Probability

PROBABILITY
COUNTING
TECHNIQUES
Counting Technique
In the previous lesson, we learned that
the classical approach to assigning
probability to an event involves
determining the number of elements
in the event and the sample space.
There are many situations in which it
would be too difficult and/or too
tedious to list all of the possible
outcomes in a sample space.
2
Counting Technique
In this lesson, we will learn various ways
of counting the number of elements in
a sample space without actually having
to identify the specific outcomes. The
specific counting techniques we will
explore include the multiplication rule,
permutations and combinations.
3
MULTIPLICATION RULE
If there are:
n1 outcomes of a random experiment E1
n2 outcomes of a random experiment E2... and
...nm outcomes of a random experiment Em
then there are n1 × n2 × … × nm outcomes of the
composite experiment
4
MULTIPLICATION RULE
The hardest part of using the Multiplication Principle is

determining, ni, the number of possible outcomes for
each random experiment (procedure) performed. You'll
want to pay particular attention to:
✗ whether replication is permitted
✗ whether other restrictions exist
5
MULTIPLICATION RULE
Example 1:
A menu offers a choice of 3 salads, 8 main dishes, and 5
desserts. How many different meals consisting of one
salad, one main dish, and one dessert are possible?
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MULTIPLICATION RULE
Example 1
A menu offers a choice of 3 salads, 8 main dishes, and 5 desserts.
How many different meals consisting of one salad, one main dish,
and one dessert are possible?
Solution
There are three tasks, picking a salad, a main dish, and a dessert. The salad task
can be done 3 ways, the main dish task can be done 8 ways, and the dessert
task can be done 5 ways. The ways to pick a salad, main dish, and dessert are
3 salad x 8 main x 5 dessert =120 different meals
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MULTIPLICATION RULE
Example: 2
How many three letter “words” can be made from the
letters a, b, and c with no letters repeating? A “word” is
just an ordered group of letters. It doesn’t have to be a real
word in a dictionary.
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MULTIPLICATION RULE
Example 2:
How many three
letter “words” can
be made from the
letters a, b, and c
with no letters
repeating? A “word”
is just an ordered
group of letters. It
doesn’t have to be a
real word in a
dictionary.
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MULTIPLICATION RULE
Solution
There are three tasks that must be done in this case. The tasks are to pick the
first letter, then the second letter, and then the third letter. The first task can
be done 3 ways since there are 3 letters. The second task can be done 2 ways,
since the first task took one of the letters. The third task can be done 1 ways,
since the first and second task took two of the letters. There are
3 first letter x 2 second letter x 1 third letter = 6 different words
10
In Example2, the solution was found
by find 3 ∗ 2 ∗ 1 = 6. Many counting
problems involve multiplying a list
of decreasing numbers. This is
called a factorial. There is a
special symbol for this and a
special button on your calculator.
11
FACTORIAL
𝑛!=𝑛(𝑛−1)(𝑛−2)⋯(3)(2)(1)
As an example:
5! = 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 = 120
8! = 8 ∗ 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 = 40320
0 factorial is defined to be 0!=1 and 1 factorial is defined to be 1!=1.
12
FACTORIAL
𝑛!=𝑛(𝑛−1)(𝑛−2)⋯(3)(2)(1)
Give the factorial of the following:

11! = 39,916,800
13! = 6,227,020,800
13
Sometimes you are trying to select r objects
from n total objects. The number of ways to do
this depends on if the order you choose
the r objects matters or if it doesn’t. As an
example if you are trying to call a person on
the phone, you have to have their number in the
right order. Otherwise, you call someone you
didn’t mean to. In this case, the order of the
numbers matters.
14
If however you were picking random numbers for
the lottery, it doesn’t matter which number you
pick first. As long as you have the same numbers
that the lottery people pick, you win. In this
case the order doesn’t matter.
15
Permutations
Combinations
16
PERMUTATION
17
PERMUTATION
- is an arrangement of all or part of a set of objects,
with regard to the order of the arrangement. For
example, suppose we have a set of three letters: A,
B, and C. we might ask how many ways we can arrange 2
letters from that set.
- Formula:
n = of the set from which elements are permuted.

r = size of each permutation
n, r are non negative integers
18
PERMUTATION
Example:
A computer scientist is trying to discover the

keyword for a financial account. If the keyword
consists only of 10 lower case characters (e.g., 10
characters from among the set: a, b, c... w, x, y,
z) and no character can be repeated, how many
different unique arrangements of characters exist?
19
PERMUTATION
Example:
A computer scientist is trying to discover the keyword for a

financial account. If the keyword consists only of 10 lower case
characters (e.g., 10 characters from among the set: a, b, c... w, x,
y, z) and no character can be repeated, how many different unique
arrangements of characters exist?
Solution:
Step 1:
Determine whether the question pertains to permutations or
combinations. Since changing the order of the potential keywords
would create a new possibility, this is a permutations problem.
20
PERMUTATION
Example:
A computer scientist is trying to discover the keyword for a

financial account. If the keyword consists only of 10 lower case
characters (e.g., 10 characters from among the set: a, b, c... w, x,
y, z) and no character can be repeated, how many different unique
arrangements of characters exist?
Solution:
Step 2:
Determine n and r
n = 26 since the computer scientist is choosing from 26 possibilities (e.g.,
a, b, c... x, y, z).
r = 10 since the computer scientist is choosing 10 characters.
21
PERMUTATION
Example: Solution:
A computer scientist is Apply the formula:
trying to discover the
keyword for a financial
account. If the keyword
consists only of 10 lower
case characters (e.g., 10
characters from among the
set: a, b, c... w, x, y,
z) and no character can be
repeated, how many
different unique
arrangements of characters
exist?
22
COMBINATION
- is a selection of all or part of a set of objects,
without regard to the order in which objects are
selected. For example, suppose we have a set of three
letters: A, B, and C. we might ask how many ways we
can select 2 letters from that set.
- Formula:
n = the number of objects to choose from.

r = the number of objects selected
23
COMBINATION
Example:
How many different groups of 10 students

can a teacher select from her classroom
of 15 students?
24
COMBINATION
Example:
How many different groups of 10 students can a teacher select from

her classroom of 15 students?
Solution:
Step 1:
Determine whether the question pertains to permutations or
combinations. Since changing the order of the selected students
would not create a new group, this is a combinations problem.
25
COMBINATION
Example:
How many different groups of 10 students can a teacher select from

her classroom of 15 students?
Solution:
Step 1:
Determine n and r
n = 15 since the teacher is choosing from 15 students.
r = 10 since the teacher is selecting 10 students.
26
COMBINATION
Example:
How many different groups

of 10 students can a
teacher select from her
classroom of 15 students?
Solution:
Apply the formula:
27
https://www.tutorialspoint.com/statistics/permutation.htm
https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-
combinations/permutation-lib/v/permutation-formula?modal=1
https://statisticsbyjim.com/probability/permutations-probabilities/
https://byjus.com/maths/permutation-and-
combination/#:~:text=A%20permutation%20is%20an%20act,the%20objects%20does%20not%2
0matter.
https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_Statistics_Using_Techn
ology_(Kozak)/04%3A_Probability/4.04%3A_Counting_Techniques
https://online.stat.psu.edu/stat414/lesson/3/3.2
28

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The hardest part of using the Multiplication Principle is

0 factorial is defined to be 0!=1 and 1 factorial is defined to be 1!=1.

Give the factorial of the following:

n = of the set from which elements are permuted.

A computer scientist is trying to discover the

A computer scientist is trying to discover the keyword for a

A computer scientist is trying to discover the keyword for a

n = the number of objects to choose from.

How many different groups of 10 students

How many different groups of 10 students can a teacher select from

How many different groups of 10 students can a teacher select from

n = 15 since the teacher is choosing from 15 students.

r = 10 since the teacher is selecting 10 students.

How many different groups

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