Conditional Probability. Conditional probability and the simplest basic formulas. The theorem of multiplication of the probabilities of events, one of which takes place under the condition of the other

4. Conditional Probability. Probability multiplication theorem.

In many problems it is necessary to find the probability of combining events BUT and AT if the probabilities of events are known BUT and AT.

Consider the following example. Let two coins be thrown. Find the probability of the appearance of two coats of arms. We have 4 equally probable pairwise incompatible outcomes that form a complete group:

Now let us know that the coat of arms fell on the first coin. How will the probability that the coat of arms will appear on both coins change after this? Since the coat of arms fell on the first coin, now the full group consists of two equally probable incompatible outcomes:

new event probability BUT, assuming that an event has occurred B, we will denote P B (A).

In this way, P(A)=1/4; P B (A) \u003d 1/2

Multiplication theorem. The probability of combining events A and B is equal to the product of the probability of one of them by the conditional probability of the other, calculated on the assumption that the first event has taken place, i.e.

Proof. Let us prove the validity of relation (4) based on the classical definition of probability. Let the possible outcomes E 1, E 2, ..., E N of this experience form a complete group of equally probable pairwise incompatible events, of which the event A favor M outcomes, and let from these M outcomes L outcomes favor the event B. Obviously, the combination of events A and B favor L from N possible test results. This gives ; ;
In this way,
Swapping places A and B, similarly we get
The multiplication theorem can be easily generalized to any finite number of events. So, for example, in the case of three events A 1, A2, A 3 we have *
In general

From relation (6) it follows that from two equalities (8) one is a consequence of the other.

Let, for example, the event A- the appearance of the coat of arms during a single toss of a coin, and the event B- the appearance of a card of a diamond suit when a card is removed from the deck. Obviously the events A and B independent.

If the events are independent A to B formula (4) will take a simpler form:

* Event A 1 A 2 A 3 can be represented as a combination of two events: events C=A 1 A 2 and events A 3.

Consider events A and B associated with the same experience. Let it become known from some sources that the event B occurred, but it is not known which of the elementary outcomes that make up the event B, happened. What can be said in this case about the probability of an event A?

Event Probability A, calculated under the assumption that the event B happened, it is customary to call the conditional probability and denote P(A|B).

conditional probability P(A|B) developments A subject to the event B within the framework of the classical scheme, it is natural to define the probability as the ratio NAB outcomes that favor the joint implementation of events A and B, to the number NB outcomes favoring the event B, that is

If we divide the numerator and denominator of this expression by the total number N elementary outcomes, we get

Definition. Conditional probability of an event A subject to the event B is called the ratio of the probability of the intersection of events A and B to the probability of an event B:

At the same time, it is assumed that P(B) ≠ 0.

Theorem. Conditional Probability P(A|B) has all the properties of unconditional probability P(A).

The meaning of this theorem is that the conditional probability is the unconditional probability given on the new space Ω 1 elementary outcomes coinciding with the event B.

Example. From the urn in which a=7 whites and b=3 black balls, two balls are drawn at random without replacement. Let the event A 1 is that the first ball drawn is white, and A2- the second ball is white. Wanted to find P(A 2 |A 1).

Method 1.. By definition of conditional probability

Method 2.. Let's move on to a new space of elementary outcomes Ω 1. Since the event A 1 happened, this means that in the new space of elementary outcomes, the total number of equally possible outcomes NΩ 1 =a+b-1=9, and the event A2 favors it N A 2 \u003d a-1 \u003d 6 outcomes. Consequently,

Theorem [multiplication of probabilities]. Let the event A=A 1 A 2 …A n and P(A)>0. Then the equality is true:

P(A) = P(A 1) P(A 2 |A 1) P(A 3 |A 1 A 2) … P(A n |A 1 A 2 …A n-1).

Comment. From the commutativity property of an intersection, one can write

P(A 1 A 2) = P(A 1) P(A 2 |A 1)

P(A 1 A 2) = P(A 2) P(A 1 |A 2).

Example. Letters forming the word "NIGHTINGALE" are written on 7 cards. The cards are shuffled and three cards are randomly removed from them and laid out from left to right. Find the probability that the word "VOL" will be obtained (the event A).

Let the event A 1- the letter "B" is written on the first card, A2- the letter "O" is written on the second card, A2- on the third card - the letter "L". Then the event A- intersection of events A 1, A2, A 3. Consequently,

P(A) = P(A 1 A 2 A 3) = P(A 1) P(A 2 |A 1) P(A 3 |A 1 A 2).

P(A1)=1/7; if event A 1 happened, then on the remaining 6 cards “O” occurs twice, which means P(A 2 |A 1)=2/6=1/3. Likewise, P(A 3 |A 1)=2/6=1/3. Consequently,

Definition. Developments A and B, having a non-zero probability, are called independent if the conditional probability A on condition B coincides with the unconditional probability A or if the conditional probability B on condition A coincides with the unconditional probability B, that is

P(A|B) = P(A) or P(B|A) = P(B),

otherwise the events A and B called dependent.

Theorem. Developments A and B, which have a non-zero probability, are independent if and only if

P(AB) = P(A) P(B).

Thus, we can give an equivalent definition:

Definition. Developments A and B are called independent if P(AB) = P(A) P(B).

Example. From a deck of cards containing n=36 cards, one card is drawn at random. Denote by A an event corresponding to the fact that the extracted map will be a peak, and B- an event corresponding to the appearance of a "lady". Determine if events are dependent A and B.

P(A)=9/36=1/4, P(B)=4/36=19, P(AB)=1/36, . Therefore, the events A and B independent. Likewise, .

Let BUT and AT are the two events considered in this test. In this case, the occurrence of one of the events may affect the possibility of the occurrence of another. For example, the occurrence of an event BUT can influence the event AT or vice versa. To take into account such dependence of some events on others, the concept of conditional probability is introduced.

Definition. If the probability of an event AT is located under the condition that the event BUT happened, then the resulting probability of the event AT called conditional probability developments AT. The following symbols are used to denote such a conditional probability: R BUT ( AT) or R(AT / BUT).

Remark 2. In contrast to the conditional probability, the “unconditional” probability is also considered, when any conditions for the occurrence of some event AT missing.

Example. An urn contains 5 balls, 3 of which are red and 2 are blue. In turn, one ball is drawn from it with a return and without a return. Find the conditional probability of drawing a red ball for the second time, provided that the first time taken is: a) a red ball; b) a blue ball.

Let the event BUT is drawing the red ball for the first time, and the event AT– extracting the red ball for the second time. It's obvious that R(BUT) = 3 / 5; then in the case when the ball taken out for the first time is returned to the urn, R(AT)=3/5. In the case when the drawn ball is not returned, the probability of drawing a red ball R(AT) depends on which ball was drawn for the first time - red (event BUT) or blue (event). Then in the first case R BUT ( AT) = 2 / 4, and in the second ( AT) = 3 / 4.

The theorem of multiplication of the probabilities of events, one of which takes place under the condition of the other

The probability of the product of two events is equal to the product of the probability of one of them by the conditional probability of the other, found under the assumption that the first event occurred:

R(A ∙ B) = R(BUT) ∙ R BUT ( AT) . (1.7)

Proof. Indeed, let n- the total number of equally probable and incompatible (elementary) outcomes of the test. Let it go n 1 - the number of outcomes that favor the event BUT, which occurs at the beginning, and m- the number of outcomes in which the event occurs AT assuming that the event BUT has come. In this way, m is the number of outcomes that favor the event AT. Then we get:

Those. the probability of the product of several events is equal to the product of the probability of one of these events by the conditional probabilities of the others, and the conditional probability of each subsequent event is calculated on the assumption that all previous events have occurred.

Example. There are 4 masters of sports in a team of 10 athletes. By drawing lots, 3 athletes are selected from the team. What is the probability that all the selected athletes are masters of sports?

Solution. Let us reduce the problem to the “urn” model, i.e. Let's assume that there are 4 red balls and 6 white ones in an urn containing 10 balls. 3 balls are drawn at random from this urn (selection S= 3). Let the event BUT consists in extracting 3 balls. The problem can be solved in two ways: by the classical scheme and by formula (1.9).

The first method based on the combinatorics formula:

The second method (by formula (1.9)). 3 balls are drawn consecutively from the urn without replacement. Let BUT 1 - the first drawn ball is red, BUT 2 - the second drawn ball is red, BUT 3 - the third drawn ball is red. Let also the event BUT means that all 3 drawn balls are red. Then: BUT = BUT 1 ∙ (BUT 2 / BUT 1) ∙ BUT 3 / (BUT 1 ∙ BUT 2), i.e.

Example. Let from the set of cards a, a, r, b, o, t cards are drawn one at a time. What is the probability of getting the word " Work” when sequentially folding them into one line from left to right?

Let AT- the event at which the declared word is obtained. Then by formula (1.9) we get:

R(AT) = 1/6 ∙ 2/5 ∙ 1/4 ∙ 1/3 ∙ 1/2 ∙ 1/1 = 1/360.

The probability multiplication theorem takes on its simplest form when the product is formed by events independent of each other.

Definition. Event AT called independent from the event BUT if its probability does not change regardless of whether the event occurred BUT or not. Two events are called independent (dependent) if the occurrence of one of them does not change (changes) the probability of occurrence of the other. Thus, for not dependent events p(B/A) = R(AT) or = R(AT), and for dependent events R(AT/A)

Event. Space of elementary events. Certain event, impossible event. Joint, non-joint events. Equivalent events. Complete group of events. Operations on events.

Event is a phenomenon that can be said to be going on or not happening, depending on the nature of the event itself.

Under elementary events associated with a particular test understand all the indecomposable results of that test. Each event that can occur as a result of this test can be considered as a certain set of elementary events.

Space of elementary events is called an arbitrary set (finite or infinite). Its elements are points (elementary events). Subsets of the space of elementary events are called events.

a certain event an event is called which, as a result of this test, will definitely occur; (denoted by E).

Impossible event an event is called such an event that, as a result of a given test can't happen; (denoted U). For example, the appearance of one of the six points during one throw dice- a reliable event, and the appearance of 8 points is impossible.

The two events are called joint(compatible) in a given experience, if the appearance of one of them does not exclude the appearance of the other.

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.

Form start

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Topic number 2. Axiomatic definition of probability. Classical, statistical, geometric definition of the probability of an event. Probability properties. Theorems of addition and multiplication of probabilities. independent events. Conditional Probability. The probability that at least one of the events will occur. Total Probability Formula. Bayes formula

A numerical measure of the degree of objective possibility of an event occurring is called the probability of an event. This definition, which qualitatively reflects the concept of the probability of an event, is not mathematical. To make it so, it is necessary to define it qualitatively.

According to classical definition the probability of event A is equal to the ratio of the number of cases favorable to it to the total number of cases, that is:

Where P(A) is the probability of event A.

Number of cases favorable to event A

The total number of cases.

Statistical definition of probability:

The statistical probability of an event A is the relative frequency of occurrence of this event in the tests performed, that is:

Where is the statistical probability of event A.

Relative frequency (frequency) of the event A.

Number of trials in which events A appeared

The total number of trials.

Unlike the "mathematical" probability, considered in the classical definition, the statistical probability is a characteristic of an experimental, experimental.

If there is a proportion of cases that favor event A, which is determined directly, without any trials, that is, the proportion of those trials actually performed in which event A appeared.

Geometric definition of probability:

The geometric probability of an event A is the ratio of the measure of the area favoring the occurrence of event A to the measure of all areas, that is:

In the one-dimensional case:

It is necessary to estimate the probability of hitting a point on CD/

It turns out that this probability does not depend on the location of CD on the segment AB, but depends only on its length.

The probability of hitting a point does not depend on the shapes or on the location of B on A, but depends only on the area of ​​this segment.

Conditional Probability

The probability is called conditional , if it is calculated under certain conditions and denoted:

This is the probability of event A. It is calculated under the condition that event B has already happened.

Example. We make a test, we extract two cards from the deck: The first probability is unconditional.

We calculate the probability of drawing an ace from the deck:

We calculate the occurrence of 2-ace from the deck:

A*B - joint occurrence of events

– probability multiplication theorem

Consequence:

The multiplication theorem for the joint occurrence of events has the form:

That is, each subsequent probability is calculated taking into account that all previous conditions have already occurred.

Event Independence:

Two events are called independent if the occurrence of one does not contradict the occurrence of the other.

For example, if aces are drawn repeatedly from the deck, then they are independent of each other. Again, that is, the card was looked at and returned back to the deck.

Joint and non-joint events:

joint 2 events are called if the occurrence of one of them does not contradict the occurrence of the other.

The theorem of addition of probabilities of joint events:

The probability of the occurrence of one of the two joint events is equal to the sum of the probabilities of these events without their joint occurrence.

For three joint events:

Events are called inconsistent if no two of them can appear simultaneously as a result of a single test of a random experiment.

Theorem: The probability of occurrence of one of two incompatible events is equal to the sum of the probabilities of these events.

The probability of the sum of events:

Probability addition theorem:

The probability of the sum of a finite number of incompatible events is equal to the sum of the probabilities of these events:

Corollary 1:

The sum of the probabilities of events forming a complete group is equal to one:

Consequence 2:

Comment: It should be emphasized that the considered addition theorem is applicable only for incompatible events.

Probability of opposite events:

Opposite two unique possible events that form a complete group are called. One of two opposite events is denoted by BUT, the other - through .

Example: Hitting and missing when shooting at a target are opposite events. If A is a hit, then a miss.

Theorem: The sum of the probabilities of opposite events is equal to one:

Note 1: If the probability of one of two opposite events is denoted by p, then the probability of the other event is denoted by q Thus, by virtue of the previous theorem:

Note 2: When solving problems to find the probability of an event A, it is often advantageous to first calculate the probability of the event , and then find the desired probability using the formula:

Probability of at least one event occurring:

Let us assume that as a result of an experiment one, some part or no event may appear.

Theorem: The probability of occurrence of at least one event from a set of independent events is equal to the difference between unity and their probability of not occurrence of events.

The total probability formula allows you to find the probability of an event A, which can occur only with each of n mutually exclusive events that form a complete system if their probabilities are known, and conditional probabilities developments A with respect to each of the events of the system are equal to .

Events are also called hypotheses, they are mutually exclusive. Therefore, in the literature you can also find their designation not by the letter B, but with a letter H(hypothesis).

To solve problems with such conditions, it is necessary to consider 3, 4, 5, or in the general case n the possibility of an event A- with every event.

Using the theorems of addition and multiplication of probabilities, we obtain the sum of the products of the probability of each of the events of the system by conditional probability developments A for each event in the system. That is, the probability of an event A can be calculated by the formula

or in general

,

which is called total probability formula .

Total probability formula: examples of problem solving

Example 1 There are three identical-looking urns: in the first one there are 2 white balls and 3 black ones, in the second one - 4 white ones and one black one, in the third one - three white balls. Someone randomly approaches one of the urns and takes one ball out of it. Taking advantage total probability formula, find the probability that the ball is white.

Solution. Event A- the appearance of a white ball. We put forward three hypotheses:

First urn selected;

The second urn is chosen;

The third urn has been chosen.

Conditional event probabilities A for each of the hypotheses:

, , .

We apply the total probability formula, as a result - the required probability:

.

Example 2 At the first plant, out of every 100 light bulbs, an average of 90 standard ones are produced, at the second - 95, at the third - 85, and the products of these factories account for 50%, 30% and 20%, respectively, of all electric bulbs supplied to stores in a certain area. Find the probability of purchasing a standard light bulb.

Solution. Let us denote the probability of acquiring a standard light bulb as A, and the events that the purchased light bulb was manufactured at the first, second and third factories, respectively, through . By condition, the probabilities of these events are known: , , and the conditional probabilities of the event A regarding each of them: , , . These are the probabilities of acquiring a standard light bulb, provided that it is manufactured at the first, second, and third factories, respectively.

Event A will occur if an event occurs or K- the bulb is made at the first factory and is standard, or an event L- the bulb is made at the second factory and is standard, or an event M- the bulb is manufactured at the third factory and is standard. Other possibilities for the occurrence of the event A no. Therefore, the event A is the sum of events K, L and M that are incompatible. Applying the probability addition theorem, we represent the probability of an event A as

and by the probability multiplication theorem we get

that is, a special case of the total probability formula.

Substituting the probabilities into the left side of the formula, we obtain the probability of the event A :

Example 3 The aircraft is landing at the airport. If the weather allows, the pilot lands the plane, using, in addition to instruments, also visual observation. In this case, the probability of a successful landing is . If the airfield is overcast with low clouds, then the pilot lands the plane, orienting himself only on instruments. In this case, the probability of a successful landing is ; . Devices that provide blind landing have reliability (probability of failure-free operation) P. In the presence of low cloudiness and failed blind landing instruments, the probability of a successful landing is ; . Statistics show that in k% of landings, the airfield is covered with low clouds. Find full probability of the event A- safe landing of the aircraft.

Solution. Hypotheses:

There is no low cloud cover;

There is low cloud cover.

The probabilities of these hypotheses (events):

;

Conditional Probability.

The conditional probability is again found by the formula for the total probability with hypotheses

Blind landing devices work;

Blind landing instruments failed.

The probabilities of these hypotheses are:

According to the total probability formula

Example 4 The device can operate in two modes: normal and abnormal. Normal mode is observed in 80% of all cases of operation of the device, and abnormal - in 20% of cases. Probability of device failure in a certain time t equal to 0.1; in the abnormal 0.7. Find full probability device failure in time t.

Solution. We again denote the probability of device failure as A. So, regarding the operation of the device in each mode (events), the probabilities are known by condition: for the normal mode it is 80% (), for the abnormal mode - 20% (). Event Probability A(that is, the failure of the device) depending on the first event (normal mode) is 0.1 (); depending on the second event (abnormal mode) - 0.7 ( ). We substitute these values ​​into the total probability formula (that is, the sum of the products of the probability of each of the events of the system and the conditional probability of the event A regarding each of the events of the system) and we have the required result.