DS

1. Prepared by Dr Ajay Parulekar

2. Topic Objectives
Prepared by Dr Ajay Parulekar
 To get acquainted with the basic terminology of
probability theory
 To learn use of probability theory through real life
business problems
 To understand basic probability distributions (Normal,
Binomial & Poisson)
 To learn use of these probability distributions through
real life business problems

3. Introduction
 We don’t know for sure what is going to happen in future
 We use probability to build tools to describe and understand apparent
randomness
 There is always a chance element / risk associated with happening of
any event.This is reflected by way of probability
 The probability of an outcome is the proportion of times the outcome
would occur if we observed the random process an infinite number of
times
 Thus probability is a numerical statement about the likelihood that an
event will occur
 Two basic rules regarding the mathematics of probability are,
 Probability (P) of an event always lies between 0 and 1 i.e.
0 ≤ P ≤ 1
 Sum of probabilities for all possible outcomes of an activity / event
must be equal to 1
Prepared by Prof.Dr Ajay Parulekar

4. Introduction
A few definitions:
 Random Experiments – A process of obtaining information
through observation or measurement of a phenomenon whose
outcome is subject to change
 Outcome (one or more) of such random experiment is / are
called as an event. Single possible outcome of an experiment is
called as a simple event.
 The set of all possible outcomes or simple events of an
experiment is called as sample space
 Mutually exclusive events – Two or more events that can’t occur
simultaneously in a single trial of an experiment
 Collectively Exhaustive Events – A list of events when all possible
events that can occur from an experiment include every possible
outcome
Prepared by Prof.Dr Ajay Parulekar

5. Fundamentals
 Independent and Dependent Events – Two events are said to be
independent when outcome of one event does not affect and is
not affected by, the other event. Two events are said to be
dependent when outcome of one event is affected by the other
event
 Compound Events – When two or more events occur in
connection with each other then, their simultaneous occurrence
is called a compound event
 Equally Likely Events – Two or more events are said to be equally
likely if each has an equal chance to occur
 Complementary Events – If E is any subset of the sample space,
then its complement denoted by Ē contains all the elements of
the sample space then,
Ē = S – E = All sample elements not in E
Prepared by Prof.Dr Ajay Parulekar

6. Probability
Probability – If a trial results in ‘n’ exhaustive cases which are
mutually exclusive and equally likely and out of which ‘m’ are
favourable to the happening of event A, then probability ‘p’ of
the happening of eventA is given by,
p = P(A) = m / n
If Ā denotes the non happening ofA, then its probability‘q’ is
q = P(Ā) = (n – m) / n = 1 – P(A)
Note,
P(A) + P(Ā) = 1
Prepared by Prof.Dr Ajay Parulekar

7. Adding mutually exclusive events / non
mutually exclusive events
Prepared by Dr Ajay Parulekar
 Addition also known as union of two events
 For mutually exclusive events (those which can’t happen
together at same time),
 P (A or B) = P(A) + P(B)… probability of drawing a spade
or club out of deck of cards
 For not mutually exclusive events,
 P (A or B) = P(A) + P(B) – P(A and B)… probability of
drawing a 5 and drawing a diamond out of deck of cards

8. Example 1
Prepared by Dr Ajay Parulekar
In a certain hospital 60% of the patients are suffering from
typhoid, 50% are suffering from cholera and 30% are suffering
from both. If a patient is selected at random, what is the
probability that,
i) He is suffering from typhoid or cholera
ii) He is suffering from only cholera

9. Example 2
Prepared by Dr Ajay Parulekar
A share broker purchases 3 stocksA, B & C for one week
trading purpose.As per share broker, the probabilities that A, B
& C will appreciate in value over a period of one week are 0.8,
0.7 & 0.6 resp.What is the chance that,
i) All three stocks will increase in value
ii) At least two stocks will increase in value
iii) Only one will increase in value

10. Conditional Probability
Prepared by Dr Ajay Parulekar
 P (A/B) is conditional probability of occurrence of eventA given
that event B has already occurred is computed as,
P(A/B) = P(A and B) / P(B)
For e.g.
What is the chance that a faculty is insane?
What is the chance that faculty is insane given that he teaches DS?
Sane Insane Total
TeachesDS 5 95 100
DoesnotteachDS 145 5 150
Total 150 100 250

11. Bayes’ Theorem
Prepared by Dr Ajay Parulekar
 IfA₁ ,A₂, … An are mutually exclusive and collectively
exhaustive events and B is any other event that occurs in
conjunction with events A₁ ,A₂, … An
 P (Ak / B) = P (Ak ) X P (B / Ak ) / ∑ P (Ak ) X P (B / Ak )
Where value of k varies from 1 to n

12. Example 3
Prepared by Dr Ajay Parulekar
It is known that 40% of the students in a certain college are
girls and 50% of the students are above the median height. If
2/3 of the boys are above median height, what is the probability
that a randomly selected student who is below the median
height is a girl?

13. Example 4
Prepared by Dr Ajay Parulekar
A production company, manufactures a product using three
machines M1, M2 & M3. M1 produces 50%, M2 produces
30% and M3 produces 20% of the total production. Past
experience reveals that M1 produces 4% defectives as against
M2 5% defectives and M3 producing 6% defectives. At the end
of the day from total production, 1 unit is selected at random
and is found to be defective. What is the probability that it was
produced by machine M1?

14. PROBABILITY DISTRIBUTION
Probability Distribution
The probability distribution for a discrete random variable is a
mutually exclusive listing of all possible outcome of that variable
and the corresponding probabilities of occurrences of these
outcomes. P(x) which is a function of variable x is called as its
probability function.
Binomial Probability Distribution
This distribution describes discrete data resulting from an
experiment called as Bernoulli process (a process in which each
trial has only two possible outcomes namely a success with
constant probability ‘p’ and a failure with constant probability
‘q’).
Thus,
p+q = 1
Prepared by Prof.Dr Ajay Parulekar

15. PROBABILITY DISTRIBUTION
Binomial Probability Distribution
If n= number of independent trials and ‘r’ denotes number
of successes in these‘n’ trials then,
Probability of getting‘r’ successes in‘n’ trials is given as,
P(r) = nCrprqn-r
where r = 0,1,2, … n
Mean of distribution E(r) = np and
variance = npq
standard deviation = √npq
Prepared by Prof.Dr Ajay Parulekar

16. PROBABILITY DISTRIBUTION
Poisson Probability Distribution
 This is another type of discrete probability distribution which
can be considered as limiting case of Binomial Distribution with
following conditions,
i. The number of trials i.e. n is very large
ii. constant probability of success for each trial (p) is very small
(close to zero) or very large (close to 1)
mean = np and
Probability function becomes,
P(r) = (e-mmr) / r!
where r = 0,1,2, …
P(r) stands for probability of getting r successes
e is the base of natural logarithm e = 2.7183 and
m = mean = np
Prepared by Prof.Dr Ajay Parulekar

17. PROBABILITY DISTRIBUTION
For Poisson Distribution,
variance = m
Also if there are ‘N’ sets of ‘n’ trials then the expected
frequency of occurrence of r successes is,
f(r) = N X P(r)
Prepared by Prof.Dr Ajay Parulekar

18. PROBABILITY DISTRIBUTION
 Normal Probability Distribution
This is the most important continuous probability distribution. Also
known as Gaussian Probability Distribution.
Probability function is given by,
P(x) =( 1/σ√2 Π) e-1/2((x – m)/σ)2 for -∞ <x <∞
where m is the mean and σ is the standard deviation of the
distribution
 Standard Normal Probability Distribution
we put z = (x – m)/ σ
where‘z’ is called as Standard NormalVariate or Z score,
P(z) = ( 1/√2 Π) e-1/2(z)2
-∞ <z <∞ and
mean = 0 and σ = 1
Prepared by Prof.Dr Ajay Parulekar

19. Examples
Prepared by Dr Ajay Parulekar
 The mean of a binomial distribution is 4 and variance is 4/3.
Find the probability of getting i) no success ii) at least 5
successes
 The incidence of a certain disease is such that on an average
20% of workers suffer from it. If 10 workers are selected at
random, find the probability that more than 2 workers suffer
from the disease
 If 2% of electric bulb manufactured are defective, find the
probability that in a sample of 200 bulbs, less than 2 are
defective? (Given e-4 = 0.0183, e-2 = 0.1141)