Roberto Will Randomly Choose 1 Book to Read
Lecture 6
Probability and combinatorial analysis
Classical definition of probability
Example. Let an urn contain 6 identical, carefully shuffled balls, and ii of them are blood-red, three – blue and 1 –
white. Obviously, the possibility to have out at random from the urn a colour ball (i.due east. red or blue) is more
than the possibility to extract a westhite brawl.
Is it possible to depict this possibility by a number? It appears it is possible. This number is said to exist
the probability of an outcome (appearance of a colour ball). Thus, the probability is the number describing
the degree of possibility of an appearance of an upshot.
Allow the event A be an appearance of a colour ball. Wdue east phone call each of possible results of a trial (the trial is an
extracting a ball from the urn) by simple upshot. We announce simple events by 1, 2, 3 and et
cetera. In our exaplenty the following vi elementary events are possible: 1 – the white ball has appeared;
2, three – a red ball has appeared; iv, 5, 6 – a blueish ball has appeared. These events course a complete
grouping of pairwise incompatible events (it necessarily will be appeared only one ball) and they are equally
possible (a ball is randomly extracted; the balls are identical and carefully shuffled).
Nosotros telephone call those unproblematic events in which the result interesting for us occurs, every bit favorable to this consequence. In
our example the following 5 events favor to the result A (appearance of a colour ball): two, iii, 4, 5, vi.
In this sense the effect A is subdivided on some elementary events; an elementary effect is non subdivided
into other events. It is the stardom between the event A and an elementary issue.
The ratio of the number of favorable to the consequence A elementary events to their total number is said to be
the probability of the upshot A and information technology is denoted by P(A) . In the considered example we have 6 elementary
events; 5 of them favor to the event А. Therefore, the probability that the taken ball volition be colour is equal
to P(A) = 5/6. This number gives such a quantitative estimation 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 northwardumber of favorable elementary events for this issue to
their total number of all as possible incompatible elementary events forming a complete group.
Thus, the probability of the effect A is adamant 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. Hither we suppose that elementary events are incompatible, equally possible and form a
complete grouping.
The definition of probability implies the following its backdrop:
Property ane. The probability of a reliable effect is equal to one.
In fact, if an event is reliable, each elementary issue of a trial favors to the consequence. In this case k = n
and consequently P(A) = yard/n = n/due north = one.
Property 2. The probability of an impossible event is equal to 0.
Indeed, if an event is impossible and then none of elementary events of a trial favors to the event. In this
example g = 0 and consequently P(A) = m/northward = 0/northward = 0.
Holding 3. The probability of a random event is the positive number between 0 and ane.
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 < m/north < 1 and consequently 0 < P(A) < i.
Thus, the probability of an arbitrary consequence A satisfies the double inequality:
0 P(A) i
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