Probability is a numerical measure of how likely an event is to occur. It is used in business to quantify uncertainty and make predictions. Some common applications in business include predicting sales based on price changes, estimating increases in productivity from new methods, and assessing the likelihood of investments being profitable. Probability is calculated on a scale of 0 to 1, with values closer to 1 indicating an event is more certain or likely to occur.

2. Probability
Why Probability in Business?
- What are the chances that sales will decrease if we
increase the prices?
- What is the likelihood a new assembly method will
increase productivity?
- What is the chance that a new investment is
profitable?

3. Probability Definition
• Is a numerical measure of the likelihood that an
event will occur.
• Probability values are always assigned on a
scale from 0 to 1.
• A probability near zero indicates an event is
unlikely to occur; a probability near 1 indicates an
event is almost certain to occur.
• Ex. Weather report

4. ● A quantitative measure of uncertainty
● A measure of the strength of belief in the
occurrence of an uncertain event
● A measure of the degree of chance or
likelihood of occurrence of an uncertain event
● Measured by a number between 0 and 1 (or
between 0% and 100%)

5. Experiment
Is a process that generates well-defined outcomes.
Ex.
Experiment Experimental outcomes
Toss a coin Head, Tail
Conduct a sales call Purchase, No purchase
Roll a die 1,2,3,4,5,6
Play a cricket game Win, lose, tie

6. Sample space
The sample space for an experiment is the set of all
experimental outcomes.
Ex. Tossing a coin
s= (head, tail)
Rolling a die
s=(1,2,3,4,5,6)
Workout- Game
An experimental outcome is also called as a sample point.

7. Counting Rules, Combinations and Permutations
Multiple-step experiment
The experiment of tossing two coins can be thought of
as two-step experiment.
Sample space S= (H,H),(H,T),(T,H),(T,T)
Four experimental outcomes are possible
The counting rule for multiple-step experiment makes it
possible to determine the number of experimental
outcomes without listing them.

8. Counting rule for multiple-step experiment
If an experiment can be described as a sequence of k
steps with n1 possible outcomes on the first step , n2
possible outcomes on the second step an so on , then
the total number of experimental outcome is given by
(n1) (n2)….(nk).
Tossing two coin n1=2 and n2=2 then using counting
rule (2)(2)=4.
Ex. Tree diagram and tossing of 2 coins.

9. Combinations
A second useful counting rule enables us to count
the number of experimental outcomes when n objects
are to be selected from a set of N objects.
where: N! = N(N - 1)(N - 2) . . . (2)(1)
n! = n(n - 1)(n - 2) . . . (2)(1)
0! = 1
Ex: In a group of five parts how many combinations of two parts can
be selected
10 outcomes are possible. If five parts are A,B,C,D,E then 10
combinations of outcomes are AB,AC,AD,AE,BC,BD,BE,CD,CE,DE

10. Ex.1
Simple random sampling uses a sample of size n form a population of
size N to obtain data that can be used to make inferences about the
characteristics of a population. Suppose that, from a population of 50
bank accounts, we want to take a random sample of four accounts in
order to learn about the population. How many different random
samples of four accounts are possible?.
Ex.2. How many ways a 6 member team can be formed having 3 men
and 3 ladies from a group of 6 men and 7 ladies? 6c3+7c3=700

11. Permutations
A third useful counting rule enables us to count
the number of experimental outcomes when n
objects are to be selected from a set of N objects,
where the order of selection is important.
where: N! = N(N - 1)(N - 2) . . . (2)(1)
n! = n(n - 1)(n - 2) . . . (2)(1)
0! = 1
Ex. Same problem as combinations
Ordering is important ex:
AB,BA,AC,CA,AD,DA,AE,EA,BC.CB.BD.DB.BE.EB.CD.DC.CE.EC.DE.ED.

12. • Permutation : Permutation means arrangement of things.
The word arrangement is used, if the order of things is
considered.
• Combination: Combination means selection of things. The
word selection is used, when the order of things has no
importance.
• Example: Suppose we have to form a number of
consisting of three digits using the digits 1,2,3,4, To form
this number the digits have to be arranged. Different
numbers will get formed depending upon the order in
which we arrange the digits. This is an example
of Permutation.
• Now suppose that we have to make a team of 11 players
out of 20 players, This is an example of combination,
because the order of players in the team will not result in
a change in the team. No matter in which order we list
out the players the team will remain the same!

13. Assigning probabilities
How probabilities can be a assigned to experimental outcomes.
1. Classical
2. Relative frequency
3. Subjective
Basic requirements for assigning probabilities
1. The probability assigned to each experimental
outcome must be between 0 and 1, inclusively.
0 < P(Ei) < 1 for all i
where:
Ei is the ith experimental outcome
and P(E ) is its probability

14. 2. The sum of the probabilities for all experimental
outcomes must equal 1.
P(E1) + P(E2) + . . . + P(En) = 1
where:
n is the number of experimental outcomes

15. Classical Method
The classical method is appropriate when all the experimental
outcomes are equally likely.
If n experimental outcomes are possible, a probability of 1/n is
assigned to each experimental outcome.
Ex. Tossing a coin and rolling a die
Relative frequency
When data are available to estimate the proportion of the time the
experimental outcome will occur if the experiment is repeated a large
number of times.
Ex Number waiting Number of days outcome occurred Relative frequency
0 2 .10
1 5 .25
2 6 .30
3 4 .20
4 3 .15
total 20

16. Subjective method
Subjective probability result from intuition, educated
guesses, and estimates. For instance, given a patient’s
health and extent of injuries, a doctor may feel a
patient has 90% chance of a full recovery. A business
analyst may predict that the chance of the employees
of a certain company going on strike is .25
Ex.
The probability of your phone ringing during the class is
is 0.1
This probability is most likely based on an educated
guess. It is an example of subjective probability.

17. Exercise
1. Tossing a coin three times
a. develop a tree diagram
b. list the experimental outcomes
c. What is the probability for each experimental outcome. What method you use.
2. An experiment with three outcomes has been repeated 50 times and it was
learned that E1 occurred 20 times E2 occurred 13 times and E3 occurred 17
times. Assign probabilities to the outcomes. What method did you use?.
3. A decision maker subjectively assigned the following probabilities to the four
outcomes of an experiment P(E1)=.10, P(E2)=.15, P(E3)=.40 and P(E4)=.20. Are
the probability assignment valid?

18. Events: Definition
●Sample Space or Event Set
Set of all possible outcomes (universal set) for a given
experiment
●E.g.: Roll a regular six-sided die
S = {1,2,3,4,5,6}
●Event
Collection of outcomes having a common characteristic
●E.g.: Even number
A = {2,4,6}
Event A occurs if an outcome in the set A occurs
●Probability of an event
Sum of the probabilities of the outcomes of which it consists
●P(A) = P(2) + P(4) + P(6)

19. •For example:
Roll a die
• Six possible outcomes {1,2,3,4,5,6}
• If each is equally-likely, the probability of each is 1/6 = 0.1667 =
16.67%
• Probability of each equally-likely outcome is 1 divided by the
number of possible outcomes
Event A (even number)
• P(A) = P(2) + P(4) + P(6) = 1/6 + 1/6 + 1/6 = 1/2
for e in A

20. Complement of a Set
A
S
Venn Diagram illustrating the Complement of an event

21. ✔ Intersection (And)
– a set containing all elements in both A and B
✔ Union (Or)
– a set containing all elements in A or B or
both
Complements - Probability of not A
Basic Definitions (Continued)

22. Sets: A Intersecting with B
A
B
S

23. Sets: A Union B
A
B
S
2-23

24. Rules of Probability:
1. General Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)
If A and B are mutually exclusive, then
P(A and B) = 0, so the rule can be simplified
P(A or B) = P(A) + P(B)

25. Mutually Exclusive or Disjoint Sets
A
B
S
Sets have nothing in common
2-25

26. Ex.1 Study by personal manager of a software company. The study
shows 30% of the employees who left the firm within two years did
so primarily because they were dissatisfied with their salary, 20% left
for work assignment and 12% indicated dissatisfaction with both their
salary and work assignment.
What is the prob. that an employee who leaves within two years does
so because of dissatisfaction with salary, work assignment or both?

28. Ex.2 A survey of magazine subscribers showed that 45.8% rented a car
during the past 12 months for business reasons, 54% rented a car
during the past 12 months for personal reasons, and 30% rented a car
during the past 12 months for both business and personal reasons.
1. What is the probability that a subscriber rented a car during the pas
12 months for business or personal reasons?
2. What is the probability that a subscriber did not rent a car during
the past 12 months for either business or personal reasons?

29. Conditional probability:
2.A conditional probability is the probability of one event,
given that another event has occurred
The conditional
probability of A
given that B has
occurred
The conditional
probability of B
given that A has
occurred
Where P(A and B) = joint probability of A and B

30. School quality School cost or
convenience
Other Totals
Full Time 421 393 76 890
Part time 400 593 46 1039
Totals 821 986 122 1929

31. • Marginal Probabilities
• P(S) P(C) P(O) P(F) P(P)
• Joint Probabilities
• P(S and F)
• P(S and P)
• P(C and F)
• P(C and P)

32. Rules of conditional probability:
If events A and B are statistically independent:
so
so
Conditional Probability (continued)

33. 3. Multiplication Rule– for two events A and B
3. Two events A and B are statistically independent if
the probability of one event is unchanged by the
knowledge that other even occurred. That is:
3. Then the multiplication rule for two statistically
independent events is:

34. Some more definitions
Set - a collection of elements or objects of interest
✔Empty set (denoted by ∅)
●a set containing no elements
✔Universal set (denoted by S)
●a set containing all possible elements
✔Complement (Not). The complement of A is
●a set containing all elements of S not in A

35. Multiplication rule Examples
When asked to find the probability of A and B, we want to find out
the probability of events A and B happening.
Suppose we roll one die followed by another and want to find
the probability of rolling a 4 on the first die and rolling an even number
on the second die. Notice in this problem we are not dealing with the
sum of both dice. We are only dealing with the probability of 4 on one
die only and then, as a separate event, the probability of an even
number on one die only.

36. P(4) = 1/6
P(even) = 3/6
So P(4 even) = (1/6)(3/6) = 3/36 = 1/12
While the rule can be applied regardless of dependence or
independence of events, we should note here that rolling a 4 on
one die followed by rolling an even number on the second die
are independent events. Each die is treated as a separate thing
and what happens on the first die does not influence or effect
what happens on the second die.
This is our basic definition of independent events: the outcome of
one event does not influence or effect the outcome of another
event.

37. Suppose you have a box with 3 blue marbles, 2 red marbles,
and 4 yellow marbles. You are going to pull out one marble,
record its color, put it back in the box and draw another
marble. What is the probability of pulling out a red marble
followed by a blue marble?
The multiplication rule says we need to find P(red) P(blue).
P(red) = 2/9
P(blue) = 3/9
P(red blue) = (2/9)(3/9) = 6/81 = 2/27
Independent or not ?

38. Workout
1.There are 11 marbles in a bag. Two are yellow, five are
pink and four are green. Suppose you pull out one marble,
record its color, put it back in the bag and then pull out
another marble. What is the probability of P(yellow and
pink)

39. Consider the same box of marbles as in the previous example.
However in this case, we are going to pull out the first marble,
leave it out, and then pull out another marble. What is
the probability of pulling out a red marble followed by a blue
marble?
We can still use the multiplication rule which says we need to find
P(red) P(blue). But be aware that in this case when we go to pull
out the second marble, there will only be 8 marbles left in the
bag.
P(red) = 2/9
P(blue) = 3/8
P(red blue) = (2/9)(3/8) = 6/72 = 1/12
The events in this example were dependent. When the first
marble was pulled out and kept out, it effected the probability of
the second event. This is what is meant by dependent events.

41. Rule of Subtraction
The probability of an event ranges from 0 to 1.
The sum of probabilities of all possible events equals 1.
The rule of subtraction follows directly from these
properties.
Rule of Subtraction The probability that event A will occur
is equal to 1 minus the probability that event A
will not occur.
P(A) = 1 - P(A')
Suppose, for example, the probability that Bill will graduate
from college is 0.80. What is the probability that Bill will
not graduate from college? Based on the rule of
subtraction, the probability that Bill will not graduate is
1.00 - 0.80 or 0.20.

42. Summary
Two events are mutually exclusive or disjoint if they cannot
occur at the same time.
The probability that Event A occurs, given that Event B has
occurred, is called a conditional probability. The conditional
probability of Event A, given Event B, is denoted by the
symbol P(A|B).
The complement of an event is the event not occuring. The
probability that Event A will not occur is denoted by P(A').
The probability that Events A and B both occur is the
probability of the intersection of A and B. The probability of
the intersection of Events A and B is denoted by P(A ∩ B). If
Events A and B are mutually exclusive, P(A ∩ B) = 0.

43. The probability that Events A or B occur is the probability of
the union of A and B. The probability of the union of Events
A and B is denoted by P(A ∪ B) .
If the occurrence of Event A changes the probability of Event
B, then Events A and B are dependent. On the other hand, if
the occurrence of Event A does not change the probability
of Event B, then Events A and B are independent.

44. PROBABILITY REVISION USING BAYES’
THEOREM

45. Example

46. • A local bank reviewed its credit card policy with the intention of
recalling some of its credit cards. In the past approximately 5% of
cardholders defaulted, leaving the bank unable to collect the
outstanding balance. Hence, management established a prior
probability of .05 that any particular cardholder will default. The bank
also found that the probability of missing a monthly payment is .20 for
customers who do not default. Of course, the probability of missing a
monthly payment for those who default is 1.
• a. Given that a customer missed one or more monthly payments,
compute the posterior probability that the customer will default.
• b. The bank would like to recall its card if the probability that a
customer will default is greater than .20. Should the bank recall its card
if the customer misses a monthly payment? Why or why not?

47. • A consulting firm submitted a bid for a large research project. The
firm’s management initially felt they had a 50-50 chance of getting
the project. However, the agency to which the bid was submitted
subsequently requested additional information on the bid. Past
experience indicates that for 75 percent of successful bids and 40
percent of unsuccessful bids, the agency requested additional
information
• A. What is the prior probability of the bid being successful
• B. What is the conditional probability of a request for additional
information given that the bid will ultimately be successful
• C. Compute the posterior probability that the bid will be successful
given a request for additional information

48. • In August 2012, tropical storm Isaac formed in the Caribbean and was headed
for the Gulf of Mexico. There was an initial probability of .69 that Isaac would
become a hurricane by the time it reached the Gulf of Mexico (National
Hurricane Center website, August 21, 2012).
• a. What was the probability that Isaac would not become a hurricane but
remain a tropical storm when it reached the Gulf of Mexico?
• b. Two days later, the National Hurricane Center projected the path of Isaac
would pass directly over Cuba before reaching the Gulf of Mexico. How did
passing over Cuba alter the probability that Isaac would become a hurricane by
the time it reached the Gulf of Mexico? Use the following probabilities to
answer this question. Hurricanes that reach the Gulf of Mexico have a .08
probability of having passed over Cuba. Tropical storms that reach the Gulf of
Mexico have a .20 probability of having passed over Cuba.
• c. What happens to the probability of becoming a hurricane when a tropical
storm passes over a landmass such as Cuba?