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10.1 – Counting by Systematic Listing
One-Part Tasks
The results for simple, one-part tasks can often be listed easily.
Heads or tails
Tossing a fair coin:
Rolling a single fair die
1, 2, 3, 4, 5, 6
Consider a club N with four members:
N = {Mike, Adam, Ted, Helen}
or
N = {M, A, T, H}
In how many ways can this group select a president?
There are four possible results:
M, A, T, and H.
10.1 – Counting by Systematic Listing
Product Tables for Two-Part Tasks
Determine the number of two-digit numbers that can be
written using the digits from the set {2, 4, 6}.
The task consists of two parts:
1. Choose a first digit
2. Choose a second digit
The results for a two-part task can be pictured in a product
table.
First
Digit
2
4
6
2
22
42
62
Second Digit
4
24
44
64
6
26
46
66
9 possible numbers
10.1 – Counting by Systematic Listing
Product Tables for Two-Part Tasks
What are the possible outcomes of rolling two fair die?
10.1 – Counting by Systematic Listing
Product Tables for Two-Part Tasks
Find the number of ways club N can elect a president and
secretary.
N = {Mike, Adam, Ted, Helen} or N = {M, A, T, H}
The task consists of two parts:
1. Choose a president 2. Choose a secretary
M
Pres.
M
A
T
H
Secretary
A
T
H
MM
MA
MT
MH
AM
AA
AT
AH
TM
TA
TT
TH
HM
HA
HT
HH
12 outcomes
10.1 – Counting by Systematic Listing
Product Tables for Two-Part Tasks
Find the number of ways club N can elect a two member
committee.
N = {Mike, Adam, Ted, Helen} or N = {M, A, T, H}
2nd Member
1st
Member
M
A
T
H
M
A
T
H
MM
MA
MT
MH
AM
AA
AT
AH
TM
TA
TT
TH
HM
HA
HT
HH
6 committees
10.1 – Counting by Systematic Listing
Tree Diagrams for Multiple-Part Tasks
A task that has more than two parts is not easy to analyze with
a product table. Another helpful device is a tree diagram.
Find the number of three digit numbers that can be written
using the digits from the set {2, 4, 6} assuming repeated digits
are not allowed.
A product table will not work for more than two digits.
Generating a list could be time consuming and disorganized.
10.1 – Counting by Systematic Listing
Tree Diagrams for Multiple-Part Tasks
Find the number of three digit numbers that can be written
using the digits from the set {2, 4, 6} assuming repeated digits
are not allowed.
1st #
2
4
6
2nd #
3rd #
4
6
246
6
4
264
2
6
426
6
2
462
2
4
624
4
2
642
6 possibilities
10.1 – Counting by Systematic Listing
Other Systematic Listing Methods
There are additional systematic ways to produce complete
listings of possible results besides product tables and tree
diagrams.
How many triangles (of any size) are in the figure below?
D
E
One systematic approach is begin with A, and
proceed in alphabetical order to write all 3-letter
combinations (like ABC, ABD, …), then cross
out ones that are not triangles and those that
repeat.
C
F
A
B
Another approach is to “chunk” the figure to
smaller, more manageable figures.
There are 12 triangles.
10.2 – Using the Fundamental Counting Principle
Uniformity Criterion for Multiple-Part Tasks:
A multiple part task is said to satisfy the uniformity criterion if
the number of choices for any particular part is the same no
matter which choices were selected for previous parts.
Uniformity exists:
Find the number of three letter combinations that can be written using the
letters from the set {a, b, c} assuming repeated letters are not allowed.
2 dimes and one six-sided die numbered from 1 to 6 are tossed. Generate
a list of the possible outcomes by drawing a tree diagram.
Uniformity does not exists:
A computer printer allows for optional settings with a panel of three onoff switches. Set up a tree diagram that will show how many setting are
possible so that no two adjacent switches can be on?
10.2 – Using the Fundamental Counting Principle
Uniformity
Find the number of three letter combinations that can be written using the
letters from the set {a, b, c} assuming repeated letters are not allowed.
1st letter
a
b
c
2nd letter
3rd letter
b
c
abc
c
b
acb
a
c
bac
c
a
bca
a
b
cab
b
a
cba
6 possibilities
10.2 – Using the Fundamental Counting Principle
Uniformity
2 dimes and one six-sided die numbered from 1 to 6 are tossed. Generate
a list of the possible outcomes by drawing a tree diagram.
Die #
1
2
3
4
5
6
Dime
d1
d2
d1
d2
d1
d2
d1
d2
d1
d2
d1
d2
1 d1
1 d2
2 d1
2 d2
3 d1
3 d2
4 d1
4 d2
5 d1
5d2
6 d1
6 d2
12 possibilities
10.2 – Using the Fundamental Counting Principle
Uniformity does not exist
A computer printer is designed for optional settings with a panel of three
on-off switches. Set up a tree diagram that will show how many setting
are possible so that no two adjacent switches can be on? (o = on, f = off)
1st switch
2nd switch
3rd switch
o
o
o
f
o
f
o
f
f
o
f
o
f
f
10.2 – Using the Fundamental Counting Principle
Fundamental Counting Principle
The principle which states that all possible outcomes in a
sample space can be found by multiplying the number of ways
each event can occur.
Example:
At a firehouse fundraiser dinner, one can choose from 2 proteins (beef
and fish), 4 vegetables (beans, broccoli, carrots, and corn), and 2 breads
(rolls and biscuits). How many different protein-vegetable-bread
selections can she make for dinner?
Proteins
2
Vegetables

4
Breads

16 possible selections
2
=
10.2 – Using the Fundamental Counting Principle
Example
At the local sub shop, customers have a choice of the following: 3 breads
(white, wheat, rye), 4 meats (turkey, ham, chicken, bologna), 6
condiments (none, brown mustard, spicy mustard, honey mustard,
ketchup, mayo), and 3 cheeses (none, Swiss, American). How many
different sandwiches are possible?
Breads
3
Meats

4
Condiments Cheeses
6

3 =

216 possible sandwiches
10.2 – Using the Fundamental Counting Principle
Example:
Consider the set of digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
(a) How many two digit numbers can be formed if repetitions are allowed?
1st digit
9
2nd digit

10
=
90
(b) How many two digit numbers can be formed if no repetitions are allowed?
1st digit
9
2nd digit

9
=
81
(c) How many three digit numbers can be formed if no repetitions are allowed?
1st digit
9
2nd digit

9
3rd digit

8
=
648
10.2 – Using the Fundamental Counting Principle
Example:
(a) How many five-digit codes are possible if the first two digits are letters and
the last three digits are numerical?
1st digit
26
2nd digit
3rd digit
4th digit
10  10 
 26

676000 possible five-digit codes
5th digit
10
(a) How many five-digit codes are possible if the first two digits are letters and
the last three digits are numerical and repeats are not permitted?
1st digit
26
2nd digit
3rd digit
4th digit
10  9
 25


468000 possible five-digit codes
5th digit
8
10.2 – Using the Fundamental Counting Principle
Factorials
For any counting number n, the product of all counting numbers from n
down through 1 is called n factorial, and is denoted n!.
For any counting number n, the quantity n factorial is calculated by:
n! = n(n – 1)(n – 2)…(2)(1).
Definition of Zero Factorial:
0! = 1
Examples:
b) (4 – 1)!
a) 4!
4321
24
3!
321
6
c)
=
54
= 20
10.2 – Using the Fundamental Counting Principle
Arrangements of Objects
Factorials are used when finding the total number of ways to
arrange a given number of distinct objects.
The total number of different ways to arrange n distinct
objects is n!.
Example:
How many ways can you line up 6 different books on a shelf?
6

5

4

3
720 possible arrangements

2

1
10.2 – Using the Fundamental Counting Principle
Arrangements of n Objects Containing Look-Alikes
The number of distinguishable arrangements of n objects,
where one or more subsets consist of look-alikes (say n1 are of
one kind, n2 are of another kind, …, and nk are of yet another
kind), is given by
n!
n1 !n2 !
nk !
.
Example:
Determine the number of distinguishable arrangements of the letters of
the word INITIALLY.
9 letters
9!
3!  2!
with 3 I’s
and 2 L’s
30240 possible arrangements
10.3 – Using Permutations and Combinations
Permutation: The number of ways in which a subset of
objects can be selected from a given set of objects, where
order is important.
Given the set of three letters, {A, B, C}, how many possibilities are there
for selecting any two letters where order is important?
(AB, AC, BC, BA, CA, CB)
Combination: The number of ways in which a subset of
objects can be selected from a given set of objects, where
order is not important.
Given the set of three letters, {A, B, C}, how many possibilities are there
for selecting any two letters where order is not important?
(AB, AC, BC).
10.3 – Using Permutations and Combinations
Factorial Formula for Permutations
n!
.
n Pr 
(n  r )!
Factorial Formula for Combinations
n Pr
n!

.
n Cr 
r ! r !(n  r )!
10.3 – Using Permutations and Combinations
Evaluate each problem.
a) 5P3
b) 5C3
c) 6P6
d) 6C6
543
60
10
720
1
10.3 – Using Permutations and Combinations
How many ways can you select two letters followed by three
digits for an ID if repeats are not allowed?
Two parts:
1. Determine the set of two letters. 2. Determine the set of three digits.
26P2
10P3
2625
1098
650
720
650720
468,000
10.3 – Using Permutations and Combinations
A common form of poker involves hands (sets) of five cards each, dealt
from a deck consisting of 52 different cards. How many different 5-card
hands are possible?
Hint: Repetitions are not allowed and order is not important.
52C5
2,598,960
5-card hands
10.3 – Using Permutations and Combinations
Find the number of different
subsets of size 3 in the set:
{m, a, t, h, r, o, c, k, s}.
9C3
Find the number of arrangements
of size 3 in the set:
{m, a, t, h, r, o, c, k, s}.
9P3
987
504
84
Different subsets
arrangements
10.3 – Using Permutations and Combinations
Guidelines on Which Method to Use
11.1 – Probability – Basic Concepts
Probability
The study of the occurrence of random events or phenomena.
It does not deal with guarantees, but with the likelihood of an
occurrence of an event.
Experiment:
- Any observation or measurement of a random phenomenon.
Outcomes:
- The possible results of an experiment.
Sample Space:
- The set of all possible outcomes of an experiment.
Event:
- A particular collection of possible outcomes from a sample space.
11.1 – Probability – Basic Concepts
Example:
If a single fair coin is tossed, what is the probability that it will land
heads up?
Sample Space: S = {h, t}
Event of Interest: E = {h}
P(heads) = P(E) =
1/
2
The probability obtained is theoretical as no coin was actually
flipped
Theoretical Probability:
P(E) =
number of favorable outcomes
total number of outcomes
=
n(E)
n(S)
11.1 – Probability – Basic Concepts
Example:
A cup is flipped 100 times. It lands on its side 84 times, on its
bottom 6 times, and on its top 10 times. What is the probability
that it lands on it top?
P(top) =
number of top outcomes
total number of flips
=
10
100
=
1
10
The probability obtained is experimental or empirical as the cup
was actually flipped.
Empirical or Experimental Probability:
P(E)
͌
number of times event E occurs
number of times the experiment was performed
11.1 – Probability – Basic Concepts
Example:
There are 2,598,960 possible five-card hand in poker. If there are
36 possible ways for a straight flush to occur, what is the
probability of being dealt a straight flush?
P(straight flush) =
=
number of possible straight flushes
total number of five-card hands
36
2,598,960
=
0.0000139
This probability is theoretical as no cards were dealt.
11.1 – Probability – Basic Concepts
Example:
A school has 820 male students and 835 female students. If a
student is selected at random, what is the probability that the
student would be a female?
P(female) =
=
P(female) =
number of possible female students
total number of students
835
820 + 835
=
835
1655
=
167
331
0.505
This probability is theoretical as no experiment was performed.
11.1 – Probability – Basic Concepts
The Law of Large Numbers
As an experiment is repeated many times over, the
experimental probability of the events will tend closer and
closer to the theoretical probability of the events.
Flipping a coin
Spinner
Rolling a die
11.1 – Probability – Basic Concepts
Odds
A comparison of the number of favorable outcomes to the
number of unfavorable outcomes.
Odds are used mainly in horse racing, dog racing, lotteries and
other gambling games/events.
Odds in Favor: number of favorable outcomes (A) to the number
of unfavorable outcomes (B).
A to B
A: B
Example:
What are the odds in favor of rolling a 2 on a fair six-sided die?
1:5
What is the probability of rolling a 2 on a fair six-sided die?
1/
6
11.1 – Probability – Basic Concepts
Odds
Odds against: number of unfavorable outcomes (B) to the number
of favorable outcomes (A).
B to A
B:A
Example:
What are the odds against rolling a 2 on a fair six-sided die?
5:1
What is the probability against rolling a 2 on a fair six-sided die?
5/
6
11.1 – Probability – Basic Concepts
Odds
Example:
Two hundred tickets were sold for a drawing to win a new
television. If you purchased 10 tickets, what are the odds in favor
of you winning the television?
10 Favorable outcomes
200 – 10 = 190 Unfavorable outcomes
10 : 190 = 1 : 19
What is the probability of winning the television?
10/
1/
=
= 0.05
200
20
11.1 – Probability – Basic Concepts
Converting Probability to Odds
Example:
The probability of rain today is 0.43. What are the odds of rain
today?
P(rain) = 0.43
Of the 100 total outcomes, 43 are favorable for rain.
Unfavorable outcomes: 100 – 43 = 57
The odds for rain today:
43 : 57
The odds against rain today:
57 : 43
11.1 – Probability – Basic Concepts
Converting Odds to Probability
Example:
The odds of completing a college English course are 16 to 9. What
is the probability that a student will complete the course?
The odds for completing the course: 16 : 9
Favorable outcomes + unfavorable outcomes = total outcomes
16 + 9 = 25
P(completing the course) =
= 0.64