The Product Rule:
Use & Misuse
The Product Rule
If
a procedure has
1. A 1st stage with s1 outcomes
2. A 2nd stage with s2 outcomes
and the composite outcomes are distinct
then
The procedure has s1 x s2 composite outcomes.
Copyright © Peter Cappello 2011
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Visualizing the Procedure
– The procedure for constructing a composite outcome
requires a selection at each stage.
– Visualize all invocations of the procedure as a tree.
• Each level in the tree corresponds to a stage.
• Each leaf in the tree corresponds to a composite
outcome: Each leaf must be distinct.
Copyright © Peter Cappello 2011
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Example 1
Let B = { 0, 1 } and V = { a, e, i, o, u }.
How many 1-to-1 functions, f, are there from B to V?
Solution:
1. Select the vowel for f( 0 ) (5 choices).
2. Select the vowel for f( 1 ) (4 choices).
Thus, there are 5x4 1-to-1 functions from B to V.
Copyright © Peter Cappello 2011
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Example 1 continued
Visualize the selection process as a tree.
1. Pick f(0)
a
e
i
o
u
2. Pick f(1)
(a,e)
(e,a)
(i,a)
Copyright © Peter Cappello 2011
(o,a)
(u,a)
(u,o)
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Misuse of the Product Rule
The set of 5 vowels has how many subsets of 2 letters?
Erroneous solution:
1. Pick the 1st letter (5 choices).
2. Pick the 2nd letter (4 choices).
There are 5 x 4 subsets of 2 letters. Not!
–
Visualize the selection process above as a tree.
–
The composite outcomes are not distinct.
• Each leaf appears twice (e.g., ae, ea)
• The same set of 2 vowels is counted twice.
To use the product rule properly, it is necessary that:
Each component of the composite outcome is associated with 1 stage
of the selection process.
If it cannot be so associated, the product rule is used incorrectly.
Copyright © Peter Cappello 2011
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Subset Example continued
Visualize the selection process as a tree.
1. Pick “1st” vowel
a
e
i
o
u
2. Pick “2nd”
{a,e}
{e,a}
{i,a}
Copyright © Peter Cappello 2011
{o,a}
{u,a}
{u,o}
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Proper Use of the Sum Rule
•
The subsets are pairwise disjoint.
•
The union of the subsets includes every element
that you want to count.
Sound familiar?
Copyright © Peter Cappello 2011
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Proper Use of the Sum Rule
Let S1, S2, …, Sn be subsets of S.
|S| = |S1| + |S2| + …+ |Sn |  S1, S2, …, Sn partition S.
Copyright © Peter Cappello 2013
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