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3.2. Probability addition theorem

Lecture



The probability addition theorem is formulated as follows.

The probability of the sum of two incompatible events is equal to the sum of the probabilities of these events:

  3.2.  Probability addition theorem . (3.2.1)

Let us prove the probability addition theorem for the case scheme. Let the possible outcomes of the experience be reduced to a set of cases, which, for clarity, we depict in the form of n points:

  3.2.  Probability addition theorem

Suppose that of these cases   3.2.  Probability addition theorem favorable event   3.2.  Probability addition theorem , but   3.2.  Probability addition theorem - event   3.2.  Probability addition theorem . Then

  3.2.  Probability addition theorem

Since the events   3.2.  Probability addition theorem and   3.2.  Probability addition theorem incompatible, there are no such cases that are favorable and   3.2.  Probability addition theorem and   3.2.  Probability addition theorem together. Therefore, the event   3.2.  Probability addition theorem favorable   3.2.  Probability addition theorem cases and

  3.2.  Probability addition theorem

Substituting the obtained expressions into formula (3.2.1), we obtain the identity. The theorem is proved.

Let us generalize the addition theorem to the case of three events. Denoting an event   3.2.  Probability addition theorem letter   3.2.  Probability addition theorem , and adding to the amount of another event   3.2.  Probability addition theorem it is easy to prove that

  3.2.  Probability addition theorem

Obviously, using the full induction method, one can generalize the addition theorem to an arbitrary number of incompatible events. Indeed, suppose that it is valid for n events:

  3.2.  Probability addition theorem

and prove that it will be valid for   3.2.  Probability addition theorem events:

  3.2.  Probability addition theorem

Denote:

  3.2.  Probability addition theorem

We have:

  3.2.  Probability addition theorem .

But since for n events we consider the theorem already proved, then

  3.2.  Probability addition theorem ,

from where

  3.2.  Probability addition theorem ,

Q.E.D.

Thus, the addition theorem is applicable to any number of incompatible events. It is more convenient to write in the form:

  3.2.  Probability addition theorem . (3.2.2)

We point out the consequences arising from the probability addition theorem.

Corollary 1. If events   3.2.  Probability addition theorem form a complete group of incompatible events, the sum of their probabilities is equal to one:

  3.2.  Probability addition theorem .

Evidence. Since the events   3.2.  Probability addition theorem form a complete group, the appearance of at least one of them is a reliable event:

  3.2.  Probability addition theorem .

Because   3.2.  Probability addition theorem - incompatible events, the probability addition theorem is applicable to them

  3.2.  Probability addition theorem ,

from where

  3.2.  Probability addition theorem ,

Q.E.D.

Before deriving the second consequence of the addition theorem, we define the concept of "opposing events."

Opposite events are two incompatible events that make up the full group.

Event opposite to event   3.2.  Probability addition theorem it is customary to denote   3.2.  Probability addition theorem .

Examples of opposite events.

one)   3.2.  Probability addition theorem - hit with a shot,   3.2.  Probability addition theorem - miss when fired;

2)   3.2.  Probability addition theorem - loss of the coat of arms when throwing a coin,   3.2.  Probability addition theorem - loss of numbers when throwing a coin;

3)   3.2.  Probability addition theorem - trouble-free operation of all elements of the technical system,   3.2.  Probability addition theorem - failure of at least one element;

four)   3.2.  Probability addition theorem - detection of at least two defective items in the inspection lot,   3.2.  Probability addition theorem - detection of no more than one defective product.

Corollary 2. The sum of the probabilities of opposite events is equal to one:

  3.2.  Probability addition theorem .

This consequence is a special case of Corollary 1. It is highlighted especially because of its great importance in the practical application of probability theory. In practice, it is often easier to calculate the probability of the opposite event.   3.2.  Probability addition theorem than the probability of a direct event   3.2.  Probability addition theorem . In these cases, calculate   3.2.  Probability addition theorem and find   3.2.  Probability addition theorem .

Consider a few examples on the application of the addition theorem and its consequences.

Example 1. There are 1000 tickets in the lottery; 500 rubles are won for one ticket, 100 rubles for 100 tickets, 20 rubles for 50 tickets, 5 rubles for 100 tickets, other tickets are non-winning. Someone buys one ticket. Find the probability of winning at least 20 rubles.

Decision. Consider the events:

  3.2.  Probability addition theorem - win at least 20 rubles.,

  3.2.  Probability addition theorem - win 20 rubles.

  3.2.  Probability addition theorem - win 100 rubles.

  3.2.  Probability addition theorem - win 500 rubles.

Obviously

  3.2.  Probability addition theorem .

By the addition theorem

  3.2.  Probability addition theorem .

Example 2. Three ammunition depots are bombed, one bomb being dropped. The probability of getting into the first warehouse is 0.01; in the second 0,008; in the third 0.025. When hit in one of the warehouses all three explode. Find the probability that the warehouses will be blown up.

Decision. Consider the events:

  3.2.  Probability addition theorem - explosion of warehouses,

  3.2.  Probability addition theorem - getting into the first warehouse,

  3.2.  Probability addition theorem - getting into the second warehouse,

  3.2.  Probability addition theorem - getting into the third warehouse.

Obviously

  3.2.  Probability addition theorem .

Since when dropping a single bomb event   3.2.  Probability addition theorem incompatible then

  3.2.  Probability addition theorem .

Example 3. The circular target (Fig. 3.2.1) consists of three zones: I, II and III. The probability of hitting the first zone with one shot is 0.15, the second is 0.23, and the third is 0.17. Find the probability of a miss.

  3.2.  Probability addition theorem

Fig. 3.2.1.

Decision. Denote   3.2.  Probability addition theorem - miss   3.2.  Probability addition theorem - hit. Then

  3.2.  Probability addition theorem ,

Where   3.2.  Probability addition theorem - hit respectively in the first, second and third zones

  3.2.  Probability addition theorem ,

from where

  3.2.  Probability addition theorem .

As already mentioned, the probability addition theorem (3.2.1) is valid only for incompatible events. In the event that events   3.2.  Probability addition theorem and   3.2.  Probability addition theorem are joint, the probability of the sum of these events is expressed by the formula

  3.2.  Probability addition theorem . (3.2.3)

The validity of formula (3.2.3) can be clearly seen by considering Figure 3.2.2.

  3.2.  Probability addition theorem

Fig. 3.2.2.

Similarly, the probability of the sum of three joint events is calculated by the formula

  3.2.  Probability addition theorem .

The validity of this formula also clearly follows from the geometric interpretation (Fig. 3.2.3).

  3.2.  Probability addition theorem

Fig. 3.2.3.

The method of complete induction can prove the general formula for the probability of the sum of any number of joint events:

  3.2.  Probability addition theorem , (3.2.4)

where amounts apply to different index values   3.2.  Probability addition theorem , etc.

Formula (3.2.4) expresses the probability of the sum of any number of events in terms of the probabilities of the products of these events, taken one, two, three, etc.

A similar formula can be written for the production of events. Indeed, from fig. 3.2.2 it is immediately clear that

  3.2.  Probability addition theorem . (3.2.5)

From fig. 3.2.3 shows that

  3.2.  Probability addition theorem . (3.2.6)

The general formula expressing the probability of the product of an arbitrary number of events in terms of the probabilities of the sums of these events, taken one, two, three, etc., has the form:

  3.2.  Probability addition theorem . (3.2.7)

Formulas of the type (3.2.4) and (3.2.7) find practical application in converting various expressions containing probabilities of sums and products of events. Depending on the specifics of the task, in some cases it is more convenient to use only sums, and in others only the works of events: similar formulas serve to convert one into another.

Example. The technical device consists of three units: two units of the first type -   3.2.  Probability addition theorem and   3.2.  Probability addition theorem - and one unit of the second type -   3.2.  Probability addition theorem . Aggregates   3.2.  Probability addition theorem and   3.2.  Probability addition theorem duplicate each other: if one of them fails, it automatically switches to the second. Unit   3.2.  Probability addition theorem not duplicated. In order for the device to stop working (failed), it is necessary that both units fail simultaneously.   3.2.  Probability addition theorem and   3.2.  Probability addition theorem or unit   3.2.  Probability addition theorem . Thus, a device failure is an event.   3.2.  Probability addition theorem - is presented in the form:

  3.2.  Probability addition theorem ,

Where   3.2.  Probability addition theorem - unit failure   3.2.  Probability addition theorem ,   3.2.  Probability addition theorem - unit failure   3.2.  Probability addition theorem ,   3.2.  Probability addition theorem - unit failure   3.2.  Probability addition theorem .

Required to express the probability of an event   3.2.  Probability addition theorem through probabilities of events that contain only sums, and not products of elementary events   3.2.  Probability addition theorem ,   3.2.  Probability addition theorem and   3.2.  Probability addition theorem .

Decision. By the formula (3.2.3) we have:

  3.2.  Probability addition theorem ; (3.2.8)

according to the formula (3.2.5)

  3.2.  Probability addition theorem ;

according to the formula (3.2.6)

  3.2.  Probability addition theorem .

Substituting these expressions into (3.2.8) and making abbreviations, we get:

  3.2.  Probability addition theorem .


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis