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Handout 6. Probability 4.54.6
Probability Concepts:
•
Mutually Exclusive Events
– If A occurs, then B cannot occur
– A and B have no common elements
Black
Cards
Red
Cards
A card cannot be
Black and Red at the
same time.
A
B
Independent and Dependent Events
Independent:
Occurrence of one does not
influence the probability of
occurrence of the other
Dependent:
Occurrence of one affects the probability of the other
•
Independent Events
A = heads on one flip of a fair coin
B = heads on second flip of same coin
Result of second flip does not
depend on the result of
the first flip.
•
Dependent Events
A = rain forecasted on the news
B = take umbrella to work
Probability of the second event is
affected by the
occurrence of the first event.
Conditional Probability
Example 6.1
No CD
CD
Total
AC
.2
.5
.7
No AC
.2
.1
.3
Total
.4
.6
1.0
• Of the cars on a used car lot,
70%
have air conditioning (AC)
and
40%
have a CD player (CD).
20%
of the cars have both.
What is the probability that a car has a CD player, given that it
has AC ? i.e., we want to find
P(CD  AC)
.2857
.7
.2
P(AC)
AC)
and
P(CD
AC)

P(CD
=
=
=
Given AC, we only consider the top row (70% of the cars). Of these, 20% have a CD
player.
20% of 70% is about 28.57%.What is the probability that the car has air
conditioning , given that it has CD?
Conditional Probabilities
•
The probability that A
occurs, given that event B
has occurred is called the
conditional probability
of A given B and is
defined as
•
Think about events A and B
spaced in
time (B has
occured).
0
)
(
if
)
(
)
(
)

(
≠
=
B
P
B
P
AB
P
B
A
P
“given”
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•
Multiplication law:
P(AB)=P(B)P(AB) or
P(AB)=P(A)P(BA)
Two events A and B are independent
when
(1) P(AB)=P(A). That is, the probability that A
occurs does not change with the occurrence of B,
or (2) P(BA)=P(B)
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 Spring '08
 Jeneralczuk
 Statistics, Mutually Exclusive, Probability

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