Chapter 3
Independence
•
Suppose that it is known that event A has occurred.
We update the probability of the event B
from P(B) to P(B

A) =
P(BA)
P(A)
•
P(B

A) can be larger, smaller, or the same as P(B)
•
If P(B

A) = P(B), then the occurrence of A does not change our estimate of the probability of B
•
P(B

A) = P(B)
⇔
P(B
c

A) = P(B
c
)
•
The occurrence of A does not change the odds favoring B
— P(B):P(B
c
) = P(B

A):P(B
c

A)
•
Equivalently, the occurrence of B does not change the odds favoring A
•
In such cases, A and B are said to be
independent
events
•
The formal definition of independence uses A and B in more symmetric fashion
•
If P(B

A) = P(AB)/P(A) = P(B), then P(AB) = P(A)P(B)
•
Definition:
Events A and B are said to be (stochastically) independent if
P(AB) = P(A)P(B)
•
Do not confuse the notions of independent events and mutually exclusive events
•
P(AB) = P(A)P(B)
≠
0 for independent events, so they
cannot be
mutually exclusive
•
P(AB) = 0
≠
P(A)P(B) for mutually exclusive events, so they
cannot be
independent
•
Magic mantra to memorize
•
Independent events cannot be mutually exclusive
Mutually exclusive events cannot be independent
•
Note:
There are trivial uninteresting exceptions when either P(A) or P(B) equals 0
•
If A and B are independent events, then so are A and B
c
, A
c
and B, and A
c
and B
c
•
P(AB
c
) = P(A) – P(AB) = P(A) – P(A)P(B) = P(A)[(1 – P(B)] = P(A)P(B
c
)
•
The other two results can be proved similarly (Try them!)
•
Independence of events is of great help in calculations;
P(AB) is just P(A)P(B) for independent events A and B
•
For example, P(A
∪
B) = P(A) + P(B) – P(AB) = P(A) + P(B) – P(A)P(B)
•
P(A
∪
B
c
) = P(A)+P(B
c
) – P(AB
c
) = P(A) + P(B
c
) – P(A)P(B
c
)
•
Even though independence is a great help in calculations, it cannot and should not be used
indiscriminately (e.g., whenever you are stuck in solving a problem and can’t figure out a way
to proceed)
•
DO
NOT
ASSUME
that events are independent unless the problem explicitly says so
•
On homework and exams, if you are asked to determine whether or not A and B are
independent events, just find P(AB) and compare it to the product P(A)P(B)
•
But this is not the way the concept of independence is used in probability theory
•
Physical independence versus stochastic independence
•
It might be reasonable to assume on the basis of study and preliminary analysis of the physical
phenomenon under consideration that two events are
physically independent
•
Physical independence is an
assumption
that we justify based purely on physical
considerations
•
We conclude from physical principles that occurrence (or nonoccurrence) of one event seems
to have no influence on the occurrence of another event
•
Example:
two successive tosses of a fair coin.
Does the occurrence of a head on the first toss
influence the result of the second toss?
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 Fall '06
 SCHWAGER
 Probability, Probability theory, University of Illinois, Engineering Applications

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