The sum of all probabilities of events in the sample space equals 1. For example, if the experiment is tossing a coin with Event A = heads and Event B = tails, then A and B represent the entire sample space. Means, P(A) + P(B) = 0.5 + 0.5 = 1.
Example.In the previously proposed example of calculating the probability of removing a red pen from a robe pocket (this is event A), which contains two blue and one red pens, P(A) = 1/3 ≈ 0.33, the probability of the opposite event - drawing a blue pen - will be
Before moving on to the main theorems, we introduce two more complex concepts - the sum and product of events. These concepts are different from the usual concepts of sum and product in arithmetic. Addition and multiplication in probability theory - symbolic operations, subject to certain rules and facilitating the logical construction of scientific conclusions.
Amount several events is an event consisting in the occurrence of at least one of them. That is, the sum of two events A and B is called event C, which consists of the occurrence of either event A, or event B, or events A and B together.
For example, if a passenger is waiting at a tram stop for one of two routes, then the event he needs is the appearance of a tram on the first route (event A), or a tram on the second route (event B), or the joint appearance of trams on the first and second routes (event WITH). In the language of probability theory, this means that the event D needed by the passenger consists in the occurrence of either event A, or event B, or event C, which will be symbolically written in the form:
D=A+B+C
The product of two eventsA And IN is an event consisting of the joint occurrence of events A And IN. The product of several events the joint occurrence of all these events is called.
In the above example with a passenger, the event WITH(joint appearance of trams on two routes) is the product of two events A And IN, which is symbolically written as follows:
Let's say that two doctors separately examine a patient to identify a specific disease. During inspections, the following events may occur:
Discovery of diseases by the first doctor ( A);
Failure to detect the disease by the first doctor ();
Detection of the disease by a second doctor ( IN);
Failure to detect the disease by the second doctor ().
Consider the event that the disease will be detected during examinations exactly once. This event can be realized in two ways:
The disease will be discovered by the first doctor ( A) and will not detect the second ();
Diseases will not be detected by the first doctor () and will be detected by the second ( B).
Let us denote the event under consideration by and write it symbolically:
Consider the event that the disease will be detected during examinations twice (by both the first and the second doctor). Let's denote this event by and write: .
We denote the event that neither the first nor the second doctor discovers the disease by and write it down: .
Joint and non-joint events.
The two events are called joint in a given experiment, if the appearance of one of them does not exclude the appearance of the other. Examples : Hitting an indestructible target with two different arrows and getting the same number of points on both dice.
The two events are called incompatible(incompatible) in a given experiment if they cannot occur together in the same trial. Several events are called incompatible if they are pairwise incompatible. Examples of incompatible events: a) hit and miss with one shot; b) a part is randomly taken from a box with parts - the events “a standard part is taken out” and “a non-standard part is taken out” c) the ruin of the company and its profit.
In other words, events A And IN are compatible if the corresponding sets A And IN have common elements, and are inconsistent if the corresponding sets A And IN have no common elements.
When determining the probabilities of events, the concept is often used equally possible events. Several events in a given experiment are called equally possible if, according to the conditions of symmetry, there is reason to believe that none of them is objectively more possible than the others (the loss of heads and tails, the appearance of a card of any suit, the choice of a ball from an urn, etc.)
Each trial is associated with a number of events, which, generally speaking, can occur simultaneously. For example, when throwing a dice, the event is the roll of a two, and the event is the roll of an even number. Obviously, these events are not mutually exclusive.
Let all possible test results be realized in a number of uniquely possible particular cases, mutually exclusive of each other. Then
ü each test outcome is represented by one and only one elementary event;
ü every event associated with this test is a set of finite or infinite number of elementary events;
ü an event occurs if and only if one of the elementary events included in this set is realized.
An arbitrary but fixed space of elementary events can be represented as a certain area on the plane. In this case, elementary events are points of the plane lying inside. Since an event is identified with a set, all operations that can be performed on sets can be performed on events. By analogy with set theory, we construct algebra of events. In this case, the following operations and relationships between events can be defined:
AÌ B(set inclusion relation: set A is a subset of the set IN) – event A entails event B. In other words, the event IN occurs whenever an event occurs A. Example - rolling a two results in rolling an even number of points.
(set equivalence relation) – event identically or equivalent event. This is possible if and only if and simultaneously , i.e. each occurs whenever the other occurs. Example – event A – breakdown of the device, event B – breakdown of at least one of the blocks (parts) of the device.
() – sum of events. This is an event consisting in the fact that at least one of two events or (logical "or") has occurred. In general, the sum of several events is understood as an event consisting of the occurrence of at least one of these events. Example – the target is hit by the first weapon, the second or both simultaneously.
() – product of events. This is an event consisting of the joint occurrence of events and (logical “and”). In general, the production of several events is understood as an event consisting of the simultaneous occurrence of all these events. Thus, events are incompatible if their production is an impossible event, i.e. . Example – event A is the removal of a card of the diamond suit from the deck, event B is the removal of an ace, then the appearance of the ace of diamonds has not occurred.
A geometric interpretation of operations on events is often useful. Graphic illustrations of operations are called Venn diagrams.
Definition 1. They say that in some experience an event A entails behind the appearance of an event IN, if upon the occurrence of an event A the event comes IN. Notation for this definition A Ì IN. In terms of elementary events, this means that each elementary event included in A, is also included in IN.
Definition 2. Events A And IN are called equal or equivalent (denoted A= IN), If A Ì IN And INÌ A, i.e. A And IN consist of the same elementary events.
Reliable event is represented by the embracing set Ω, and the impossible event is represented by an empty subset Æ in it. Incompatibility of events A And IN means that the corresponding subsets A And IN do not intersect: A∩IN = Æ.
Definition 3. The sum of two events A And IN(denoted WITH= A + IN) is called an event WITH, consisting of coming at least one of the events A or IN(the conjunction "or" for amount is keyword), i.e. comes or A, or IN, or A And IN together.
Example. Let two shooters shoot at a target at the same time, and the event A is that the 1st shooter hits the target, and the event B– that the 2nd shooter hits the target. Event A+ B means that the target is hit, or, in other words, that at least one of the shooters (1st shooter or 2nd shooter, or both shooters) hit the target.
Similarly, the sum of a finite number of events A 1 , A 2 , …, A n (denoted A= A 1 + A 2 + … + A n) the event is called A, consisting of the occurrence of at least one from events A i ( i = 1, … , n), or an arbitrary collection A i ( i = 1, 2, … , n).
Example. The sum of events A, B, C is an event consisting of the occurrence of one of the following events: A, B, C, A And IN, A And WITH, IN And WITH, A And IN And WITH, A or IN, A or WITH, IN or WITH,A or IN or WITH.
Definition 4. The product of two events A And IN called event WITH(denoted WITH = A ∙ B), consisting in the fact that as a result of the test, the event also occurred A, and event IN simultaneously. (The conjunction “and” for producing events is the key word).
Similar to the product of a finite number of events A 1 , A 2 , …, A n (denoted A = A 1 ∙A 2 ∙…∙ A n) the event is called A, consisting in the fact that as a result of the test all specified events occurred.
Example. If events A, IN, WITH there is the appearance of a “coat of arms” in the first, second and third trials, respectively, then the event A× IN× WITH There is a drop of the “coat of arms” in all three trials.
Remark 1. For incompatible events A And IN equality is true A ∙ B= Æ, where Æ is an impossible event.
Note 2. Events A 1 , A 2, … , A n form a complete group of pairwise incompatible events if .
Definition 5. Opposite events two uniquely possible incompatible events that form a complete group are called. Event opposite to event A, denoted by . Event opposite to event A, is an addition to the event A to the set Ω.
For opposite events, two conditions are simultaneously satisfied A∙= Æ and A+= Ω.
Definition 6. By difference events A And IN(denoted A– IN) is called an event consisting in the fact that the event A will come, and the event IN - no and it is equal A– IN= A× .
Note that events A + B, A ∙ B, , A – B it is convenient to interpret graphically using Euler–Venn diagrams (Fig. 1.1).
|
Rice. 1.1. Operations on events: negation, sum, product and difference |
Let us formulate the example this way: let experience G consists of shooting at random in the area Ω, the points of which are elementary events ω. Let getting into the region Ω be a reliable event Ω, and let getting into the region A And IN– respectively events A And IN. Then the events A+B(or AÈ IN– light area in the figure), A ∙ B(or AÇ IN - area in the center), A – B(or A\IN - light subregions) will correspond to the four images in Fig. 1.1. In the conditions of the previous example with two shooters shooting at a target, the product of events A And IN there will be an event C = AÇ IN, consisting of hitting the target with both arrows.
Remark 3. If operations on events are represented as operations on sets, and events are represented as subsets of some set Ω, then the sum of events A+B matches the union AÈ IN these subsets, and the product of events A ∙ B- intersection A∩IN these subsets.
Thus, operations on events can be associated with operations on sets. This correspondence is shown in table. 1.1
Table 1.1
|
Designations |
Probability language |
Set theory language |
|
Space element. events |
Universal set |
|
|
Elementary event |
Element from the universal set |
|
|
Random event |
Subset of elements ω from Ω |
|
|
Reliable event |
The set of all ω |
|
|
Impossible event |
Empty set |
|
|
AМ В |
A entails IN |
A– subset IN |
|
A+B(AÈ IN) |
Sum of events A And IN |
Union of sets A And IN |
|
A× V(AÇ IN) |
Producing Events A And IN |
Intersection of sets A And IN |
|
A – B(A\IN) |
Event difference |
Set difference |
Actions on events have the following properties:
A + B = B + A, A ∙ B = B ∙ A(commutative);
(A + B) ∙ C = A× C + B× C, A ∙ B + C =(A+C) × ( B + C) (distribution);
(A + B) + WITH = A + (B + C), (A ∙ B) ∙ WITH= A ∙ (B ∙ C) (associative);
A + A = A, A ∙ A = A;
A + Ω = Ω, A∙ Ω = A;
Addition rule- if element A can be chosen in n ways, and element B can be chosen in m ways, then A or B can be chosen in n + m ways.
^ Multiplication rule - if element A can be chosen in n ways, and for any choice of A, element B can be chosen in m ways, then the pair (A, B) can be chosen in n·m ways.
Rearrangement. Permutation of a set of elements is the arrangement of elements in a certain order. Thus, all different permutations of a set of three elements are
The number of all permutations of elements is denoted by . Therefore, the number of all different permutations is calculated by the formula
Accommodation. The number of placements of a set of elements by elements is equal to
^ Placement with repetition. If there is a set of n types of elements, and you need to place an element of some type in each of the m places (the types of elements can coincide in different places), then the number of options for this will be n m.
^ Combination. Definition. Combinations of various elements according toelements are called combinations that are made up of data elements by elements and differ in at least one element (in other words,-element subsets of a given set of elements). butback="" onclick="goback(684168)">^ " ALIGN=BOTTOM WIDTH=230 HEIGHT=26 BORDER=0>
Space of elementary events. Random event. Reliable event. Impossible event.
Random event - any subset of the space of elementary events.
^ Reliable event - will definitely happen as a result of the experiment.
Impossible event - will not occur as a result of the experiment.
Actions on events: sum, product and difference of events. Opposite event. Joint and non-joint events. Full group events.
^ Incompatible events – if they cannot occur simultaneously as a result of the experiment. They say that several incompatible events form full group of events, if one of them appears as a result of the experiment.
If the first event consists of all elementary outcomes except those included in the second event, then such events are called opposite.
The sum of two events A and B is an event consisting of elementary events belonging to at least one of the events A or B. ^ The product of two events A and B – an event consisting of elementary events belonging simultaneously to A and B. Difference A and B – an event consisting of elements of A that do not belong to event B.
Classical, statistical and geometric definitions probabilities. Basic properties of event probability.
, g – part of figure G. ^ Basic properties of probability: 1) 0≤Р(А)≤1, 2) The probability of a reliable event is 1, 3) The probability of an impossible event is 0.
The theorem for adding the probabilities of incompatible events and its consequences.
Conditional probability. Independent events. Multiplying the probabilities of dependent and independent events.
^ Multiplying probabilities: For addicts. Theorem. P(A∙B) = P(A)∙P A (B). Comment. P(A∙B) = P(A)∙P A (B) = P(B)∙P B (A). Consequence. P(A 1 ∙…∙A k) = P(A 1)∙P A1 (A 2)∙…∙P A1-Ak-1 (A k). For independents. P(A∙B) = P(A)∙P(B).
^Ttheorem for adding probabilities of joint events. Theorem . The probability of the occurrence of at least one of two joint events is equal to the sum of the probabilities of these events without the probability of their joint occurrence
Total probability formula. Bayes formulas.
H 1, H 2 ...H n - form a complete group - hypotheses.
Event A can occur only if H 1, H 2 ...H n appears,
Then P(A)=P(N 1)*P n1 (A)+P(N 2)*P n2 (A)+…P(N n)*P n n (A)
^ Bayes' formula
Let N 1, N 2 ...H n be hypotheses, event A can occur under one of the hypotheses
P(A)= P(N 1)* P n1 (A)+P(N 2)*P n2 (A)+…P(N n)*P n n (A)
Let's assume that event A has occurred.
How did the probability H 1 change due to the fact that A occurred? Those. R A (H 1)
P(A* N 1)=P(A)* P A (N 1)= P(N 1)* P n1 (A) => P A (N 1)= (P(N 1)* P n1 ( A))/ P(A)
H 2, H 3 ...H n are determined similarly
General view:
P A (N i)= (P (N i)* P n i (A))/ P (A) , where i=1,2,3…n.
The formulas allow us to overestimate the probabilities of hypotheses as a result of the fact that the result of the tests that resulted in the occurrence of event A becomes known.
“Before” testing – a priori probabilities - P(N 1), P(N 2)…P(N n)
“After” the test - posterior probabilities - P A (N 1), P A (N 2) ... P A (N n)
Posterior probabilities, as well as prior ones, add up to 1.
9.Bernoulli and Poisson formulas.
Bernoulli's formula
Let n trials be carried out, in each of which event A may or may not appear. If the probability of event A in each of these trials is constant, then these trials are independent with respect to A.
Consider n independent trials, in each of which A can occur with probability p. This sequence of tests is called a Bernoulli circuit.
Theorem: the probability that in n trials event A will occur exactly m times is equal to: P n (m)=C n m *p m *q n - m
Number m 0 - the occurrence of event A is called the most probable if the corresponding probability P n (m 0) is not less than other P n (m)
P n (m 0)≥ P n (m), m 0 ≠ m
To find m 0 use:
np-q≤ m 0 ≤np+q
^ Poisson's formula
Consider Bernoulli's test:
n is the number of tests, p is the probability of success
Let p be small (p→0) and n be large (n→∞)
average number of occurrences of success in n trials
We add λ=n*p → p= λ into Bernoulli’s formula:
P n (m)=C n m *p m *(1-q) n-m ; C n m = n!/((m!*(n-m)!) →
→ P n (m)≈ (λ m /m!)*e - λ (Poisson)
If p≤0.1 and λ=n*p≤10, then the formula gives good results.
10. Local and integral theorems of Moivre-Laplace.
Let n be the number of tests, p be the probability of success, n be large and tend to infinity. (n->∞)
^ Local theorem
Р n (m)≈(f(x)/(npg)^ 1/2, where f(x)= (e - x ^2/2)/(2Pi)^ 1/2
If npq≥ 20 – gives good results, x=(m-np)/(npg)^ 1/2
^ Integral theorem
P n (a≤m≤b)≈ȹ(x 2)-ȹ(x 1),
where ȹ(x)=1/(2Pi)^ 1/2 * 0 ʃ x e (Pi ^2)/2 dt – Laplace function
x 1 =(a-np)/(npq)^ 1/2, x 2 =(b-np)/(npq)^ 1/2
Properties of the Laplace function
ȹ(x) – odd function: ȹ(-x)=- ȹ(x)
ȹ(x) – increases monotonically
values ȹ(x) (-0.5;0.5), and lim x →∞ ȹ(x)=0.5; lim x →-∞ ȹ(x)=-0.5
P n (│m-np│≤Ɛ) ≈ 2 ȹ (Ɛ/(npq) 1/2)
P n (ɑ≤m/n≤ƥ) ≈ ȹ(z 2)- ȹ(z 1), where z 1=(ɑ-p)/(pq/n)^ 1/2 z 2=(ƥ -p )/(pq/n)^ 1/2
P n (│(m/n) - p│≈ ∆) ≈ 2 ȹ(∆n 1/2 /(pq)^ 1/2)
11. Random variable. Types of random variables. Methods for specifying a random variable.
SV is a function defined on a set of elementary events.
X,Y,Z – NE, and its value is x,y,z
Random They call a quantity that, as a result of testing, will take one and only one possible value, not known in advance and depending on random reasons that cannot be taken into account in advance.
NE discrete, if the set of its values is finite or countable (they can be numbered). It takes on distinct, isolated possible values with defined probabilities. The number of possible values of a discrete SV can be finite or infinite.
NE continuous, if it takes all possible values from a certain interval (on the entire axis). Its meanings may differ very little.
^ Law of distribution of discrete SV M.B. given by:
1.table
| X | x 1 | x 2 | … | x n |
| P(X) | p 1 | p 2 | … | p n |
(distribution series)
X=x 1) are inconsistent
р 1 + р 2 +… p n =1= ∑p i
2.graphic
Probability distribution polygon
3.analytical
P=P(X)
12. Distribution function of a random variable. Basic properties of the distribution function.
The distribution function of SV X is a function F(X), which determines the probability that SV X will take a value less than x, i.e.
x x = cumulative distribution function
A continuous SV has a continuous, piecewise differentiable function.