Probability

Basic Probability Concepts
 Random Experiment:
 Possible Outcome:
 Sample Space:

 Event:
Simple or Elementary Event:
 Mutually Exclusive Events:
 Independent Events: Dependent Events:
 Exhaustive Events:
 Equally Likely Events:


 Types of Probability:
1. 1. Classical Approach ( Priori Probability):
2. Relative Frequency Theory of Probability:
3. Subjective Approach:
4. Axiomatic Approach:

Theorems of Probability
Probability Theorems

Addition Multiplication Theorem Bayes’
Theorem Theorem
Independent
Dependent
Variables
Variables
Mutually Partially
Exclusive Overlapping
Events Events

Basic Probability Concepts
 Experiment:
 Any operation or process that results in two or more
outcomes is called as experiment.
 Example: Tossing a coin is an experiment or trial, where
the outcome head or trial is unpredictable.
 Random Experiment:
 Any well – defined process of observing a given chance
phenomena through a services of trials that are finite or
infinite and each of which leads to a single outcome is
known as a random experiment.
 Example: Drawing a card from a pack of 52 cards. This
is also a chance phenomena with only one outcome.
 A random experiment is different from experiments
under control conditions because the observation in a
random experiment involves chance phenomena and is
not performed under controlled conditions.

 Possible Outcome:
 The result of a random experiment is called an
outcome.
 Example: Tossing a coin and getting head or tail up is
an outcome.
 Event:
 Any possible outcome of a random experiment is
called an event.
 Performing an experiment is called trial and outcomes
are termed as events.
 Simple or Elementary Event:
 An event is called simple if it corresponds to a single
possible outcome.
 Example: Rolling two dice then the event of getting a

 Six on either the first or second die is a
compound event.
 Sample Space:
 The set or aggregate of all possible outcomes is
known as sample space.
 Example: When we roll a die, the possible
outcomes are 1, 2, 3, 4, 5, and 6. Thus all the
outcomes 1, 2, 3, 4, 5, and 6 are sample space.
And each possible outcome or element in a
sample space called sample point.
 Favorable Event:
 The number of outcome which result in the
happening of a desired event are called
favorable cases to the event.

 Example: In a single throw of a dice the number of
favorable cases of getting an odd number are three.
 Mutually Exclusive Events:
 Two events are said to be mutually exclusive if the
occurrence of one of them excludes the possibility of the
occurrence of the other in a single observation. The
occurrence of one event prevents the occurrence of the
other event.
 Example: If a coin is tossed, either the head can be up or
tail can be up, but both can not be at the same time.
 Independent Events:
 A set of events is said to be independent, if the
occurrence of any one of them does not, in any way,
affect the occurrence of any other in the last.
 Example: When a coin is tossed twice, the result of the
second toss will in no way be affected by the result of the
first toss.

 Dependent Events:
 Two events are said to be dependent, if the occurrence
or non-occurrence of one event in any trial affects the
probability of the other subsequent trials.
 If the occurrence of one event affects the happening of
the other events, then they are said to be dependent
events.
 Example: The probability of drawing a king from a pack
of 52 cards is 4/52, the card is not put back, then the
probability of drawing a king again is 3/51. Thus the
outcome of the first event affects the outcome of the
second event and they are dependent.
 The total number of possible outcomes of a random
experiment is called exhaustive events.

 Example: In tossing a coin, the possible
outcome are head or tail, exhaustive events are
two.
 The outcomes are said to be equally likely when
one does not occur more often than the others.
 Two or more events are said to be equally likely
if the chance of their happening is equal.
 Example: In a throw of a die the coming up of 1,
2, 3, 4, 5, 6 is equally likely.
 Types of Probability:
1. Classical Approach ( Priori Probability):

 According to this approach, the probability is the
ratio of favorable events to the total no. of
equally likely events.
 In tossing a coin the probability of the coin
coming down ids 1, of the head coming up is ½
and of the tail coming up is ½.
 The probability of one event as ‘P’ (success)
and of the other event as ‘q’ (failure) as there is
no third event.
No. of favorable cases
p=
Total number of equally likely
cases

 If an event can occur in ‘a’ ways and fail to occur
in ‘b’ ways and these are equally to occur, then
the probability of the event occurring, a/a+b is
denoted by p. Such probabilities are known as
unitary or theoretical or mathematical probability.
 p is the probability of the event happening and q
is the probability of its not happening.
p = a/a+b and q = b/a+b
Hence p+q = (a+b)/(a+b)
Therefore p+q = 1
 Probabilities can be expressed either as ratio,
fraction or percentage, such as ½ or 0.5 or 50%.

 Example: Tossing of a coin.
 Limitations:
 This definition is confined to the problems
of games of chance only and can not
explain the problem other than the games
of chance.
 This method can not be applied, when the
outcomes of a random experiment are not
equally likely.
 The classical definition is applicable only
when the events are mutually exclusive.

Relative Frequency Theory of
Probability:
 In this approach, the probability of happening of
an event is determined on the basis of past
experience or on the basis of relative frequency
of success in the past.
 Example: If a machine produces 100 articles in
the past, 2 articles were found to be defective,
then the probability of the defective articles is
2/100 or 2%.
 The relative frequency obtained on the basis of
past experience can be shown to come to very
close to the classical probability.
 Example: If a coin is tossed for 6 times, we may

not get exactly 3 heads and 3 tails. But if it is
tossed for larger number of times say 10000
times, we can expect heads and tails very close
to 50%.
 There are certain laws, according to which the
‘occurrence’ or ‘non-occurrence’ of the events
take place. Posterior probabilities, also called
empirical probabilities are based on experiences
of the past and on experiments conducted.
 Limitations:
 The experimental conditions may not remain
essentially homogeneous and identical in a large
number of repetitions of the experiment.


 The relative frequency m/n, may not attain a
unique value no matter however large.
 Probability p(A) defined can never be obtained
in practice. We can only attempt at a close
estimate of p(A) by making N sufficiently large.
 Subjective Approach:
 The subjective theory of probability is also
known as subjective theory of probability.
 This theory is commonly used in business
decision making.
 The decision reflects the personality of the
decision maker.
 Persons may arrive at different probability
assignment because of differences in value at
experience etc. The personality of the decision
maker is reflected in a final decision.

 Example: A student would top in B. Com Exam
this year.
 A subjective would assign a weight between
zero and one to this event according to his belief
for its possible occurrence.
 Axiomatic Approach:
 The probability calculations are based on the
axioms. The axiomatic probability includes the
concept of both classical and empirical
definitions of probability.
 The approach assumes finite sample spaces
and is based on the following three axioms:
i) The probability of an event ranges from 0 to 1.
If the event cannot take place its probability shall
be ‘0’ and if it is bound to occur its probability is
‘1’.

ii) The probability of the entire sample space is 1, i.e.
p(S)=1.
iii) If A and B are mutually exclusive events then the
probability of occurrence of either A or B denoted by
p(AUB) = p(A) + p(B)

Addition Theorem
1. Mutually Exclusive Events:
 If two events are mutually exclusive, then the
probability of the occurrence of either A or B is
the sum of the probabilities A and B. Thus,
 P(A or B) = P (A) + P(B)
 Example: A bag contains 4 white and 3 black
and 5 red balls. What is the probability of getting
a white or red ball at a random in a single draw?
 Solution: The probability of getting a white ball
= 4/12

 The probability of getting a red ball is = 5/12
 The probability of getting a white or red ball
= 4/12 + 5/12
= 9/12
Non-Mutually Exclusive Events:
In such cases where the events are not mutually
exclusive, the probability of one of them
occurring is the sum of the marginal probabilities
of the events minus the joint probability of the
occurrence of the events.
P(A or B) = p(A) + p(B) – p(A and B)
Example: Two students A & B work independently
on a problem. The probability that A will solve it
is 3/4 and the probability that B will solve it is
2/3.

What is the probability that problem will be solved?
The probability that A will solve the problem = 3/4
The probability that B will solve the problem= 2/3
The events are not mutually exclusive as both of them
may solve the problem.
The probability that problem will be solved
=3/4 + 2/3 – (3/4 X 2/3)
=11/12
Multiplication:
When it is desired to estimate the chances of the
happening of successive events, the separate
probabilities of these successive events are multiplied.

 If two events A and B are independent, then the
probability that both will occur is equal to the
product of the respective probabilities.
P(A and B) = p(A) X P(B)
 Example: In two tosses of a fair coin, what are
the chances of head in both?
Probability of head in first toss = 1/2
Probability of head in the second toss = 1/2
Probability of head in both tosses = 1/2 X1/2
= 1/4
Conditional Probability:

The multiplication theorem explained above is
not applicable in case of dependent events.

 If the events are dependent, the probability is
conditional.
 Two events A and B are dependent, B occurs
only when A is known to have occurred (or vice-
versa).
 P(B/A) means the probability of B given that A
has occurred.
 P(A/B) = p(AB)/p(B)
 P(A/B) is the probability of A given that B has
occurred.
 P(B/A) = p(AB)/p(A)
 The general rule of multiplication in its modified
form in terms of conditional probability becomes:

 p(A and B) = p(B) X p(A/B)
 p(A and B) = p(A) X p(B/A)
 Example: A bag contains 5 white 3 black balls.
Two balls are drawn at random one after the
other without replacement. Find the probability
that both balls drawn are black.
Probability that both balls drawn are black is
given by p(AB) = p(A) X p(B)
= 3/8 X 2/7 = 3/28
Bayes’ Theorem:
 It is also known as the inverse probability.

 Probabilities can be revised when new
information pertaining to a random experiment is
obtained.
 One of the important applications of the
conditional probability is in the computation of
unknown probabilities, on the basis of the
information supplied by the experiment or past
records. That is, the applications of the results of
probability theory involves estimating unknown
probabilities and making decisions of new
sample information.
 Quite often the businessman has the extra
information on a particular event either through a
personnel belief or from the past history of the
events.

 Revision of probability arises from a need to
make better use of experimental information.
 Probabilities assigned on the basis of personal
experiences, before observing the outcomes of
the experiment are called prior probabilities.
 Example: Probabilities assigned to past sales
records.
 When the probabilities are revised with the use
of Bayes’ rule, they are called posterior
probabilities.
 It is useful in solving practical business problems
in the light of additional information.

 Thus probability of the theorem has been mainly
because of its usefulness in revising a set of old
probability (prior probability) in the light of
additional information made available and to
derive a set of new probability (i.e. posterior
probability)
 Illustration:
Assume that a factory has two machines. Past
records show that machine 1 produces 30% of
the items of output and machine 2 produces
70% of the items. Further, 5% of the item
produced by machine 1 were defective and only
1% produced by machine 2 were defective .

If a defective item is drawn at random, what is
the probability that the defective item was
produced by machine 1 or machine 2?
Let A1 = the event of drawing an item
produced by machine 1,
A2 = the event of drawing an item
produced by machine 1,
B = the event of drawing a defective item
produced either by machine 1 or
machine 2
Then from the first information,
p(A1) = 30 % = .30
p(A2) = 70% = .70
From the additional information

 P(B/A1) = 5% = .05
 P(B/A2) = 1% = .1
 The required value are tabulated below:
1 2 3 4
Events Prior Conditional Joint Posterior
probability probability probability probability
p(A1) p(B/A1) p(A1/B)

A1 .30 .05 .015 .015/.022 = .682

A2 .70 .1 .007 .007/.022 = .318

.100 P(B) = .022 1.000


Without the additional information, we may be
inclined to say that the defective item is drawn
from machine 2 output since
P(A1) = 70% is larger than P(A1) = 30%
With the additional information, we may give a
better answer. The probability that the defective
item was produced by machine 1 is .682 or
68.2% and that by machine 2 is only .318 or
31.8%. We may say that the defective item is
more likely drawn from the output produced by
machine 1.
The above answer may be checked by actual no.
of items as follows:

 If 10000 items were produced by two machines in a
given period, the no. of items produced by machine 1
is
10000 X 30% = 3000
and the no. of items produced by machine 2
is 10000 X 70% = 7000
 The number of defective items produced by machine 1
is 3000 X 5% = 150
and the number of defective items produced by
machine 2 is 7000 X 1%= 70
The probability that a defective item was produced by
machine 1 is = 150/150+70 = .682
and by machine 2 is = 70/150+70 = .318

Probability

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