Use amp Misuse Copyright Peter Cappello 2011 2 The Product Rule If a procedure has A 1 st stage with s 1 outcomes A 2 nd stage with s 2 outcomes and the composite outcomes are ID: 269434
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Slide1
The Product Rule: Use & MisuseSlide2
Copyright © Peter Cappello 20112
The Product Rule
If
a procedure has
A 1
st
stage with
s
1
outcomes
A 2
nd
stage with
s
2
outcomes
and
the composite outcomes are
distinct
then
The procedure has
s
1
x
s
2
composite outcomes.Slide3
Copyright © Peter Cappello 20113
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
.Slide4
Copyright © Peter Cappello 20114
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
:
Select the vowel for
f(
0
)
(
5 choices
).
Select the vowel for
f(
1
)
(
4 choices
).
Thus, there are
5x4
1-to-1 functions from
B
to
V
.Slide5
Copyright © Peter Cappello 20115
Example 1 continued
Visualize the selection process as a tree.
a
e
i
o
u
(a,e)
(u,o)
(e,a)
(i,a)
(o,a)
(u,a)
1. Pick
f(
0
)
2. Pick
f(
1
)Slide6
Copyright © Peter Cappello 20116
Misuse of the Product Rule
The set of
5
vowels has how many
subsets
of
2
letters?
Erroneous solution
:
Pick the 1
st
letter (
5 choices
).
Pick the 2
nd
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.Slide7
Copyright © Peter Cappello 20117
Subset Example continued
Visualize the selection process as a tree.
a
e
i
o
u
{a,e}
{u,o}
{e,a}
{i,a}
{o,a}
{u,a}
1. Pick
“1
st
” vowel
2. Pick
“2
nd
”Slide8
Copyright © Peter Cappello 20118
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
?Slide9
Proper Use of the Sum RuleLet S1
,
S
2
, …,
S
n
be subsets of S.
|S| = |
S
1
| + |S
2
| + …+ |
S
n
|
S1, S2, …, Sn
partition S.Copyright © Peter Cappello 2013
9