PROBABILITY
INTRODUCTION
The theory of probability consist of
 Statistical approach
 Classical approach
Statistical approach
It is also known as repeated experiments and observe
frequency approach . In this ,probability is defined as
the ratio of observed to the total frequency .
Classical approach.
In this , probability is defined as the ratio
of favourable number of outcomes to the total number of
equally likely outcomes(elementary outcomes).
DEFICIENCIES & LIMITATIONS
• These approaches cannot be applied
experiments which have large no. of outcomes.
to
the
• The classical definition of probability cannot be applied
whenever it is not possible to make a simple enumeration
of cases which can be considered equally likely.
The statistical definition has difficulties from a
mathematical point of view because an actual limiting
number may not really exist. For this reason modern
probability theory has been developed axiomatically.
This theory of probability was developed by A.N
Kolmogrov (1903 -1987)Russian Mathematician in
1933 .
In order to understand this approach we must know
about same basic terms viz .random experiment ,
elementary events ,sample space , compound events
etc
He laid down
certain axioms to
interpret
probability , in his
book “Foundation
of probability”
published in 1933.
:KOLMOGROV
Random experiments
The word experiment means an operation which
can produce some well defined outcomes . There are two
types of experiments viz.
(i) Deterministic experiments .
(ii) Random or probability experiments
Deterministic experiments.
The experiments which have a fixed outcome or result no
matter any number of times they are repeated .
For example: We can definitely say that the sum of
measures of angles of a triangle is 180degree.
RANDOM EXPERIMENT
When an experiment is repeated under
identical conditions do not produce the outcome every
time but the outcome in a trial is one of the several
outcomes ,then such an experiment is called random
experiments.
For example:
• Rolling an unbiased die .
• Drawing a card from a well
shuffled pack of cards.
Sample space
The set of all possible outcomes of a random experiments
is called sample space. It is denoted by S.
ILLUSTRATION
 Random experiment of tossing a coin.
The possible outcomes of the experiments are H
and T.
The sample space associated with this experiment is
given by :
S = {H,T}
EVENT
A subset of sample space associated with
random experiments is called a event .
For example :the random of throwing a die .
S={1,2,3,4,5,6}
S has 26=64 subsets .Each one of these 64
subsets is associated with random
experiment of through a die .
Types of events
1. Impossible events & sure events
The empty set and sample space S describes the event. In fact
empty set is called impossible event and the whole sample
space is called sure event
E.g. while rolling a die and getting a number divisible by 7.let a
be this event . We know that none of the possible outcomes
Compound event
An event associated with a random experiment is a compound
event , if it is the disjoint union of two or more elementary event
.
For e.g.
In a single throw of an ordinary die there are six elementary
events and the total number of events is 26 =64 so 26 –
(6+1)= 57 is the total number of compound events.
MUTUALLY EXCLUSIVE EVENTS
Two or more events are said to be mutually exclusive ( or
incompatible ) when only one of the events can occuring a
random experiment and two(or more) events cannot
simultaneously.
If two events associated with a random experiment are
mutually exclusive, then the subsets of the sample space
representing the two events are disjoint.
Illustration ; In tossing a coin, the two events represented by
the outcomes “ Head “ and “ Tail “ are mutually exclusive . For
, if “head “ turns up , we cannot get “ tail “ and vice versa
EXHAUSTIVE EVENTS .
A set of events is said to be exhaustive if at least one
of them occurs in a random experiment. The
elementary events in a sample space are exhaustive.
All elementary events associated with a random
experiment from a set of exhaustive events.
 Illustration ; Consider the experiment of drawing a
card from a well shuffled deck of playing cards. Let A
be the event “ card is red “, B be the event “ card is
black “ “Clearly, A and Bare exhaustive events
PROBABILITY THEORY
SET NOTATION
Sample space
S
Outcome of a random experiment
ω
Event A
A c S
Event A has occurred
ω €A
Event A has not occurred
ω€A
Event” A or B “
AUB
Supplementary problems
1. A coin is tossed . If it shows head , then a die is thrown
.but if it shows tail , then another coin is tossed. Describe
the sample space ?
Ans- (H1,H2,HH3,H4,H5,H6,THT,TT,).
2. From a bag containing 3 red,2 blue balls two balls are
drawn at random . Write down the sample space and then
find the set representing the event “ I ball is red and the
other blue “.
AXIOMATIC APPROACH TO PROBABILITY
AXIOMATIC APPROACH IS ANOTHER WAY OF DESCRIBING
PROBABILITY OF AN EVENT. IN THIS APPROACH SOME
AXIOMS OR RULES ARE DEPICTED TO ASSIGN PROBABILITIES.
For example, in ‘ a coin tossing ‘ experiment we can assign the
number ½ to each of the outcomes H and T.
P( H) = ½ and P ( T ) = ½
Applications
Probability theory is applied in everyday life
in risk assessment and in trade on financial
markets.
Governments apply probabilistic methods
in environmental regulations where it is called
pathway analysis. A good example is the effect
of the perceived probability of any widespread
Middle East conflict on oil prices—which have
ripple effects in the economy as a whole
If there are n elementary events associated
with a random experiment and m of them are
favorable to an event A , then the probability
of happening or occurrence of A is denoted
by P( A ) and is defined as the ratio m/n.
Thus, P(A)=m/n
WORK SHEET
1.One card is drawn from a pack of 52 cards. The probability
that it is the card of a king or spade is
a) 1/26 b) 3/26 c)4/13 d)3/13
2.Two dice are thrown simultaneously . The probability of
obtaining a total score of 5 is
a) 1/18
b)1/12
c)1/9
d) none of these
3.A coin is tossed until two heads or a tail is realized . Specify
the sample space of the experiment.
4. A coin is tossed two times in succession .What is the
probability that the first toss resulted in a head ?
5. From the pack of 52 cards three cards are drawn at random
.Find the probability of drawing exactly two aces.
6. Four coins are tossed simultaneously . What is the
probability of getting at least one head?
7.The probability of A,B,C solving a problem are 1/3 ,2/7and
3/8 respectively .If all the three try to solve the problem
simultaneously ,find the probability that exactly one of them
solve it?
8. Two cards are drawn from a well shuffled pack of
52cards without replacement. Find the probability that
neither a jack nor a card of spade is drawn.
9.The probability that a student will get A,B.,C or D grade
are 0.4,0.35,0.15 and 0.1 respectively. Find the
probability that she will get
a)B or C
b) at most C
10.The odds in favour of an event are 3:5. Find the
probability of occurrence of this event.