Probability is a chance of occurrence

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Probability is a chance of occurrence
. Sample space versus an event (they are just like a population versus a sample) A sample space (S) is a set of all possible events or all possible outcomes of arandom experiment. An event is each possible outcome of a variable.(mostly denoted by capital capital letters, say A or B or C etc.Example 1. When you roll a die as an experiment, all possible outcomes are
S
={
1,2,3,4,5,6
}
Let event
A
=
{
1,2
}
and
B
={
2,4 6
}
Assigning probability to an event.
Assigning probability to event A. Using the previous example (example 1)
S
={
1,2,3,4,5,6
}
the possibility or chance of getting event A is
{
1,2
}
out of
{
1,2,3,4,5,6
}
.
Number of elements in A is 2,
n
(
A
)=
2
and number of elementsin S is 6,
n
(
S
)=
6
so chance of event A occurring is given by:
P
(
A
)
=
n
(
A
)
n
(
S
) =
26
=
13
=
0.33
Do the same for all events. Assigning probability to event B.
S
={
1,2
,
3,4
,
5,6
}
the possibility or chance of getting event B is
{
2,4,6
}
out of
{
1,2,3,4,5,6
}
.
Number of elements in B is 3,
n
(
B
)=
3
and number of elementsis S is 6,
n
(
S
)=
6
so chance of event A occurring is given by:
P
(
B
)
=
n
(
B
)
n
(
S
) =
36
=
12
=
0.5
The probability of a sample space is always equal to one,
P
(
S
)
=
n
(
S
)
n
(
S
)=
66
=
1
Rules of probabilities
.Let A and B be any events. Complement can be denoted by
A
c
or
´
A
or
A
'
“or” means union
∪
(A or B means all elements in A and B)“and” means intersection
∩
(A and B common elements in A and B)1.Complementary:
P
(
A
c
)
=
1
−
P
(
A
)
or
P
(
B
c
)
=
1
−
P
(
B
)
2.Additional:
P
(
A
∪
B
)=
P
(
A
)
+
P
(
B
)−
P
(
A ∩B
)
3.Conditional:
P
(
A
/
B
)
=
P
(
A∩B
)
P
(
B
)
or
(
B
/
A
)
=
P
(
A∩B
)
P
(
A
)
.Note that
P
(
A∩B
)
=
P
(
B∩A
)
and
P
(
A
∪
B
)
=
P
(
B
∪
A
)
Conditions of probabilities
1.Mutually exclusive : there is no intersection between the events, that is
P
(
A∩B
)
=
0
2.Independent : one event does not depend on the other to occur, that is
P
(
A∩B
)
=
P
(
A
)
×P
(
B
)
Let A and B be mutually exclusive events.
1.
Complementary:
P
(
A
c
)
=
1
−
P
(
A
)
or
P
(
B
c
)
=
1
−
P
(
B
)
Is not affected by the condition because it does not contain
P
(
A∩B
)
.
2.
Additional
:
P
(
A
∪
B
)
=
P
(
A
)
+
P
(
B
)
−
P
(
A∩B
)
=
P
(
A
)
+
P
(
B
)
−
0
¿
P
(
A
)
+
P
(
B
)
3.
Conditional:
P
(
A
/
B
)
=
P
(
A∩B
)
P
(
B
) =
0
P
(
B
)=
0
or
P
(
B
/
A
)
=
P
(
A∩B
)
P
(
A
) =
0
P
(
A
)=
0
.
Let A and B be independent events.1.Complementary:
P
(
A
c
)
=
1
−
P
(
A
)
or
P
(
B
c
)
=
1
−
P
(
B
)
Is not affected by the condition because it does not contain
P
(
A∩B
)
.
2.Additional
:
P
(
A
∪
B
)
=
P
(
A
)
+
P
(
B
)
−
P
(
A∩B
)
=¿
P
(
A
)
+
P
(
B
)
−[
P
(
A
)
×P
(
B
)
]
3.
Conditional:
P
(
A
/
B
)
=
P
(
A∩B
)
P
(
B
) =
P
(
A
)
×P
(
B
)
P
(
B
) =
P
(
A
)
or
P
(
B
/
A
)
=
P
(
A∩B
)
P
(
A
) =
P
(
A
)
×P
(
B
)
P
(
A
) =
P
(
B
)
.
Contingency table
Events
B
´
B
total
A
P
(
A∩B
)
P
(
A∩
´
B
)
P
(
A
)
´
A
P
( ´
A∩B
)
B
´
A∩
´¿
P
¿
P
( ´
A
)
total
P
(
B
)
P
( ´
B
)
1

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