Events
Event. Elementary event. Space of elementary events.
Certain event. Impossible event. Identical events.
Sum, product, difference of events. Complementary events.
Mutually exclusive events. Equally likely events.
An event in probability theory is any fact, which may occur as a result of an experiment with a random outcome or may not. The simplest result of such experiment is called an elementary event ( for instance, an appearance of heads or tails at throwing of a coin, shooting hit, an appearance of an ace at taking a card out of a pack, a random appearance of number at throwing of a die etc.).
A set of all elementary events E is called a space of elementary events . So, this space consists of six elementary events at throwing of a die and 52 elementary events at taking a card out of a pack. An event can consist of one or several elementary events, for example, an appearance of two aces one after the other at taking a card out of a pack, or an appearance of the same number at triple throwing of a die. Then it's possible to define an event as an arbitrary subset of a space of elementary events.
A certain event is all space of elementary events. Thus, a certain event is an event, which must happen as a result of the experiment without fail. Such event at throwing of a die is a fall of the die on one of its faces.
An impossible event
(
)
is called an empty subset of a space of elementary events.
That is,
an impossible event
cannot happen as a result of the experiment. So, such event at throwing of a die
is a fall of the die on its edge.
Events
A
and
B
are
called
identical events
(
A
=
B
), if the event
A
occurs if and only if the
event
B
occurs. An event
A
involves
the event
B
(
À
Â
), if the condition
"the event B occurred
"
follows from the condition
"the
event A occurred
"
.
An event
C
is called
a sum of the events
A
and
B
(
Ñ
=
À
Â
), if the event
C
happens if and only if
either the event
A
happens or the event B happens.
An event
C
is called
a product of the events
A
and
B
(
Ñ
=
À
Â
),
if the event
C
happens if and only if both
event
A
and event B happen.
An event C is called a difference of the events A and B ( Ñ = À – Â ), if the event C happens if and only if the event A happens and the event B doesn't.
An event A' is a complementary event to the event A , if the event A doesn't occur. So, shooting hit and miss are complementary events.
Events
A
and
B
are
called
mutually exclusive
(
À
Â
=
),
if
their simultaneous occurrence
is impossible.
For
instance,
an
occurrence both heads and tails
at throwing of a coin.
If several events can happen as a result of an experiment, and each of them isn't more possible than others according to objective conditions, then such events are called equally likely events . Examples of equally likely events are: an appearance of two, an ace and a knave at taking a card out of a pack, an appearance of any number from 1 to 6 at throwing of a die etc.