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Lecture 6

Probability and combinatorial analysis

Classical definition of probability

Example. Let an urn contain half dozen identical, carefully shuffled balls, and 2 of them are red, 3 – blue and 1 –

white. Obviously, the possibility to take out at random from the urn a colour ball (i.e. red or bluish) is more

than the possibility to extract a due westhite ball.

Is it possible to describe this possibility by a number? It appears it is possible. This number is said to be

the probability of an consequence (appearance of a color ball). Thus, the probability is the number describing

the degree of possibility of an appearance of an event.

Permit the upshot A be an appearance of a colour ball. We call each of possible results of a trial (the trial is an

extracting a ball from the urn) by elementary result. Nosotros denote simple events by  1,  2,  3 and et

cetera. In our example the post-obit half-dozen elementary events are possible:  ane – the white ball has appeared;

 2,  three – a red ball has appeared;  4,  5,  six – a blueish ball has appeared. These events class a complete

grouping of pairwise incompatible events (information technology necessarily will exist appeared only one ball) and they are equally

possible (a ball is randomly extracted; the balls are identica50 and carefully shuffled).

We phone call those unproblematic events in which the event interesting for us occurs, as favorable to this outcome. In

our example the following 5 events favor to the event A (appearance of a color brawl):  2,  three,  4,  5,  6.

In this sense the event A is subdivided on some simple events; an unproblematic upshot is not subdivided

into other events. Information technology is the distinction betwixt the effect A and an elementary event.

The ratio of the number of favorable to the event A unproblematic events to their full number is said to be

the probability of the event A and it is denoted by P(A) . In the considered case we accept 6 elementary

events; v of them favor to the event А. Therefore, the probability that the taken ball will be colour is equal

to P(A) = 5/vi. This number gives such a quantitative estimation of the degree of possibility of anorth

appearance of a colour ball that we wanted to detect.

The probability of the event A is the ratio of the number of favorable elementary events for this event to

their total number of all as possible incompatible simple events forming a consummate group.

Thus, the probability of the consequence A is determined past the formula:

where thou is the number of elementary events favorable to A ; n is the number of all possible elementary

events of a trial. Here we suppose that uncomplicated events are incompatible, equally possible and form a

complete group.

The definition of probability implies the following its backdrop:

Property 1. The probability of a reliable outcome is equal to ane.

In fact, if an event is reliable, each elementary result of a trial favors to the event. In this example m = north

and consequently P(A) = thousand/due north = n/north = 1.

Property two. The probability of an incommunicable event is equal to 0.

Indeed, if an issue is impossible then none of unproblematic events of a trial favors to the outcome. In this

case g = 0 and consequently P(A) = m/n = 0/n = 0.

Belongings iii. The probability of a random event is the positive number betwixt 0 and 1.

In fact, a random event is favored only part of the total number of elementary events of a trial. In this

case 0 < grand < northward; and so 0 < m/north < 1 and consequently 0 < P(A) < ane.

Thus, the probability of an arbitrary event A satisfies the double inequality:

0  P(A)  i

Source: https://www.studocu.com/row/document/%D2%9Baza%D2%9Bstan-britan-tekhnikaly%D2%9B-universiteti/mathematics-for-economists/l6-st-2019-lecture-notes-6/6032040

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