STAT 200 Chapter 4
Probability: The Study of Randomness
Probability concepts
(Section 4.2)
•
A sample space
S
is the set of of all possible outcomes of a random phenomenon.
•
An event is an outcome or a combination of outcomes from a random phenomenon.
We denote an event by an uppercase letter, e.g., A, B, C.
e.g., Rolling a die: rolling a ‘6’ is an event. Rolling a ‘5’ is another event. Rolling two
‘4”s in two rolls is also an event.
•
The notation
P
(
A
) denotes the probability of an event A that will occur.
Properties of
P
(
A
):
1. 0
≤
P
(
A
)
≤
1
P
(
A
) = 0 implies event
A
is impossible.
P
(
A
) = 1 implies event
A
is certain.
The larger the
P
(
A
), the higher the chance that the event
A
will occur.
2. The sum of the probabilities of all the nonoverlapping
events in the sample space
is equal to 1.
The sample space contains all possible outcomes of a random phenomenon, so by
the above property,
P
(
S
) = 1.
Terminologies and probability rules
(Sections 4.2 and 4.5)
•
Some terminologies and notations:
A or B
(A and B are events) The notation “A or B” means either event A or event
B or both occur.
A and B
(A and B are events) The notation “A and B” means event A and event B
occur together.
complement of an event
The complement of an event A is the event that A does
not occur. We denote the complement of A by
A
c
.
disjoint
Two events A and B are said to be disjoint if the occurrence of A prevents
the occurrence of B. In other words, the two events cannot occur together.
independent
Two events A and B are said to be independent if the occurrence of A
does not alter the probability that B will occur.
1
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View Full Documentconditional probability of B given A
(A and B are events) The notation
P
(
B

A
)
denotes the probability of event B given that event A has occurred.
P
(
B

A
) =
P
(
A
and
B
)
P
(
A
)
,
P
(
A
)
>
0
•
If all the possible outcomes in a sample space are equally likely, then the probability
of an event
A
is given by:
P
(
A
) =
number of outcomes in event A
total number of outcomes in the sample space
•
Determining
P
(
A
or
B
) and
P
(
A
and
B
):
1. If events A and B are disjoint, then the probability that event A or event B or
both will occur is
P
(
A
or
B
) =
P
(
A
) +
P
(
B
)
[Addition Rule]
Otherwise,
P
(
A
or
B
) =
P
(
A
)+
P
(
B
)

P
(
A
and
B
)
[General Addition Rule, Section 4.5]
2. If events A and B are independent, then the probability that event A and event
B will occur together is
P
(
A
and
B
) =
P
(
A
)
×
P
(
B
)
[Multiplication Rule]
Otherwise,
P
(
A
and
B
)
6
=
P
(
A
)
×
P
(
B
)
,
and
P
(
A
and
B
) =
P
(
A
)
×
P
(
B

A
)
[General Multiplication Rule, Section 4.5]
For a series of events
A
i
’s (1
,
2
,...,m
), they are independent if and only if
P
(
A
i
and
A
j
) =
P
(
A
i
)
×
P
(
A
j
) for all
i
6
=
j
P
(
A
i
and
A
j
and
A
k
) =
P
(
A
i
)
×
P
(
A
j
)
×
P
(
A
k
) for all
i
6
=
j
6
=
k
.
.
. =
.
.
.
P
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 Spring '08
 KARIM
 Normal Distribution, Probability, Probability distribution, Probability theory, probability density function

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