Probability – Basic Concepts Types of Probability Rules Joint Probability Marginal Probability Conditional Probability

1. Module 3-
Probability
Mr.Cyriac Jose

2. Contents
• Probability – Basic Concepts
• Types of Probability
• Rules
• Joint Probability
• Marginal Probability
• Conditional Probability

3. Basic Concepts of Probability
• Probability is the likelihood that an outcome occurs.
Probabilities are expressed as values between 0 and
1.
• An experiment is the process that results in an
outcome.
• The outcome of an experiment is a result that we
observe.
• The sample space is the collection of all possible
outcomes of an experiment.

22. OR

32. Thanks

33. Question 1
Freshveggies.com, an e-commerce vegetable retailer, has recently started their operations in Kottayam. They
have a dedicated customer service center which has by now received 112 customer’s complaints. 78 customers
complained about late delivery of vegetables and 40 complained about poor quality of vegetables received.
a) Calculate the probability that a customer complaint will be about both late delivery and poor quality.
b) What is the probability that a complaint is only about poor quality of vegetables?

34. Answer to Question 1
• Let X = late delivery of vegetables ; Y = poor quality of vegetables
Given,
• n(X) = 78
• n(Y) = 40
• n(XUY) = 112
a)probability that a customer complaint will be about both late delivery and poor quality
• n(X∩Y) = 118 – 112 =6
• P(X∩Y) = n(X∩Y)/Total number of complaints = 6/112 = 0.0535
b)probability that a complaint is only about poor quality of vegetables
• 1 – P(X) = 1 – 78/112 = 0.3035

35. Question 2
A Business School summarized the gender and UG
course of its incoming class as follows
a. Construct the Joint Probability Table
b. Calculate the Marginal Probabilities
c. What is the probability that a Female student is an Engineer or Science Graduate
Gender
U.G. Course
Engineer
ing
Science
Humaniti
es
Arts
Vocation
al
Male 67 46 27 95 15
Female 33 26 31 53 7

36. Answer to Question 2
Gende
r
UG Degree
Engin
eerin
g
Scien
ce
Human
ities
Arts
Vocat
ional
Male
0.167
5
0.115
0.067
5
0.237
5
0.037
5
0.62
5
Femal
e
0.082
5
0.065
0.077
5
0.132
5
0.017
5
0.37
5
0.25 0.18 0.145 0.37 0.055 1
C . 14.75%

37. Question 3
• 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 items produced by
machine 1 were defective and only 1% produced
by machine 2 were defective. If the defective
item is drawn at random,
• what is the probability that the defective
item was produced by machine 1.
• While calculation clearly mention which are
marginal, conditional probabilities and joint
probabilities.

38. Answer to Question 3
• P(Machine 1): 0.3 (30%) - This is the marginal probability of an
item being produced by machine 1, regardless of whether it's
defective.
• P(Machine 2): 0.7 (70%) - This is the marginal probability of an
item being produced by machine 2, regardless of whether it's
defective.
• P(Defective | Machine 1): 0.05 (5%) - This is the conditional
probability of an item being defective, given that it was produced by
machine 1.
• P(Defective | Machine 2): 0.01 (1%) - This is the conditional
probability of an item being defective, given that it was produced by
machine 2.
• P(Defective): We need to calculate this as the probability of a
defective item being produced by either machine 1 or machine 2.

39. Answer to Question 3
• Calculation of P(Defective):
• P(Defective) = (P(Machine 1) * P(Defective | Machine 1)) +
(P(Machine 2) * P(Defective | Machine 2))
• P(Defective) = (0.3 * 0.05) + (0.7 * 0.01)
• P(Defective) = 0.015 + 0.007
• P(Defective) = 0.022

40. Answer to Question 3
Calculation of P(Defective | Machine 1):
• This is the conditional probability we want to find. We can use the
formula:
• P(Defective | Machine 1) = (P(Machine 1) * P(Defective ∩ Machine
1)) / P(Defective)
• We already know all the values needed from the previous
calculations.
• P(Defective | Machine 1) = (0.3 * 0.05) / 0.022
• P(Defective | Machine 1) ≈ 0.6818
Interpretation:
• Therefore, the probability that a defective item came from
machine 1, given that a defective item was drawn, is
approximately 0.6818 or 68.18%. This is the conditional
probability we were looking for.

42. What is Binomial Distribution? List the four criteria that a Binomial
Distribution must follow. Give 1 example in business analytics that
can be associated with Binomial Distribution.
Answer :
The binomial distribution is a special discrete distribution where
there are two distinct complementary outcomes, a “success” and a
“failure”. [1 mark]
Four conditions to be satisfied: [ 0.5 mark for each correct condition
= 2 marks]
• The number of observations n is fixed.
• Each trial results in one of the two outcomes, called success
and failure.
• he probability of "success" p is the same for each outcome.
• Each observation is independent.
Example: [1 marks for any correctly written example = 1 marks; may
cite any correct example]
• Customer churn where the outcomes are: (a) Customer churn and
(b) No customer churn
• Fraudulent insurance claims where the outcomes are: a)
Fraudulent claim b) Genuine claim
• Loan repayment default by a customer where the outcomes are: a)
Default and b) No default
• Employee attrition at a company where the outcomes are: a) The