Elements of combinatorics. Combinatorics

A \u003d (the point falls into circle A)

Basic concepts

B \u003d (point falls into triangle B)
IN
A
A + B \u003d (the point falls into at least one figure A and B).

Basic concepts

• A-B \u003d (the point will fall into circle A
• and will not fall into triangle B)

A
IN
A

Basic concepts

IN
A
AB \u003d (the point falls into both figures A and B).

Basic concepts

• An event is called opposite to an event if it occurs if and only if the event does not occur.

A
A \u003d (the point falls into circle A)
=(point is not in circle A)

Basic concepts

• Events A and B are called incompatible if they cannot occur together in the same experiment.

IN
A
A \u003d (the point falls into circle A)
B \u003d (point falls into triangle B)
A and B are incompatible events

Operation Properties

• A+B=B+A
• (A+B)+C=A+(B+C)
• A + Ǿ \u003d A
• A+ Ω = Ω

AB=BA
A(BC)=(AB)C
And Ǿ = Ǿ
A Ω = A
Ǿ
( A + B ) C=A C + B C
(D.z.)

Space of elementary events

• Consider a stochastic experiment.

1. Events ω mutually exclude each other.
2. As a result of the experiment, be sure
one of them comes.
3. For any event A,
upon the occurrence of the event ω , we can say that
event A occurred or did not occur.
The events ω are elementary .

Space of elementary events

• Example 3. Throwing a dice.
• =(drop number 1)
• =(drop number 2)
• - - - - - - - - - - - - - - - - -
• =(drop number 6)

Space of elementary events

• Example 4
• A factory produces N items of the same type.
• To assess the quality, m products are selected and examined.

ω is any set of m products.
- space of elementary events .

Definition of probability

• Consider the stochastic
• experiment.

• finite or countable set

elementary events.

Definition of probability

• Properties probabilities:
• 1)
• 2)

P ( Ǿ )=0