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
1)A+B
2)AB 3) A-B 4)
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.
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
.
4. Geometric definition of probability
The event A = {a point falls in the region g}. Then
,
S(G), S(g) – areas of figures.