Let's Say We Draw Two Socks From The Box. What Is The Probability That We Draw A Matching Pair?

Lecture 6

Probability and combinatorial analysis

Classical definition of probability

Case. Let an urn comprise 6 identical, carefully shuffled balls, and 2 of them are cherry-red, 3 – blue and 1 –

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

than the possibility to extract a white ball.

Is it possible to describe this possibility by a number? Information technology appears it is possible. This number is said to exist

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

the degree of possibility of an appearance of an consequence.

Let the event A be an advent of a color ball. Westdue east phone call each of possible results of a trial (the trial is an

extracting a ball from the urn) by elementary event. Nosotros denote elementary events by  1,  two,  3 and et

cetera. In our example the following half dozen elementary events are possible:  1 – the white brawl has appeared;

 2,  3 – a red ball has appeared;  iv,  5,  6 – a bluish brawl has appeared. These events form a complete

group of pairwise incompatible events (it necessarily will be appeared only one brawl) and they are equally

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

Nosotros call those uncomplicated events in which the outcome interesting for us occurs, every bit favorable to this event. In

our example the post-obit v events favor to the event A (advent of a colour ball):  2,  3,  four,  v,  6.

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

into other events. Information technology is the distinction between the outcome A and an unproblematic event.

The ratio of the number of favorable to the upshot A elementary events to their total number is said to be

the probability of the outcome A and it is denoted by P(A) . In the considered instance we have 6 elementary

events; five of them favor to the consequence А. Therefore, the probability thursdayat the taken ball will be colour is equal

to P(A) = 5/six. This number gives such a quantitative interpretation of the degree of possibility of an

appearance of a colour ball that we wanted to find.

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

their total number of all equally possible incompatible elementary events forming a complete grouping.

Thus, the probability of the issue A is adamant by the formula:

where m is the number of unproblematic events favorable to A ; north is the number of all possible elementary

events of a trial. Hither we suppose that elementary events are incompatible, equally possible and form a

complete group.

The definition of probability implies the following its properties:

Property 1. The probability of a reliable event is equal to 1.

In fact, if an effect is reliable, each elementary event of a trial favors to the event. In this case g = n

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

Property 2. The probability of an impossible event is equal to 0.

Indeed, if an event is impossible then none of elementary events of a trial favors to the event. In this

instance m = 0 and consequently P(A) = one thousand/n = 0/north = 0.

Holding 3. The probability of a random event is the positive number between 0 and i.

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

case 0 < m < n; then 0 < 1000/north < 1 and consequently 0 < P(A) < 1.

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

0  P(A)  ane

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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