Download A, B

PROBABILITY RULES AND TREES
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Rule of complement
Addition rule
Multiplication rule
Probability tree
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RULE OF COMPLEMENT
• The simplest probability rule involves the
complement of an event.
• If the probability of A is P(A), then the probability of
its complement, P(Ac), is
P(Ac)=1- P(A)
• Equivalently, the probability of an event and the
probability of its complement sum to 1.
P(A) + P(Ac)=1
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RULE OF COMPLEMENT
THE BENDRIX SITUATION
• Find P(Bc) using the rule of complements
• Does the rule of complements give the same result as it is
given by the frequencies?
Event
B
BC
Venn Diagram
Sample
Space
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ADDITION RULE
MUTUALLY EXCLUSIVE EVENTS
• We say that events are mutually exclusive if at most one
of them can occur. That is, if one of them occurs, then none
of the others can occur.
• Let A1 through An be any n mutually exclusive events. Then
the addition rule of probability involves the probability that at
least one of these events will occur.
P(at least one of A1 through An) = P(A1) + P(A2) +  + P(An)
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ADDITION RULE
EXHAUSTIVE EVENTS
• Events can also be exhaustive, which means that they
exhaust all possibilities. Probabilities of exhaustive events
add up to 1.
• If A and B are exhaustive,
P(A)+ P(B)=1
• If A, B and C are exhaustive,
P(A)+ P(B)+ P(C)=1
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ADDITION RULE
THE BENDRIX SITUATION
• Interpret the events
E1 = (A and B)
E2 = (A and BC)
Sample
Space
A
B
Venn Diagram
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ADDITION RULE
THE BENDRIX SITUATION
• Are the events E1 and E2 mutually exclusive?
• Verify the following
P(A) = P(E1)+P(E2)
Sample
Space
A
B
Venn Diagram
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ADDITION RULE
THE BENDRIX SITUATION
• Find P(A) using the relationship P(A) = P(E1)+P(E2), if the
relationship is correct
• Are the events
E1 and E2 exhaustive?
Sample
Space
A
B
Venn Diagram
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MULTIPLICATION RULE
INDEPENDENT EVENTS
• We say that two events are independent if occurrence
of one does not change the likeliness of occurrence of
the other
• If A and B are two independent events, the joint
probability P(A and B) is obtained by the
multiplication rule.
P(A and B) = P(A)P(B)
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CONDITIONAL PROBABILITY
• Probabilities are always assessed relative to the information
currently available. As new information becomes available,
probabilities often change.
• A formal way to revise probabilities on the basis of new
information is to use conditional probabilities.
• Let A and B be any events with probabilities P(A) and P(B).
Typically the probability P(A) is assessed without
knowledge of whether B does or does not occur. However if
we are told B has occurred, the probability of A might
change.
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CONDITIONAL PROBABILITY
• The new probability of A is called the conditional
probability of A given B. It is denoted P(A|B).
– Note that there is uncertainty involving the event to the
left of the vertical bar in this notation; we do not know
whether it will occur or not. However, there is no
uncertainty involving the event to the right of the
vertical bar; we know that it has occurred.
• The following formula conditional probability formula
enables us to calculate P(A|B):
P( A | B) 
P( A and B)
P( B)
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CONDITIONAL PROBABILITY
• If A and B are two mutually exclusive events, at most one
of them can occur. So,
P(A|B) =0
P(B|A) =0
• If A and B are two independent events, occurrence of one
does not change the likeliness of occurrence of the other.
So,
P(A|B) = P(A)
P(B|A) = P(B)
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MULTIPLICATION RULE
FOR ANY TWO EVENTS
• In the conditional probability rule the numerator is the
probability that both A and B occur. It must be known in
order to determine P(A|B).
• However, in some applications P(A|B) and P(B) are known;
in these cases we can multiply both side of the conditional
probability formula by P(B) to obtain the multiplication
rule.
P(A and B) = P(A|B)P(B)
• The conditional probability formula and the multiplication
rule are both valid; in fact, they are equivalent.
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MULTIPLICATION RULE
THE BENDRIX SITUATION
• Are the events A and B independent?
• Find P(A and B) using the multiplication rule
• Does the multiplication
rule give the same result
as it is given by the
A
frequencies?
Sample
Space
B
Venn Diagram
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PROBABILITY TREES
• Probability trees are useful to
– calculate probabilities
– identify all simple events
– visualize the relationship among the events
• Probability trees are useful if it is possible to
– break down simple events into stages
– identify mutually exclusive and exhaustive events at
each stage
– ascertain the probabilities of events at each stage
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PROBABILITY TREES
• A probability tree consists of some nodes and branches
• Nodes
– an initial unlabelled node called origin
– other nodes, each labeled with the event represented
by the node
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PROBABILITY TREES
• Branches
– each branch connect a pair of nodes.
– a branch from A to B implies that event B may occur
after event A
– each branch from
• origin to A is labeled with probability P(A)
• A to B is labeled with the probability P(B|A)
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PROBABILITY TREES
• Any path through the tree from the origin to a terminal
node corresponds to one possible simple event.
• All simple events and their probabilities are shown next to
the terminal nodes.
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PROBABILITY TREES
• Example 1: Construct a probability tree diagram for the
Bendrix Company.
Stage 1
P(
B)
=2
/3
B
Stage 2
=3/4
P(A|B)
P(A C|B
)=1/4
C
B
P(
3
1/
)=
BC
Simple Joint
Events Probabilities
C )=1/5
A
BA
P(BA)=0.5000
AC
BAC
P(BAC)=0.1667
A
BCA
P(BCA)=0.0667
AC
BCAC
P(BCAC)=0.0266
P(A|B
P(A C|B C
)=4/5
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PROBABILITY TREES
• Example 2: In a bag containing 7 red chips and 5 blue
chips you select 2 chips one after the other without
replacement. Construct a probability tree diagram for this
information.
Stage 1
Stage 2
=6/11
)
R
|
R
(
P
P(
R
)=
7/
12
R
P(B|R)
=5/11
/
=5
B)
P(
=7/11
P(R|B)
B
Simple Joint
Events Probabilities
RR
P(RR)=7/22
B
RB
P(RB)=35/132
R
BR
P(BR)=35/132
B
BB
P(BB)=5/33
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R
P(B|B)
=4
/11
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PROBABILITY TREES
• What is the probability of getting a red chip first and then a
blue chip?
• What is the probability of getting a blue chip first and then
a red chip?
• What is the probability of getting a red and a blue chip?
• What is the probability of getting 2 red chips?
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