Chapter I. Event and probability

1.Events. Operations above events.

Definition1.1 a). An experiment or a complex of conditions is a situation involving a chance leads to results which we call outcomes.

b). An outcome is a result of a single trial of an experiment.

c). An event is one or more outcomes of an experiment.

Definition 1.2 The event is called certain (denoted U) if it occurs always (at a certain experiment).

Definition 1.3 The event is called impossible (denoted V), if it occurs never (at a certain experiment).

Definition 1.4 Let A be an event. Then the event is called the complement of the event A (denoted ), if always either A or occurs and if A occurs, then does not occurs and inverse if occurs, then A does not occurs.

Definition 1.5 The event is called the product or intersection of events A and B (denoted ), if A and B occurs.

Definition 1.6 The event is called the sum or union of events A and B ( denoted A+B or } if A+B occurs A or B occurs.

Definition 1.7 An event is called the difference of events A and B (denoted A - B or A\B} if A - B occurs A occurs B does not occur.

Definition 1.8 a). If the occurring of an event A involves the occurring of an event B, then we write .

b). If and , then we say A is equal to B and write A=B.

Definition 1.9 Events A and B are called mutually exclusive if AB=V .

Venn chart

2. Elements of combinatorics.

Definition 2.1

If we change an order of elements of a sample and after this we receive a new sample, we call this sample ordered. Otherwise a sample is called disordered.

Definition 2.2

If any element can not be included in a sample more then once, then we call this sample a sample without repetition. Otherwise we call it a sample with repetition.

Definition 2.3

The ordered samples without repetition are called accommodations. The number of all accommodations of m from n elements we denote .

Proposition 2.1

.

Definition 2.4

An accommodation is called a permutation if m = n. The number of all permutations of n elements we denote .

.

Definition 2.5

Disordered samples without repetition are called combinations. The number of all combinations of m elements from n elements we denote .

Proposition 2.3 .

3. Classical definition of probability.

Let’s events satisfy the following conditions:

1) , where U is the certain event.

2) , where V is the impossible event. This means that are pared mutually exclusive.

3) are equally likely or equally probable.

Definition 3.1

Then the set is called the space of elementary events and is called the elementary event .

Now we define the system of events S as follows:

a). .

b). .

c). .

Definition 3.2

The system S is called the field of events.

.

Proposition 3.1 If S a field of events and , then

a). A+B = B+A, AB = BA – commutative laws.

b) A+(B+C) = (A+B)+C,

(AB)C = (A(BC) – associative lows.

c). A(B+C) = AB+AC – distributive law.

d). A+BC =(A+B)(A+C).

e). A+A=A, AA=A.

f). A+U = U, AU = A.

g). A+V = A, AV = V.

definition of probability P(A) for any event .

Definition 3.2 If and , then

.

.

Properties of P(A)

1. For any we have .

2. P(U) = 1.

3. (Theorem of addition of probabilities)

P(A) = P(B)+P(C).

4. For any we have .

5. P(V) = 0.

6. If .

7. For any we have .