5.

Elementary probability
5.1 Introduction
 Probability (p): Is a numerical description of chance occurrence of a given phenomena
under a certain condition. It is used to measure the degree of certainty.
5.2 Definition of some probability terms
Random (Probability) Experiment: It is an experiment that can be repeated any number of times
under similar conditions and it is possible to enumerate the total number of outcomes without
predicting an individual out come.
 Example 1: If a fair die is rolled once it is possible to list all the possible outcomes i.e.
1, 2, 3, 4, 5, 6 but it is not possible to predict which outcome will occur.
 Example 2: Tossing a coin two times and observing the no of heads appearing on the top.
An outcome: is the result of a single trial of a random experiment.
 Example: when a coin is tossed once, there are two possible outcomes i.e. head(H) &tail (T).
Sample space (S): is a set of all possible outcomes of a random experiment.
 Example 1: Rolling a die: { }
 Example 2: Tossing a coin once: { }.
 Example 3: Tossing a coin twice: { }.

Event : Is a subset of sample space. It is a statement of one or more outcomes of a random

experiment. They are denoted by capital letters.
 Example: Getting an odd numbers in rolling a die.
Solution: Let is an event of getting odd numbers. Then { }
Complement of an event: The complement of an event A means non- occurrence of A and
is denoted by which contains those points of the sample space which don’t belong to A.
Example: a) Find the complement of an event of getting odd numbers in rolling a die.
b) If we toss a coin two times and getting all heads.
Solution: a) Let is an event of getting odd numbers in rolling a die.
{ } { }
b) Let be an event of getting all heads in tossing a coin two times.
{ } { }
Mutually exclusive (disjoint) events: Two events which cannot happen at the same time.
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 Equally likely events: Events that have the same probability of occurring.
 Example: when a single die is rolled, each outcome has the same probability of
occurrence 1/6.
5.3 Counting Rules (Techniques)
 In order to calculate probabilities, we have to know:
 The number of elements in the event.
 The number of elements in the sample space.
 That is, in order to judge what is probable, we have to know what is possible.
 In order to determine the number of outcomes, one can use several rules of counting.
 The addition rule
 The multiplication rule
 The permutation rule
 The combination rule
 To list the outcomes of the sequence of events, a useful device called tree diagram is used.

1. Addition Principle (Rule)

If a task can be accomplished by "k distinct" procedures where the ith procedures has
alternatives , then the total number of ways of accomplishing the task is:

Example 1: A student goes to the nearest snack (Cafe) to have a breakfast. He can take tea,
coffee, or milk with bread, cake or sandwich. How many possibilities does he have?
Solution: Bread Bread Bread
Tea Cake or Coffee Cake or Milk Cake
Sandwich Sandwich Sandwich

Example 2: There are two transportation means from city A to city B, either using bus
transportation or train transportation. There are 3 buses and 2 trains. How many ways of
transportation is there from city A to city B?
Solution: A person can take any of 5 means of transportation from city A to B.

Example 3: Suppose one wants to purchase a certain commodity and this commodity is on
sale in 5 government owned shops, 3 public shops and 4 private shops. How many
alternatives are there for the person to purchase this commodity?
Solution:

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 2. Multiplication Principle (Rule)
If a choice consists of k steps of which the first can be made in ways, and the second can be
th
made in and the k can be made in ways, then the whole choice can be made in:

Note: In this case and means to multiply.

Example1: A paint manufacturer wishes to manufacture several different paints. The
categories include three types of colors (i.e. red, white, blue), two types of type (i.e. latex and
oil) and two types of use (i.e. outdoor & indoor). How many different kinds of paints can be
made if a person can select one color, one type and one use?
Latex Outdour
Solution: Indour
Oil Outdour
Red Indour
Latex Outdour
Indour
Person White Oil Outdour 12 different ways.
Indour
Blue Latex Outdour
Indour
Oil Outdour
Indour

Example 2: The digits are to be used in a . How many

different cards are possible if
(a) Repetitions are permitted?
b) Repetitions are not permitted?
Solution: 1st digit 2nd digit 3rd digit 4th digit

a.

b.
Example 3: (a) an urn contains 3 balls whose colors are red (R), black (B) and white (W). A
ball is selected, its color is noted, and it is replaced, then a 2nd ball is selected, and its color is
noted. How many color schemes are possible?
(b) If the 1st ball is not replaced. How many different outcomes are there?

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 R B
a. R B b. R W
W R
Urn R Urn B W 6 ways.
B B 9 color schemes. R
W W B
R
W B
W
a). b).
3. Permutations
 Definition: Permutation is an arrangement of "n distinct" objects in a specific order.
Permutation Rules:
1. The arrangement of n objects in a specified order using r objects at a time is called the
permutation of n objects taken r objects at a time. It is written as

2. The number of permutations of n distinct objects taken all together is or

3. The number of permutations of n objects in which are alike (the same), are alike, ...
etc, then the total number of arrangements is

Example 1: In how many ways can the letters be arranged taken two at a time.

Example 2: a) In how many ways can 3 students be arranged in rows of 3 chairs?

b) In how many ways can a student arrange his/her 4 different books on a shelf?
Solution: a)
b)
Example 3: If 2 different mathematics books, 3 different statistics books and 2 different Chemistry
books are to be arranged in a shelf, then how many different arrangements are possible if:
a. The books in each particular subject must "stand all together".
b. Only the Mathematics books must stand all together.
c. There is no restriction.
Solution: a)
;

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 b)
;
c) ;
Example 4: How many different permutations of n objects can be made from the letters in
the word MISSISSIPPI.
Solution:

Ex: If the permutation of the word WHITE is selected at random, how many of the permutations
i. Begins with a consonant?
ii. Ends with a vowel?
iii. Has a consonant and vowels alternating?
4. COMBINATIONS
 Combination is a selection of n distinct objects without regard to order.
 It is used when the order of arrangement is not important, as in the selection process.
 The number of combinations of r objects selected from n objects is denoted by ( )

Example 1: Given the letters List the number of permutations & combinations for
selecting two letters.
Solution:


 Note that in permutation AB is different from BA. But in combination AB is the same as BA.
Example 2: Out of male students and female students in Statistics department, a
consists of students and students is to be formed. In how
many ways can this be done if:
a) any male students and any female students can be included (all students are eligible).
b) One particular female must be a member.
c) Two particular male students cannot be member for some reasons.
Solution: a) b) c)

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 Example 3: A committee of 5 people must be selected from 5 men and 8 women. In how
many ways can selection be done if there are at least 3 women on the committee?
Solution: The committee may consists of 3 women and 2 men or 4 women and 1 men
or 5 women and 0 men.

5.4 Basic approaches to probability

 There are four different conceptual approaches to the study of probability theory. These are:
1. The Classical Approach.
2. The Frequents (Empirical) Approach.
3. The Axiomatic Approach.
4. The Subjective Approach.
1. The Classical Approach
 This approach is used when:
 All outcomes are equally likely.
 Total number of outcome is finite, say n.
Definition: If a random experiment with "n" equally likely outcomes is conducted and out
of these "k" outcomes are favorable to an event A, then the probability that an event A occur
denoted is defined as:

Example 1: When a single die is rolled, then what is the probability of getting an odd numbers?
Solution: let be an event that getting an odd numbers in rolling a die. Then
{ } { }

Example 2: A box of 80 candles consists of 30 defective and 50 non defective candles. If 10

of these candles are selected at random without replacement, what is the probability that:
a. All will be defective?
b. 6 will be non defective?
c. All will be non defective?
Solution: ( )

a). Let be the event that all will be defective.

 ( )( )

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 ( ) ( )
( )
b). Let be the event that 6 will be non defective.
 ( )( )

( ) ( )
( )
c). Let be the event that all will be non defective.
 ( )( )

( ) ( )
( )

Exercise: If 3 books are picked at random from a shelf containing 5 novels, 3 books of
poems, and a dictionary, then what is the probability that
a) The dictionary is selected?
b) 2 novels and 1 book of poems are selected?
2. The Frequents Approach (Empirical Probability):

 This is based on the relative frequencies of outcomes belonging to an event.

Definition: The probability of an event A is the proportion of outcomes favorable to A in
the long run when the experiment is repeated under the same condition.

In a given frequency distribution, the probability of an event A being in a given class is:

Example: If records show that 60 out of 100,000 bulbs produced are defective. What is the
probability of a newly produced bulb to be defective?
Solution: Let A - be the event that the newly produced bulb is defective.

Example 2: In a sample of 50 people, 22 had type "A", 5 had type "B", 2 had type "AB" and
21 had type "O" blood. Find the probability that a person has blood type "O"?
Solution: Let be the event that a person has blood type "O". Then

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 3. Axiomatic Approach
Let E be a random experiment and S be a sample space associated with E. With each event A
areal number called the satisfies the following properties called axioms of probability.
1. 2.
3. . 4.
5.
Remark: The Venn-diagrams can be used to solve probability problems.

S A' S
S
A
A B A
B

AUBA

A B A B

5.5 The addition Rules for Probability

1. If two events A and B are mutually exclusive, then the probability that A or B will occur is

2. If two events A and B are not mutually exclusive, then the probability that A or B will occur is
.
Example 1: If a single card is drawn from an ordinary deck and its number is noted,
then find the probability that:
a) It is an ace or a diamond. b) It is an ace or a black. c) It is an ace or a Jack.
Solution: Let ; be the event that an ace will be selected.
be the event that a diamond will be selected.
be the event that a black will be selected.
be the event that a jack will be selected.
a)

b)

c)

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 5.6. The Multiplication Rules (Theorems)
The multiplication rules can be used to find the probability of two or more events that occur in
sequence.
Independent events: Two events A and B are independent if the occurrence of "A" does not
affect the probability of "B" occurring.
Dependent events: Two events are dependent if the first event affects the outcome or
occurrence of the second event in a way the probability is changed.
5.6.1 The Multiplication Rules for Probability
1. If two events A and B are independent, then the probability of both A and B will occur is

2. If two events A and B are dependent, then the probability of both A and B will occur is

Example 1: A coin is flipped and a die is rolled. Find the probability of getting a head on the
coin and a 4 on the die.
Solution: These two events are independent since the outcome of the first event (tossing
a coin) does not affect the probability outcome of the second event (rolling a die).

Example 2: An urn contains 3 red balls, 2 blue balls and 5 white balls. A ball is selected and its
color noted. Then it is replaced. A second ball is selected and its color noted. Find the probability of:
a) Selecting two blue balls.
b) Selecting a blue ball and then a white ball.
c) Selecting a red ball and then a blue ball.
Solution: a)

b)

c)

Example 3: A card is drawn from an ordinary deck and its number noted. Then it is not
replaced. A second card is drawn and its number noted, then find the probability of:
a) Getting two Jacks (J). b) Getting an ace ( A) and a king (K) in order.
c) Getting a flower and a spade. d) Getting a red and a black in order.
Solution: a)

b)

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 c) .

d)
Exercise : If the probabilities are 0.75, 0.7 and 0.525 that a student A, B, or both can solve the
problems in a text book respectively. What is the probability that:
a) Student A can only solve the problem selected at random from the book?
b) Student B can only solve the problem selected at random from the book?
Solution:

a) A b) A
B B

a)

5.7. Conditional Probability

The conditional probability of an event is a probability obtained with the additional
information that some other event has already occurred.
 The conditional probability of event B occurring, given that event A has already
occurred, can be found by:

 The conditional probability of event A occurring, given that event B has already
occurred, can be found by:


Example: A box contains black chips and white chips. A person selects two chips without
replacement. If the probability of selecting a black chip and a white chip is and the
probability of selecting a black chip on the first draw is , find the probability of selecting the
white chip on the second draw, given that the first chip selected was a black chip.

Exercise: Let A and B are two events such that

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 5.8. Bayes' Theorem and The Total Probability Theorem
The law of total probability: Suppose are disjoint events such that
The probability of an arbitrary event A can be expressed as:

The following Figure illustrates the law for m = 5. The event A is the disjoint union of
so and for each i
the multiplication rule states .

𝐵 𝐵
b1 A vBd
𝐴 𝐵 𝐴 𝐵

𝐴 𝐵 𝐴B3 𝐵
𝐵

𝐴 𝐵
S=B
𝐵 𝐵 S
𝐴𝐴 𝐵 𝐵

Bayes' Rule: Suppose the events are disjoint and The

conditional probability of given an arbitrary event A, can be expressed as:
( )

Example 1: Box 1 contains 2 red balls and one blue ball. Box 2 contains 3 blue balls and one red
ball. A coin is tossed. If it falls heads up, Box 1 is selected and a ball is drawn. If it falls tails up,
Box 2 is selected and a ball is drawn. Then find the probability of selecting a red ball.
Solution: Let; be the event that box 1 is selected.
be the event that box 2 is selected.
be the event that a red ball is selected.
be the event that a blue ball is selected

B
R

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 R is selected, if and only if, is selected and R is selected or is selected and R is selected.

{ }

Example 2: A shipment of two boxes, each containing 6 telephones, is received by a store. Box1
contains one defective phone and box 2 contains 2 defective phones. After the boxes are unpacked, a
phone is selected and found to be defective. Then find the probability that it came from box 2.
Solution: Let be the event that box 1 is selected.
be the event that box 2 is selected.
be the event that defective phone is selected.
be the event that non defective phone is selected.

ND
D

ND

{ }

Since, is selected, iff, is selected and is selected or is selected and is selected.


{ }

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