Stat 276-Introduction to Probability and Statistics
Lecture note
2. Introduction to Probability (2 lecture hours)
2.1. Deterministic and non-deterministic models
2.2. Review of set theory: sets, union, intersection, complementation, De Morgan’s rules
2.3. Random experiments, sample space and events
2.4. Finite sample spaces and equally likely outcomes
2.5. Counting techniques
2.6. Axioms of probability
2.7. Derived theorems of probability
2.1. Deterministic and non-deterministic models
Mathematical models-One of the aims of science is to predict and describe events in the
world in which we live. One way in which this is done is to construct mathematical
models which adequately describe the real world. For example, the equation
expresses a certain relationship between the symbols and It is a mathematical
model. To use the equation to predict , the distance a body falls, as a function
of time , the gravitational constant must be known.
Deterministic model-a model in which specify that the conditions under which an
experiment is performed determine the outcome of the experiment.
Non-deterministic model (probability model) or stochastic model-used for random
experiments. Here chance plays an important role in the outcome of an experiment.
2.2. Review of set theory
A set is a well-defined collection of objects. Sets are usually denoted by capital letters A,
B etc. Examples
1. A= * + describes the set consisting of positive integers 1, 2, 3 and 4.
2. A=* + A consisting of all real numbers between 0 and 1 inclusive or
the set of all X’s where X is a real number between 0 and 1, inclusive.
The individual objects making up the collection are called members or elements of A.
Examples: 1 A 6 A
U= Universal set: the set of all objects under considerations.
Empty set or Null set: - the set containing no members. i.e * + or impossible event.
eg: A: is the set of all real numbers X satisfying the equation . A=* +
Reading assignments: - Union, intersection, De Morgan’s rules.
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2.3. Random experiments, Sample space and events
1. Experiment: Any process of observation or measurement or any process which
generates well defined outcome.
2. Probability Experiment (Random 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 with out predicting an individual out
come.
Example: If a fair coin is tossed three times, it is possible to enumerate all possible
eight sequences of head (H) and tail (T). But it is not possible to predict which sequence
will occur at any occasion.
3. Outcome: The result of a single trial of a random experiment
4. Sample Space(S): Set of all possible outcomes of a probability experiment.
Example: Sample space of a trial conducted by three tossing of a coin is
S= {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Sample space can be
Countable (finite or infinite)
Uncountable
5. Event (Sample Point): It is a subset of sample space. It is a statement about one or
more outcomes of a random experiment. It is denoted by capital letter A, B, C - - -.
For example, in the event, that there are exactly two heads in three tossing of a coin, it
would consist of three points HTH, HHT and THH.
Remark: If S (sample space) has n members with two possible outcomes in each trial
then there are exactly 2n subsets or events.
6. Equally Likely Events: Events which have the same chance of occurring.
7. Complement of an Event: the complement of an event A means non- occurrence of
A and is denoted by A' or Ac or A , contains those points of the sample space which
don’t belong to A.
8. Elementary (simple) Event: an event having only a single element or sample point.
9. Mutually Exclusive (Disjoint) Events: Two events which cannot happen at the
same time.
10. Independent Events: Two events are said to be independent if the occurrence of
one does not affect the probability of the other occurring.
11. 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.
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2.4. Counting Techniques
In order to calculate probabilities, we have to know
• The number of elements of an event
• The number of elements of 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
Permutation rule
Combination rule
Addition Rule
If event A can occur in m possible ways and event B can occur in n possible ways, there
are m + n possible ways for either event A or event B to occur, but only if there are no
events in common between them.
I.e. n (A or B) =n (A) +n (B)-n (A B)
To list the outcomes of the sequence of events, a useful device called tree diagram is used.
Example: A student goes to the nearest snack to have a breakfast. He can take tea, coffee,
or milk with bread, cake and sandwich. How many possibilities does he have?
Solutions:
cake cake
cake
Tea sandwich Coffee sandwich Milk sandwich
bread bread bread
There are nine Possibility
The Multiplication Rule:
If a choice consists of k steps of which the first can be made in n1 ways, the second can
be made in n2 ways…, the kth can be made in nk ways, then the whole choice can be
made in ( n1 n2 ,...,nk ) ways.
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Example 5.1: 1) A student has two shoes, three trousers and three jackets. In how many
can be dressed?
2) The digits 0, 1, 2, 3, and 4 are to be used in 4 digit identification card. How many
different cards are possible if
a) Repetitions are permitted.
b) Repetitions are not permitted.
Permutation
An arrangement of n objects in a specified order is called permutation of the objects.
Permutation Rules:
1. The number of permutations of n distinct objects taken all together is n!
Where n! =n*(n-1)*(n-2)*,…,*2*1.
2. 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 n P r and the
n!
formula is n Pr =
n r !
1. The number of permutations of n objects in which k1 are alike, k2 are alike ---- etc
is
n!
nPr =
k1 k 2 ....k n
2. The arrangement of n objects around a line is (n-1)! ways.
3. The number of ways of partitioning a set of n things in to k cells where there are
n1 elements in the first cell, n2 elements in the second cell,…,nk elements in the kth
cell is
n!
n 1× n 2× . . .× nk
where ∑ n i= n
Example 5.2: 1. Suppose we have a letters A, B, C, D
a) How many permutations are there taking all the four?
b) How many permutations are there two letters at a time?
2. How many different permutations can be made from the letters in the word
“MISSISSIPPI”?
3. In how many ways can people assigned 1 triple and 2 double room?
4. In how many ways can a party of 7 people arrange themselves?
a) In a row of 7 chairs?
b) Around a circular table?
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Combination
A selection of objects with out regard to order is called combination.
Example: Given the letters A, B, C, and D list the permutation and combination for
selecting two letters.
Solutions:
AB BA CA DA Permutation: AB BC
AC BC CB DB AC BD
AD BD CD DC Combination: AD DC
Note that in permutation AB is different from BA. But in combination AB is the same as
BA.
Combination Rule:
The number of combinations of r objects selected from n objects is denoted by nC r or
nr and is given by the formula:
n!
nCr =
n r !r!
Example 5.3:
1. In how many ways can a committee of 5 people be chosen out of 9 people?
2. Out of 5 Mathematician and 7 Statistician a committee consisting of 2
Mathematician and 3 Statistician is to be formed. In how many ways this can be
done if
i. There is no restriction
ii. One particular Statistician should be included
iii. Two particular Mathematicians cannot be included on the committee.
3. A committee of 5 people must be selected out 5men and 8 women. In how many
ways can be selection made if there are three women on the committee?
Definitions of probability
Probability is the chance of an outcome of an experiment. It is the measure of how
likely an outcome is to occur.
In any random experiment there is always uncertainty as to whether a particular
event will or will not occur. As a measure of the chance, or probability, with which we
can expect the event to occur, it is convenient to assign a number between 0 and 1. If
we are sure or certain that the event will occur, we say that its probability is 100% or 1,
but if we are sure that the event will not occur, we say that its probability is zero.
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1.6 Approaches to measuring Probability
There are four different conceptual approaches to study probability theory. These are:
The classical approach.
The frequencies approach.
The axiomatic approach.
The subjective approach.
1. CLASSICAL APPROACH :
If an event can occur in h different ways out of a total number of n possible ways, all of
which are equally likely, then the probability of the event is h/n.
EXAMPLE :- 1. Suppose we want to know the probability that a head will turn up in a
single toss of a coin. Since there are two equally likely ways in which the coin can come
up—namely, heads and tails (assuming it does not roll away or stand on its edge)—and
of these two ways a head can arise in only one way, we reason that the required
probability is 1/2. In arriving at this, we assume that the coin is fair, i.e., not loaded in
any way.
Example 2.A fair die is tossed once. What is the probability of getting
a) Number 4?
b) An odd number?
c) Number greater than 4?
d) Either 1 or 2 or …. Or 6
3. A box of 80 candles consists of 30 defective and 50 non defective candles. If 10 of
these candles are selected at random, what is the probability?
a) All will be defective.
b) 6 will be non defective
c) All will be non defective
2. Frequency approach :
If after n repetitions of an experiment, where n is very large, an event is observed to
occur in h of these, then the probability of the event is h/n. This is also called the
empirical probability of the event.
EXAMPLE 1. If we toss a coin 1000 times and find that it comes up heads 532 times, we
estimate the probability of a head coming up to be 532/1000=0.532.
Example 2. 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?
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Both the classical and frequency approaches have serious drawbacks, the first
because the words “equally likely” are vague and the second because the “large
number” involved is vague. Because of these difficulties, mathematicians have been
led to an axiomatic approach to probability.
3. Axiomatic Approach ( Basic notion of Probability):
Let “E” be a random experiment and S be a sample space associated with “E”. With
each event A, we associate a real number designated by P (A) and called the probability
of A satisfies the following properties:
1. 0 P A 1
2. P(S) =1
3. If A and B are mutually exclusive events, the probability that one or the other occur
equals the sum of the two probabilities. i. e. P (AuB) =P (A) +P (B)
4. If are mutually exclusive events, then
(⋃ ) ( ) ( ) ( ) ( )
For any finite n, (⋃ ) ∑ ( )
4. Subjective Approach:
It is always based on some prior body of knowledge. Hence subjective measures of
uncertainty are always conditional on this prior knowledge. The subjective
approach accepts unreservedly that different people (even experts) may have vastly
different beliefs about the uncertainty of the same event.
Example: Abebe’s belief about the chances of Ethiopia Buna club winning the FA Cup
this year may be very different from Daniel's. Abebe, using only his knowledge of the
current team and past achievements may rate the chances at 30%. Daniel, on the other
hand, may rate the chances as 10% based on some inside knowledge he has about key
players having to be sold in the next two months.
2.7. Derived theorems of probability
1) For any event A , P A 0
2) ( )
3) For any event A and B, ( ) ( ) ( ) ( )
4) PA = 1 P(A)
5) If A, B and C are any three events, then
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
6) If , then ( ) ( )
Assignment :- Prove the above theorems.
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