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.
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