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Course Instructors:
1. Dr. R. Archana Reddy
2. Mr. B. Ravindar
3. Dr. G. Ravi Kiran
Introduction:
In random phenomena, past information no matter how
voluminous, will not allow to formulate a rule to determine
precisely (uniquely) what will happen in the future.
In analyzing and interpreting data that involves an
element of “chance” or “uncertainty”
In general, the expectation is based on one’s present
knowledge and belief about the event in question. Even though,
these statements of expectation are by previous experience,
present knowledge and analytical thinking . We need a
quantitative measure to quantify the expectations.
Hence, probability is a concept which numerically
measures the degree of certainty or uncertainty of
occurrence or nonoccurrence of events.
In short, the branch of mathematics which
studies the influence of “chance” is the
theory of probability.
Basic Terminology
Experiment: An experiment is any physical action or process
that is observed and the result noted.
Ex: Tossing a coin
Firing a missile
getting up in the morning
We have two types of experiments
(i) deterministic or Predictable
(ii) Undeterministic or unpredictable or Random experiment
Def: An experiment is called deterministic if, the results can be
predicted with certainty prior to the performance of the
experiment.
Ex: Throwing a stone upwards where it is known that the stone
will definitely fall to the ground due to the force of gravitation.
Def: An experiment is called a random experiment if, when
repeated under the same conditions, it is such that the outcome
cannot predicted with certainty but all possible outcomes can be
determined prior to the performance of the experiment.
Def: A single performance of an experiment is briefly called a
trail.
Def: An event is a result of the experiment.
i) In throwing a die, getting 1 (or 2, 3 or … or 6) is an event.
ii) In drawing two cards from a pack of well shuffled cards,
“getting a king” and “getting a queen”
are events.
Sample Space: The collection of all possible outcomes of a
random experiment is called the Sample Space, denoted by S.
The elements of sample space are sample points.
Ex: A die is numbered with 1,2,3,4,5,6 on the faces. When this
die is thrown the sample space is S={1,2,3,4,5,6}
Def: A sample space is called finite sample space, if its sample
points are finite in number.
Def: A sample space is called infinite sample space, if its sample
points are infinite in number.
Def: Every non empty subset of a sample space of a random
experiment is called an event.
Note: As sample Space S is subset of itself and the empty set ϕ
also a subset of S, S and ϕ are also considered as events.
Def: The event S is called the Sure event or Certain event and the
event ϕ is called an impossible event.
Equally likely events: A set of events are said to equally likely, if no
one of them is expected to occur in preference to other in any single
trail of the random experiment.
Ex: (i) In tossing an unbiassed or uniform coin, getting head or tail
are equally likely events.
(ii) In throwing an unbiased die, all the six faces are equally likely
to occur.
Using the spinner below, answer the following questions.
Q1. Are the events spinning and landing
on 1 or 2 equally likely?
Yes. The areas of sections 1 and 2 equal.
Q2. Are the events spinning and landing on 2 or 3 equally likely
No. The areas of sections 2 and 3 are not equal.
Q3. How many times do you predict the spinner will land on each
section after 100 spins
Based on the areas of the sections, approximately 25 times each for
sections 1 and 2, 50 times for sections 3.
Mutually exclusive events: Events of random experiment are said to
be mutually exclusive, if the happening of one event, prevents the
happening of all other events. i.e., if no two or more of them can
happen simultaneously in the same trial.
Ex: When two teams E1 and E2 are playing a game, the events “E1
winning the game” and “E2 winning the game” are mutually
exclusive.
Note: Events E1, E2, …, En are mutually exclusive if and only if
Ei Ej for i j
Ex: When A, B, C, D are appearing for an examination, the event “A
passing in the examination” does not prevent the events B, C or D
passing the examination. Hence these events are not mutually
exclusive
Q: Are the births of a son or a
daughter are mutually exclusive
events?
Yes, They are mutually exclusive
Classical Definition of Probability
Def: If there are n mutually exclusive and equally likely events of
random experiment, out of which ‘s’ are favorable to a particular
event E, then we define the probability of E as, P(E)=(s / n)
= (No. of favorable events to E) / (Total no. of events of the
experiment)
This probability is also known as probability of success of E.
In this experiment ‘s’ results are favorable to E, and hence the
remaining ‘n-s’ results are not favorable to the event E. This set of
unfavorable events denoted by Ec or E
ns s
The probability P( E ) 1 1 P( E )
n n
i.e., P( E ) P( E ) 1
Axiomatic definition of probability
Probability is a number that is assigned to each member of a
collection of events from a random experiment that satisfies
the following properties.
If S is the sample space and E is any event in a random
experiment,
(i) 0 ≤ P(E) ≤ 1 for each event E in S.
(ii) P(S) = 1
(iii) If E1 and E2 are any mutually exclusive events in S, then
P(E1 U E2) = P(E1) + P(E2)
Addition theorem of Probability
Q: For any two events A an B: P(A U B) = P(A) + P(B) – P(A ∩ B)
From the fig., since (A∩B) and (Ac∩B) are disjoint,
Events (A∩B) and (Ac∩B) are mutually exclusive.
P[(A∩B) U (Ac∩B)] = P(A∩B) + P(Ac∩B)
But (A∩B) U (Ac∩B) = B A A∩B B
Threfore, P(B)= P(A∩B) + P(Ac∩B)
i.e., P(Ac∩B) = P(B) - P(A∩B) ------(1)
Further, A and (Ac∩B) are also disjoint sets
Therefore, the events A and (Ac∩B) are mutually exclusive
Therefore, P[A U (Ac∩B)] = P(A) + P(Ac∩B)
i.e., P(A U B) = P(A) + P(B) - P(A∩B)
Conditional Probability
Def: If A and E are any two events of a sample space S, then the
event of “happening of E after the happening of A” is called
conditional event and is denoted by (E/A)
Def: If E and A are any events in S, P(A) > 0, then the conditional
probability of E given A is,
P(E/A) = P(E ∩ A)/P(A)
Def: Two events are said to be independent, if the happening of an
event is not affected by the happening of the other event.
Note: If A and B are any two independent events in a sample space
‘S’ then, P(A/B) = P(A) or P(B/A) = P(B)
Note: By conditional probability, P(A/B) = P(A ∩ B)/P(B).
If A and B are independent events, then P(A ∩ B) = P(A). P(B)
This is known as special multiplication rule for independent events.
Note: If A and B are any events in S, then
P(A ∩ B) = P(A). P(B/A) if P(A)>0
= P(B). P(A/B) if P(B)>0
This is known as General multiplication rule.
Example: The probability of a hat being red is ¼, the probability of
the hat being green is ¼, and the probability of the hat being black
is ½. Then, the probability of a hat being red OR black is ¾.
Example: The probability that a US president is bearded is 14%,
the probability that a US president died in office is 19%, thus the
probability that a president both had a beard and died in office is
3%. If the two events are independent, 1.3 bearded presidents
are expected to fulfill the two conditions. In reality, 2 bearded
presidents died in office. (A close enough result.)
Baye’s Theorem
Statement: If E1, E2, …, En are mutually exclusive events with
P(Ei)≠0, (i=1, 2, …, n), then for any arbitrary event A, which is a
n
subset of Eisuch that P(A) > 0, we have
i=1
P( Ei).P( A / Ei)
P( Ei A) n
i 1
P( Ei) P( A / Ei)
All the best
