COUNTING
TECHNIQUES and
PROBABILITY
Chapters 6 and 7
WHAT IF…? WHAT WILL YOU DO?
• Tonight - Partly cloudy with isolated
thunderstorms possible. Low 77 °F (25.0 °C).
Chance of rain 60%.
• Breast cancer, the most common tumor in
women, presents a high survival percentage:
83% of patients have survived this type of
cancer after five years.
WHAT YOU SHOULD LEARN
• How to use the Fundamental Counting Principle to
find the number of ways two or more events can
occur
• How to find the number of ways a group of objects
can be arranged in order
• How to find the number of ways to choose several
objects without regard to order
• How to identify the sample space of a probability
experiment and how to identify sample events
• EXAMPLE
Suppose you have a
choice of 5 pizza toppings
and 2 kinds of crust. How
many choices do you
have?
Fundamental Counting Principle
• If one event can occur
in m ways and a
second event can
occur in n ways, the
number of ways the
two events can occur
in sequence is mn.
• In other words, the
number of ways that
events can occur in
sequence is found by
multiplying the number
of ways one event can
occur by the number of
ways the other event(s)
can occur.
• EXAMPLE
You are purchasing a
new car. The possible
manufacturers, car sizes
and colors are listed.
MANUFACTURER:
Ford, KIA, Honda
CAR SIZE:
Compact, Midsize
COLOR:
White, Red, Black, Gray
• SOLUTION
3·2·4 = 24 ways
• EXAMPLE
You are purchasing a
new car. The possible
manufacturers, car sizes
and colors are listed.
MANUFACTURER:
Ford, KIA, Honda, Toyota
CAR SIZE:
Compact, Midsize
COLOR:
White, Red, Black, Gray, Tan
• SOLUTION
4·2·5 = 40 ways
Item number 11 on page 136
• A building has 5 entrance gates and 4 exit gates.
How many different ways can a person enter
and exit from the building?
»20 ways
Item number 3 on page 135
• Using the numbers 0, 2, 3, 4 and 6, how many 3digit numbers can be formed if:
• Repetition is not allowed?
»60 three-digit numbers
• Repetition is allowed?
»125 three-digit numbers
Permutation
• It is an ordered arrangement of n different
objects.
𝑛! = 𝑛 𝑛 − 1 𝑛 − 2 𝑛 − 3 … (3)(2)(1)
n!
nPr = ------------------ , where r ≤ n
(n – r)!
Factorial
• The number of different permutations of n distinct
objects is n! read as n factorial.
• The value of n! is obtained by multiplying ALL the
integers from 1 to n.
𝑛! = 𝑛 𝑛 − 1 𝑛 − 2 𝑛 − 3 … (3)(2)(1)
Item number 8 on page 136
• For 5 finalists in a beauty contest, how many
ways can they be ranked from the 1st place to
the 5th place?
»120 ways
Item number 6 letter a on page 137
• In how many ways can 4 men and 4 women be
seated in a row of 8 chairs if they can be seated
anywhere?
»40 320 ways
Example
• The women’s hockey teams for 2010 Olympics
are Canada, Sweden, Switzerland, Slovakia,
United States, Finland, Russia and China. How
many different final standings are possible?
»40 320 ways
Permutation of n objects taken r at a time
• The number of permutations of n distinct objects taken
r at a time is
n!
nPr = ------------------ , where r ≤ n
(n – r)!
Item number 10 on page 138
• Using the letters of the English alphabet, how
many 5-letter code words can be formed from
the:
• First 10 letters
»30 240 ways
• Consonants
»2 441 880 ways
• Vowels
»120 ways
Permutation of n with Alike Objects
(Distinguishable Permutation)
• The number of distinguishable permutations of n
objects, where n1 are of one type, n2 are of another
type, and so on, is
n!
------------------ , where n1 + n2 + … + nk = n
n1!n2!...nk!
Item number 8 letter c on page 137
• Find the number of permutations in the word
STATISTICS.
»50 400
Combinations
• A combination is a selection of r objects from a group
of n objects without regard to order. The number of
combinations of r objects selected from a group of n
objects is
n!
nCr = -----------------(n – r)! r!
Item number 2 on page 138
• In how many ways can a committee of 5
students be organized from a group of 12
students?
»792 ways
Item number 2 on page 138
• Find the number of combinations of 3 face cards
from the deck of 52 playing cards?
»220 combinations
Probability is a mathematical concept that is
used to measure the certainty or
uncertainty of occurrence of a statistical
phenomena.
Definition of Terms
• A probability experiment is an action, or trial,
through which specific results (counts,
measurements or responses) are obtained.
• A sample space is the set of ALL possible
outcomes of a probability experiment.
• An event is a subset of the sample space. It
may consist one or more outcomes.
• An outcome is the result of a single trial in a
probability experiment.
• EXAMPLE of the use of
the terms: probability
experiment, sample
space, event and
outcome
• Probability Experiment
Roll a six-sided die
• Sample Space
{1, 2, 3, 4, 5, 6}
• Event
Roll an even number
• Outcome
{2}
• SURVEY
Does your favorite team’s • EXAMPLE
win or loss affect your mood? A probability experiment
Check one response:
___ YES
___ NO
___ NOT SURE
consists of recording a
response to the survey
statement and the gender
of the respondent. What
are the elements of the
sample space?
Other Concepts of Probability
• Probabilities can be written as a fraction, decimal or
percent.
• The probability that event E will occur is written as
P(E) and read as “the probability of event E.”
• There are three types of probability: CLASSICAL,
EMPIRICAL and SUBJECTIVE.
Classical Probability
• It is used when each outcome in a sample is equally
likely to occur. The classical probability for an event E is
given by
number of outcomes in E
P(E) = -------------------------------Total number of outcomes in
the sample space
Example
• You roll a six-sided die. Find the probability of each
event.
1. Event A: rolling a 3
2. Event B: rolling a 7
3. Event C: rolling a number less than 5
Exercises
• A survey of the students in a Psychology class revealed
that there were 19 females and 8 males. Of the 19
females, only 4 had no brothers or sisters, and 3 of the
males were also the only child in the household. If a
student is randomly selected from this class,
• What is the probability of obtaining a male?
• What is the probability of selecting a student who
has at least one brother or sister?
• What is the probability of selecting a female who
has no siblings?
Empirical Probability
• It is based on observations obtained from probability
experiments. The empirical probability of an event E is
the relative frequency of event E.
Frequency of event E
P(E) = -----------------------------Total frequency
Example
• A company is conducting a telephone survey of randomly
selected individuals to get their overall impressions of the
past decade (2000s). So far, 1504 people have been
surveyed. The frequency distribution shows the results.
What is the probability that the next person surveyed has a
positive overall impression of the 2000s?
Response
Number of times, f
Positive
406
Negative
752
Neither
316
Don’t know
30