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.
Permutation. A permutation of a set of elements is the arrangement of elements in a certain order. Thus, all the 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 m places (element types can match in different places), then the number of options for this will be n m .
^ Combination. Definition. Combinations from various elements according toelements are called combinations that are made up of data elements by elements and differ by at least one element (in other words,-element subsets of the given set from 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.
^ Credible event - is bound to 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. Complete group of events.
^ Incompatible events - if they cannot occur simultaneously as a result of the experiment. It is said that several disjoint 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 for 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 that belong simultaneously to A and B. The difference between A and B is an event consisting of elements A that do not belong to event B.
Classical, statistical and geometric definitions of probability. Basic properties of event probability.
, g – part of figure G. ^ Basic properties of probability: 1) 0≤P(A)≤1, 2) The probability of a certain event is 1, 3) The probability of an impossible event is 0.
The theorem of addition of probabilities of incompatible events and consequences from it.
Conditional Probability. independent events. Multiplication of probabilities of dependent and independent events.
^ Multiplication of Probabilities: For addicts. Theorem. P (A ∙ B) \u003d 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) \u003d 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 the probabilities of joint events. Theorem . The probability of the occurrence of at least one of the 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) \u003d P (N 1) * P n1 (A) + P (N 2) * P n2 (A) + ... P (N n) * P n n (A)
^ Bayes formula
Let H 1, H 2 ... H n be hypotheses, event A can occur under one of the hypotheses
P (A) \u003d P (N 1) * P n1 (A) + P (N 2) * P n2 (A) + ... P (N n) * P n n (A)
Assume that event A has occurred.
How has the probability of H 1 changed due to the fact that A has occurred? Those. R A (H 1)
R (A * H 1) \u003d R (A) * R A (H 1) \u003d R (H 1) * R n1 (A) => R A (H 1) \u003d (P (H 1) * R n1 ( A))/ P(A)
H 2 , H 3 ... H n are defined similarly
General form:
Р А (Н i)= (Р (Н i)* Р n i (А))/ Р (А) , where i=1,2,3…n.
Formulas allow you to overestimate the probabilities of hypotheses as a result of how the test result becomes known, as a result of which the event A appeared.
"Before" the test - a priori probabilities - P (N 1), P (N 2) ... P (N n)
"After" the test - a posteriori probabilities - R A (H 1), R A (H 2) ... R A (H n)
The posterior probabilities, like the prior probabilities, add up to 1.
9. Formulas of Bernoulli and Poisson.
Bernoulli formula
Let there be n trials, in each of which event A may or may not occur. If the probability of the 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. Such a sequence of tests is called the Bernoulli scheme.
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
The number m 0 - the occurrence of an 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 formula
Consider the Bernoulli test:
n is the number of trials, p is the probability of success
Let p be small (p→0) and n large (n→∞)
average number of occurrences of success in n trials
λ=n*p → p= λwe put into the Bernoulli 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 trials, 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
^ Theorem integral
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 is the Laplace function
x 1 \u003d (a-np) / (npq) ^ 1/2, x 2 \u003d (b-np) / (npq) ^ 1/2
Properties of the Laplace function
ȹ(x) – odd function: ȹ(-x)=- ȹ(x)
ȹ(x) – monotonically increasing
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 value. Types of random variables. Methods for setting a random variable.
SW is a function defined on a set of elementary events.
X,Y,Z is NE, and its value is x,y,z
Random they call a value that, as a result of tests, will take one and only one possible value, not known in advance and depending on random causes that cannot be taken into account in advance.
SW discrete, if the set of its values is finite or counted (they can be numbered). It takes on separate, isolated possible values with certain probabilities. The number of possible values of a discrete CV can be finite or infinite.
SW continuous, if it takes all possible values from some interval (on the whole axis). Its values may differ very little.
^ Discrete SW distribution law m.b. given:
1.table
| X | x 1 | x 2 | … | x n |
| P(X) | p 1 | p 2 | … | p n |
(distribution range)
X \u003d x 1) are incompatible
p 1 + p 2 +… p n =1= ∑p i
2.graphic
Probability distribution polygon
3.analytical
P=P(X)
12. The distribution function of a random variable. Basic properties of the distribution function.
The distribution function of CV X is a function F(X) that determines the probability that CV X will take a value less than x, i.e.
x x = cumulative distribution function
A continuous SW has a continuous, piecewise differentiable function.
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, rolling the same number on two dice.
The two events are called incompatible(incompatible) in a given trial if they cannot occur together in the same trial. Several events are said to be incompatible if they are pairwise incompatible. Examples of incompatible events: a) hit and miss with one shot; b) a part is randomly extracted from a box with parts - the events “standard part removed” and “non-standard part removed”; 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 probable if, according to the symmetry conditions, there is reason to believe that none of them is objectively more possible than others (falling out of a coat of arms and tails, the appearance of a card of any suit, choosing a ball from an urn, etc.)
Associated with each trial is a series of events that, generally speaking, can occur simultaneously. For example, when throwing a die, an event is a deuce, and an event is an even number of points. Obviously, these events are not mutually exclusive.
Let all possible results of the test be carried out in a number of the only possible special cases, mutually exclusive of each other. Then
ü each test outcome is represented by one and only one elementary event;
ü any 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 some 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, one constructs event algebra. 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 leads to event B. In other words, the event IN occurs whenever an event occurs A. Example - Dropping a deuce entails dropping an even number of points.
(set equivalence relation) – event identically or equivalent to event . This is possible if and only if and simultaneously , i.e. each occurs whenever the other occurs. Example - event A - failure of the device, event B - failure 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 the two events or (logical "or") has occurred. In the general case, the sum of several events is understood as an event consisting in the occurrence of at least one of these events. Example - the target is hit by the first gun, the second or both at the same time.
() – product of events. This is an event consisting in the joint implementation of events and (logical "and"). In the general case, the product of several events is understood as an event consisting in the simultaneous implementation of all these events. Thus, events and are incompatible if their product is an impossible event, i.e. . Example - event A - taking out a card of a diamond suit from the deck, event B - taking out an ace, then - the appearance of a diamond ace has not occurred.
A geometric interpretation of operations on events is often useful. The graphical illustration of operations is called Venn diagrams.
Events
Event. elemental event.
Space of elementary events.
Reliable event. Impossible event.
identical events.
Sum, product, difference of events.
opposite events. incompatible events.
Equivalent events.
Under event in probability theory is understood any fact that may or may not occur as a result of experience withrandom outcome. The simplest result of such an experiment (for example, the appearance of "heads" or "tails" when tossing a coin, hitting a target when shooting, the appearance of an ace when removing a card from the deck, randomly dropping a number when throwing a dieetc.) is calledelementary event .
The set of all elementary events E called element space tare events . Yes, at throwing a dice, this space consists of sixelementary events, and when a card is removed from the deck - from 52. An event can consist of one or more elementary events, for example, the appearance of two aces in a row when removing a card from the deck, or the loss of the same number when throwing a die three times. Then one can define event as an arbitrary subset of the space of elementary events.
a certain event the whole space of elementary events is called. Thus, a certain event is an event that must necessarily occur as a result of a given experience. When a dice is thrown, such an event is its fall on one of the faces.
Impossible event () is called an empty subset of the space of elementary events. That is, an impossible event cannot occur as a result of this experience. So, when throwing a dice, an impossible event is its fall on the edge.
Events A And IN calledidentical (A= IN) if the event Aoccurs when and only when an event occursIN .
They say that the event A triggers an event IN ( A IN), if from the condition"event A happened" should "Event B happened".
Event WITH called sum of events A And IN (WITH = A IN) if the event WITH occurs if and only if either A, or IN.
Event WITH called product of events A And IN (WITH = A IN) if the event WITH happens when and only when it happens andA, And IN.
Event WITH called difference of events A And IN (WITH = A – IN) if the event WITH happens then and Only then, when it happens event A, and the event does not occur IN.
Event A"called opposite eventAif the event didn't happen A. So, a miss and a hit when shooting are opposite events.
Events A And IN calledincompatible (A IN = ) , if their simultaneous occurrence is impossible. For example, dropping and "tails", and"eagle" when tossing a coin.
If during the experiment several events can occur and each of them, according to objective conditions, is no more possible than the other, then such events are calledequally possible . Examples of equally likely events: the appearance of a deuce, an ace and a jack when a card is removed from the deck, loss of any of the numbers from 1 to 6 when throwing a dice, etc.
Types of random events
Events are called incompatible if the occurrence of one of them excludes the occurrence of other events in the same trial.
Example 1.10. A part is taken at random from a box of parts. The appearance of a standard part excludes the appearance of a non-standard part. Events (a standard part appeared) and (a non-standard part appeared)- incompatible .
Example 1.11. A coin is thrown. The appearance of a "coat of arms" excludes the appearance of a number. Events (a coat of arms appeared) and (a figure appeared) - incompatible .
Several events form full group, if at least one of them appears as a result of the test. In other words, the occurrence of at least one of the events of the full group is reliable event. In particular, if the events that form a complete group are pairwise incompatible, then one and only one of these events will appear as a result of the test. This particular case is of greatest interest to us, since it will be used below.
Example 1.12. Purchased two tickets of the money and clothing lottery. One and only one of the following events will necessarily occur: (the winnings fell on the first ticket and did not fall on the second), (the winnings did not fall on the first ticket and fell on the second), (the winnings fell on both tickets), (the winnings did not win on both tickets). fell out). These events form full group pairwise incompatible events.
Example 1.13. The shooter fired at the target. One of the following two events is sure to occur: a hit or a miss. These two incompatible events form full group .
Events are called equally possible if there is reason to believe that none of them is no more possible than the other.
3. Operations on events: sum (union), product (intersection) and difference of events; vienne diagrams.
Operations on events
Events are denoted by capital letters of the beginning of the Latin alphabet A, B, C, D, ..., providing them with indices if necessary. The fact that the elemental outcome X contained in the event A, denote .
For understanding, it is convenient to use a geometric interpretation with the help of Vienna diagrams: let us represent the space of elementary events Ω as a square, each point of which corresponds to an elementary event. Random events A and B, consisting of a set of elementary events x i And at j, respectively, are geometrically depicted as some figures lying in the square Ω (Fig. 1-a, 1-b).
Let the experiment consist in the fact that inside the square shown in Figure 1-a, a point is chosen at random. Let us denote by A the event consisting in the fact that (the selected point lies inside the left circle) (Fig. 1-a), through B - the event consisting in the fact that (the selected point lies inside the right circle) (Fig. 1-b ).
A reliable event is favored by any , therefore a reliable event will be denoted by the same symbol Ω.
Two events are identical to each other (A=B) if and only if these events consist of the same elementary events (points).
The sum (or union) of two events A and B is called an event A + B (or ), which occurs if and only if either A or B occurs. The sum of events A and B corresponds to the union of sets A and B (Fig. 1-e).
Example 1.15. The event consisting in the loss of an even number is the sum of the events: 2 fell out, 4 fell out, 6 fell out. That is, (x \u003d even }= {x=2}+{x=4 }+{x=6 }.
The product (or intersection) of two events A and B is called an event AB (or ), which occurs if and only if both A and B occur. The product of events A and B corresponds to the intersection of sets A and B (Fig. 1-e).
Example 1.16. The event consisting of rolling 5 is the intersection of events: odd number rolled and more than 3 rolled, that is, A(x=5)=B(x-odd)∙C(x>3).
Let us note the obvious relations:
The event is called opposite to A if it occurs if and only if A does not occur. Geometrically, this is a set of points of a square that is not included in subset A (Fig. 1-c). An event is defined similarly (Fig. 1-d).
Example 1.14.. Events consisting in the loss of an even and an odd number are opposite events.
Let us note the obvious relations:
The two events are called incompatible if their simultaneous appearance in the experiment is impossible. Therefore, if A and B are incompatible, then their product is an impossible event:
The elementary events introduced earlier are obviously pairwise incompatible, that is,
Example 1.17. Events consisting in the loss of an even and an odd number are incompatible events.
