influence on it.
The probability of joint events is considered as the probability of their sum.
Probability of the sum of joint events. Example
The probability of the sum of events A and B, which are joint in relation to each other, is equal to the sum of the probabilities of the event minus the probability of their occurrence (that is, their joint occurrence):
R joint (A+B)=P(A)+P(B)- P(AB)
Let's assume that the probability of hitting the target with one shot is 0.4. Then event A is hitting the target in the first attempt, B - in the second. These events are joint, since it is possible that you can hit the target with both the first and second shots. But events are not dependent. What is the probability of the event of hitting the target with two shots (at least with one)? According to the formula:
0,4+0,4-0,4*0,4=0,64
The answer to the question is: “The probability of hitting the target with two shots is 64%.”
This formula for the probability of an event can also be applied to incompatible events, where the probability of the joint occurrence of an event P(AB) = 0. This means that the probability of the sum of incompatible events can be considered a special case of the proposed formula.
Geometry of probability for clarity
Interestingly, the probability of the sum of joint events can be represented as two areas A and B, which intersect with each other. As can be seen from the picture, the area of their union is equal to total area minus the area of their intersection. This geometric explanation makes the seemingly illogical formula more understandable. Note that geometric solutions- not uncommon in probability theory.
Determining the probability of the sum of many (more than two) joint events is quite cumbersome. To calculate it, you need to use the formulas that are provided for these cases.
Dependent Events
Events are called dependent if the occurrence of one (A) of them affects the probability of the occurrence of another (B). Moreover, the influence of both the occurrence of event A and its non-occurrence is taken into account. Although events are called dependent by definition, only one of them is dependent (B). Ordinary probability was denoted as P(B) or the probability of independent events. In the case of addicts, a new concept is introduced - conditional probability P A (B), which is the probability of a dependent event B given the event A (hypothesis) that it depends on.
But event A is also random, so it also has a probability that needs and can be taken into account in the calculations performed. The following example will show how to work with dependent events and a hypothesis.
An example of calculating the probability of dependent events
A good example for calculating dependent events would be a standard deck of cards.
Using a deck of 36 cards as an example, let’s look at dependent events. We need to determine the probability that the second card drawn from the deck will be of diamonds if the first card drawn is:
- Bubnovaya.
- A different color.
Obviously, the probability of the second event B depends on the first A. So, if the first option is true, that there is 1 card (35) and 1 diamond (8) less in the deck, the probability of event B:
R A (B) =8/35=0.23
If the second option is true, then the deck now has 35 cards, and the full number tambourine (9), then the probability of the next event B:
R A (B) =9/35=0.26.
It can be seen that if event A is conditioned on the fact that the first card is a diamond, then the probability of event B decreases, and vice versa.
Multiplying dependent events
Guided by the previous chapter, we accept the first event (A) as a fact, but in essence, it is of a random nature. The probability of this event, namely drawing a diamond from a deck of cards, is equal to:
P(A) = 9/36=1/4
Since theory does not exist on its own, but is intended to serve in practical purposes, then it is fair to note that what is most often needed is the probability of producing dependent events.
According to the theorem on the product of probabilities of dependent events, the probability of occurrence of jointly dependent events A and B is equal to the probability of one event A, multiplied by the conditional probability of event B (dependent on A):
P(AB) = P(A) *P A(B)
Then, in the deck example, the probability of drawing two cards with the suit of diamonds is:
9/36*8/35=0.0571, or 5.7%
And the probability of extracting not diamonds first, and then diamonds, is equal to:
27/36*9/35=0.19, or 19%
It can be seen that the probability of event B occurring is greater provided that the first card drawn is of a suit other than diamonds. This result is quite logical and understandable.
Total probability of an event
When a problem with conditional probabilities becomes multifaceted, it cannot be calculated using conventional methods. When there are more than two hypotheses, namely A1, A2,…, A n, ..forms a complete group of events provided:
- P(A i)>0, i=1,2,…
- A i ∩ A j =Ø,i≠j.
- Σ k A k =Ω.
So the formula full probability for event B at full group random events A1,A2,…,And n is equal to:
A look into the future
The probability of a random event is extremely necessary in many areas of science: econometrics, statistics, physics, etc. Since some processes cannot be described deterministically, since they themselves are probabilistic in nature, special working methods are required. The theory of event probability can be used in any technological field as a way to determine the possibility of an error or malfunction.
We can say that by recognizing probability, we in some way take a theoretical step into the future, looking at it through the prism of formulas.
Everything in the world happens deterministically or by chance...
Aristotle
Probability: Basic Rules
Probability theory calculates the probabilities of various events. Fundamental to probability theory is the concept of a random event.
For example, you throw a coin, it randomly falls on the coat of arms or tails. You don't know in advance which side the coin will land on. You enter into an insurance contract; you do not know in advance whether payments will be made or not.
In actuarial calculations you need to be able to estimate the probability of various events, so probability theory plays a role key role. No other branch of mathematics can deal with the probabilities of events.
Let's take a closer look at tossing a coin. There are 2 mutually exclusive outcomes: the coat of arms falls out or the tails fall out. The outcome of the throw is random, since the observer cannot analyze and take into account all the factors that influence the result. What is the probability of the coat of arms falling out? Most will answer ½, but why?
Let it be formal A indicates the loss of the coat of arms. Let the coin toss n once. Then the probability of the event A can be defined as the proportion of those throws that result in a coat of arms:
Where n total number of throws, n(A) number of coat of arms drops.
Relation (1) is called frequency events A in a long series of tests.
It turns out that in various series of tests the corresponding frequency at large n clusters around some constant value P(A). This quantity is called probability of an event A and is designated by the letter R- abbreviation for English word probability - probability.
Formally we have:
(2)
This law is called law of large numbers.
If the coin is fair (symmetrical), then the probability of getting a coat of arms is equal to the probability of getting heads and equals ½.
Let A And IN some events, for example, whether an insured event occurred or not. The union of two events is an event consisting of the execution of an event A, events IN, or both events together. The intersection of two events A And IN called an event consisting in the implementation as an event A, and events IN.
Basic Rules The calculus of event probabilities is as follows:
1. The probability of any event lies between zero and one:
2. Let A and B be two events, then:
It reads like this: the probability of two events combining is equal to the sum of the probabilities of these events minus the probability of the events intersecting. If the events are incompatible or non-overlapping, then the probability of the combination (sum) of two events is equal to the sum of the probabilities. This law is called the law addition probabilities.
We say that an event is reliable if its probability is equal to 1. When analyzing certain phenomena, the question arises of how the occurrence of an event affects IN upon the occurrence of an event A. To do this, enter conditional probability :
(4)
It reads like this: probability of occurrence A given that IN equals the probability of intersection A And IN, divided by the probability of the event IN.
Formula (4) assumes that the probability of an event IN Above zero.
Formula (4) can also be written as:
(5)
This is the formula multiplying probabilities.
Conditional probability is also called a posteriori probability of an event A- probability of occurrence A after the offensive IN.
In this case, the probability itself is called a priori probability. There are several more important formulas, which are intensively used in actuarial calculations.
Total Probability Formula
Let us assume that an experiment is being carried out, the conditions of which can be determined in advance mutually mutually exclusive assumptions (hypotheses):
We assume that there is either a hypothesis, or... or. The probabilities of these hypotheses are known and equal:
Then the formula holds full probabilities :
(6)
Probability of an event occurring A equal to the sum of the products of the probability of occurrence A for each hypothesis on the probability of this hypothesis.
Bayes formula
Bayes formula
allows you to recalculate the probability of hypotheses in the light new information which gave the result A.
Bayes' formula in a certain sense is the inverse of the total probability formula.
Consider the following practical problem.
Problem 1
Suppose there is a plane crash and experts are busy investigating its causes. 4 reasons why the disaster occurred are known in advance: either the cause, or, or, or. According to available statistics, these reasons have the following probabilities:
When examining the crash site, traces of fuel ignition were found; according to statistics, the probability of this event for one reason or another is as follows:
Question: what is the most likely cause of the disaster?
Let's calculate the probabilities of causes under the conditions of the occurrence of an event A.
From this it can be seen that the first reason is the most likely, since its probability is maximum.
Problem 2
Consider an airplane landing at an airfield.
Upon landing weather may be as follows: no low clouds (), low clouds yes (). In the first case, the probability of a safe landing is P1. In the second case - P2. It's clear that P1>P2.
Devices that provide blind landing have a probability of trouble-free operation R. If there is low cloud cover and the blind landing instruments have failed, the probability of a successful landing is P3, and P3<Р2 . It is known that for a given airfield the proportion of days in a year with low clouds is equal to .
Find the probability of the plane landing safely.
We need to find the probability.
There are two mutually exclusive options: the blind landing devices are working, the blind landing devices have failed, so we have:
Hence, according to the total probability formula:
Problem 3
An insurance company provides life insurance. 10% of those insured by this company are smokers. If the insured person does not smoke, the probability of his death during the year is 0.01. If he is a smoker, then this probability is 0.05.
What is the proportion of smokers among those insured who died during the year?
Possible answers: (A) 5%, (B) 20%, (C) 36%, (D) 56%, (E) 90%.
Solution
Let's enter the events:
The condition of the problem means that
In addition, since the events form a complete group of pairwise incompatible events, then .
The probability we are interested in is .
Using Bayes' formula, we have:
therefore the correct option is ( IN).
Problem 4
The insurance company sells life insurance contracts in three categories: standard, preferred and ultra-privileged.
50% of all insured are standard, 40% are preferred and 10% are ultra-privileged.
The probability of death within a year for a standard insured is 0.010, for a privileged one - 0.005, and for an ultra-privileged one - 0.001.
What is the probability that the deceased insured is ultra-privileged?
Solution
Let us introduce the following events into consideration:
In terms of these events, the probability we are interested in is . By condition:
Since the events , , form a complete group of pairwise incompatible events, using Bayes' formula we have:
Random variables and their characteristics
Let it be some random variable, for example, damage from a fire or the amount of insurance payments.
A random variable is completely characterized by its distribution function.
Definition. Function 
called distribution function
random variable ξ
.
Definition. If there is a function such that for arbitrary a done
then they say that the random variable ξ It has probability density function f(x).
Definition. Let . For a continuous distribution function F theoretical α-quantile is called the solution to the equation.
This solution may not be the only one.
Quantile level ½ called theoretical median , quantile levels ¼ And ¾ -lower and upper quartiles respectively.
In actuarial applications, plays an important role Chebyshev's inequality:
at any
Symbol of mathematical expectation.
It reads like this: the probability that the modulus is greater than or equal to the mathematical expectation of the modulus divided by .
Lifetime as a random variable
The uncertainty of the moment of death is a major risk factor in life insurance.
Nothing definite can be said about the moment of death of an individual. However, if we are dealing with a large homogeneous group of people and are not interested in the fate of individual people from this group, then we are within the framework of probability theory as the science of mass random phenomena that have the property of frequency stability.
Respectively, we can talk about life expectancy as a random variable T.
Survival function
Probability theory describes the stochastic nature of any random variable T distribution function F(x), which is defined as the probability that the random variable T less than number x:
.
In actuarial mathematics it is nice to work not with the distribution function, but with the additional distribution function 
.
In terms of longevity, this is the probability that a person will live to age x years.
called survival function(survival function):
The survival function has the following properties:
Life tables usually assume that there is some age limit (limiting age) (usually years) and, accordingly, at x>.
When describing mortality by analytical laws, it is usually assumed that life time is unlimited, but the type and parameters of the laws are selected so that the probability of life beyond a certain age is negligible.
The survival function has simple statistical meaning.
Let's say that we are observing a group of newborns (usually), whom we observe and can record the moments of their death.
Let us denote the number of living representatives of this group at age by . Then:
.
Symbol E here and below used to denote mathematical expectation.
So, the survival function is equal to the average proportion of those who survive to age from some fixed group of newborns.
In actuarial mathematics, one often works not with the survival function, but with the value just introduced (fixing the initial group size).
The survival function can be reconstructed from density:
Lifespan Characteristics
From a practical point of view, the following characteristics are important:
1 . Average lifetime
,
2
. Dispersion lifetime
,
Where 
,
Presented to date in the open bank of Unified State Exam problems in mathematics (mathege.ru), the solution of which is based on only one formula, which is the classical definition of probability.
The easiest way to understand the formula is with examples.
Example 1. There are 9 red balls and 3 blue balls in the basket. The balls differ only in color. We take out one of them at random (without looking). What is the probability that the ball chosen in this way will be blue?
A comment. In problems in probability theory, something happens (in this case, our action of pulling out the ball) that can have a different result - an outcome. It should be noted that the result can be looked at in different ways. “We pulled out some kind of ball” is also a result. “We pulled out the blue ball” - the result. “We pulled out exactly this ball from all possible balls” - this least generalized view of the result is called an elementary outcome. It is the elementary outcomes that are meant in the formula for calculating the probability.
Solution. Now let's calculate the probability of choosing the blue ball.
Event A: “the selected ball turned out to be blue”
Total number of all possible outcomes: 9+3=12 (the number of all balls that we could draw)
Number of outcomes favorable for event A: 3 (the number of such outcomes in which event A occurred - that is, the number of blue balls)
P(A)=3/12=1/4=0.25
Answer: 0.25
For the same problem, let's calculate the probability of choosing a red ball.
The total number of possible outcomes will remain the same, 12. Number of favorable outcomes: 9. Probability sought: 9/12=3/4=0.75
The probability of any event always lies between 0 and 1.
Sometimes in everyday speech (but not in probability theory!) the probability of events is estimated as a percentage. The transition between math and conversational scores is accomplished by multiplying (or dividing) by 100%.
So,
Moreover, the probability is zero for events that cannot happen - incredible. For example, in our example this would be the probability of drawing a green ball from the basket. (The number of favorable outcomes is 0, P(A)=0/12=0, if calculated using the formula)
Probability 1 has events that are absolutely certain to happen, without options. For example, the probability that “the selected ball will be either red or blue” is for our task. (Number of favorable outcomes: 12, P(A)=12/12=1)
We looked at a classic example illustrating the definition of probability. All similar problems of the Unified State Exam in probability theory are solved by using this formula.
In place of the red and blue balls there may be apples and pears, boys and girls, learned and unlearned tickets, tickets containing and not containing a question on some topic (prototypes,), defective and high-quality bags or garden pumps (prototypes,) - the principle remains the same.
They differ slightly in the formulation of the problem of the probability theory of the Unified State Examination, where you need to calculate the probability of some event occurring on a certain day. ( , ) As in previous problems, you need to determine what is the elementary outcome, and then apply the same formula.
Example 2. The conference lasts three days. On the first and second days there are 15 speakers, on the third day - 20. What is the probability that Professor M.’s report will fall on the third day if the order of reports is determined by drawing lots?
What is the elementary outcome here? – Assigning the professor’s report one of all possible serial numbers for the speech. 15+15+20=50 people participate in the draw. Thus, Professor M.'s report may receive one of 50 issues. This means there are only 50 elementary outcomes.
What are the favorable outcomes? - Those in which it turns out that the professor will speak on the third day. That is, the last 20 numbers.
According to the formula, probability P(A)= 20/50=2/5=4/10=0.4
Answer: 0.4
The drawing of lots here represents the establishment of a random correspondence between people and ordered places. In example 2, matching was considered from the point of view of which of the places a particular person could occupy. You can approach the same situation from the other side: which of the people with what probability could get to a specific place (prototypes , , , ):
Example 3. The draw includes 5 Germans, 8 French and 3 Estonians. What is the probability that the first (/second/seventh/last – it doesn’t matter) will be a Frenchman.
The number of elementary outcomes is the number of all possible people who could get into a given place by drawing lots. 5+8+3=16 people.
Favorable outcomes - French. 8 people.
Required probability: 8/16=1/2=0.5
Answer: 0.5
The prototype is slightly different. There are still problems about coins () and dice (), which are somewhat more creative. The solution to these problems can be found on the prototype pages.
Here are a few examples of tossing a coin or dice.
Example 4. When we toss a coin, what is the probability of landing on heads?
There are 2 outcomes – heads or tails. (it is believed that the coin never lands on its edge) A favorable outcome is tails, 1.
Probability 1/2=0.5
Answer: 0.5.
Example 5. What if we toss a coin twice? What is the probability of getting heads both times?
The main thing is to determine what elementary outcomes we will consider when tossing two coins. After tossing two coins, one of the following results can occur:
1) PP – both times it came up heads
2) PO – first time heads, second time heads
3) OP – heads the first time, tails the second time
4) OO – heads came up both times
There are no other options. This means that there are 4 elementary outcomes. Only the first one, 1, is favorable.
Probability: 1/4=0.25
Answer: 0.25
What is the probability that two coin tosses will result in tails?
The number of elementary outcomes is the same, 4. Favorable outcomes are the second and third, 2.
Probability of getting one tail: 2/4=0.5
In such problems, another formula may be useful.
If with one toss of a coin we have 2 possible outcome options, then for two tosses the results will be 2 2 = 2 2 = 4 (as in example 5), for three tosses 2 2 2 = 2 3 = 8, for four: 2·2·2·2=2 4 =16, ... for N rolls the possible results will be 2·2·...·2=2 N .
So, you can find the probability of getting 5 heads out of 5 coin tosses.
Total number of elementary outcomes: 2 5 =32.
Favorable outcomes: 1. (RRRRRR – heads all 5 times)
Probability: 1/32=0.03125
The same is true for dice. With one throw, there are 6 possible results. So, for two throws: 6 6 = 36, for three 6 6 6 = 216, etc.
Example 6. We throw the dice. What is the probability that an even number will be rolled?
Total outcomes: 6, according to the number of sides.
Favorable: 3 outcomes. (2, 4, 6)
Probability: 3/6=0.5
Example 7. We throw two dice. What is the probability that the total will be 10? (round to the nearest hundredth)
For one die there are 6 possible outcomes. This means that for two, according to the above rule, 6·6=36.
What outcomes will be favorable for the total to roll 10?
10 must be decomposed into the sum of two numbers from 1 to 6. This can be done in two ways: 10=6+4 and 10=5+5. This means that the following options are possible for the cubes:
(6 on the first and 4 on the second)
(4 on the first and 6 on the second)
(5 on the first and 5 on the second)
Total, 3 options. Required probability: 3/36=1/12=0.08
Answer: 0.08
Other types of B6 problems will be discussed in a future How to Solve article.
- Probability is the degree (relative measure, quantitative assessment) of the possibility of the occurrence of some event. When the reasons for some possible event to actually occur outweigh the opposite reasons, then this event is called probable, otherwise - unlikely or improbable. The preponderance of positive reasons over negative ones, and vice versa, can be to varying degrees, as a result of which the probability (and improbability) can be greater or lesser. Therefore, probability is often assessed at a qualitative level, especially in cases where a more or less accurate quantitative assessment is impossible or extremely difficult. Various gradations of “levels” of probability are possible.
The study of probability from a mathematical point of view constitutes a special discipline - probability theory. In probability theory and mathematical statistics, the concept of probability is formalized as a numerical characteristic of an event - a probability measure (or its value) - a measure on a set of events (subsets of a set of elementary events), taking values from
(\displaystyle 0)
(\displaystyle 1)
Meaning
(\displaystyle 1)
Corresponds to a reliable event. An impossible event has a probability of 0 (the converse is generally not always true). If the probability of an event occurring is
(\displaystyle p)
Then the probability of its non-occurrence is equal to
(\displaystyle 1-p)
In particular, the probability
(\displaystyle 1/2)
Means equal probability of occurrence and non-occurrence of an event.
The classic definition of probability is based on the concept of equal probability of outcomes. The probability is the ratio of the number of outcomes favorable for a given event to the total number of equally possible outcomes. For example, the probability of getting heads or tails in a random coin toss is 1/2 if it is assumed that only these two possibilities occur and that they are equally possible. This classical “definition” of probability can be generalized to the case of an infinite number of possible values - for example, if some event can occur with equal probability at any point (the number of points is infinite) of some limited region of space (plane), then the probability that it will occur in some part of this feasible region is equal to the ratio of the volume (area) of this part to the volume (area) of the region of all possible points.
The empirical “definition” of probability is related to the frequency of an event, based on the fact that with a sufficiently large number of trials, the frequency should tend to the objective degree of possibility of this event. In the modern presentation of probability theory, probability is defined axiomatically, as a special case of the abstract theory of set measure. However, the connecting link between the abstract measure and the probability, which expresses the degree of possibility of the occurrence of an event, is precisely the frequency of its observation.
The probabilistic description of certain phenomena has become widespread in modern science, in particular in econometrics, statistical physics of macroscopic (thermodynamic) systems, where even in the case of a classical deterministic description of the movement of particles, a deterministic description of the entire system of particles does not seem practically possible or appropriate. In quantum physics, the processes described are themselves probabilistic in nature.