Probability concepts and procedures law of profitability

Profitability

1. CHAPTER I
Probability
Concepts and procedures
Laws of Probability
Conditional and Baye’s Law

2. Experiment
An experiment is a process that produces outcomes.
•Examples of business oriented experiments with
outcomes that can be statistically analyzed might
include the following.
•Interviewing 20 randomly selected consumers and
asking them which brand of appliance they prefer
•Sampling every 200th bottle of ketchup from an
assembly line and weighing the contents
•Testing new pharmaceutical drugs on samples of
cancer patients and measuring the patients’
improvement
•Auditing every 10th account to detect any errors
•Recording the Dow Jones Industrial Average on the
first Monday of every month for 10 years

3. Event
An event is an outcome of an experiment
•The experiment defines the possibilities of the
event.
•In an experiment to roll a die, one event could
be to roll an even number and another event
could be to roll a number greater than two.

4. Sample Space
•A sample space is a complete roster or listing
of all elementary events for an experiment.
•The sample space for the roll of a single die is
{1, 2, 3, 4, 5, 6}.
•Similarly thes ample space for the roll of a pair
of dice is
(1,1) (2,1) (3,1) (4,1) (5,1) (6,1) (1,2) (2,2) (3,2)
(4,2) (5,2) (6,2) (1,3) (2,3) (3,3) (4,3) (5,3) (6,3)
(1,4) (2,4) (3,4) (4,4) (5,4) (6,4) (1,5) (2,5) (3,5)
(4,5) (5,5) (6,5) (1,6) (2,6) (3,6) (4,6) (5,6) (6,6)

5. Mutually Exclusive Events
•Two or more events are mutually exclusive events
if the occurrence of one event prevents the
occurrence of the other event(s).
•This characteristic means that mutually exclusive
events cannot occur simultaneously and therefore
can have no intersection.
•In coin toss experiment two sides (head and tail)
cannot appear at the same time.
•A manufactured part is either defective or okay: The
part cannot be both okay and defective at the same
time because “okay” and “defective” are mutually
exclusive categories.

6. Events Eand Fare
Mutually Exclusive
Events E, F and G
are Mutually
Exclusive

7. Independent Events
Two or more events are independent events if the
occurrence or nonoccurrence of one of the events
does not affect the occurrence or nonoccurrence
of the other event(s).
For example;
•Rolling dice, yield independent events; each die is
independent of the other Whether a 6 is rolled on
the first die has no influence on whether a 6 is
rolled on the second die.
•Coin tosses always are independent of each other.
The event of getting a head on the first toss of a
coin is independent of getting a head on the second
toss.

8. 8

9. Approaches of probability

10. I. Classical Probability
Consider an experiment of rolling a six-sided die. What is the probability of
the event “an even numberof spots appear face up”?
The possible outcomes are:
There are three “favorable” outcomes (a two, a four, and a six) in the
collection of six equally likely possible outcomes.

11. II. Empirical or statistical or relative frequency
•This approach assumes that the probability of
happening an event is calculated based on logical
reasoning.
•It is based on the statistical data.
•This approach is based on the principle of law of
large number (If an experiment is repeated fairly a
large number of times under the similar condition
then accurate calculation is obtained)
•The key to establishing probabilities empirically is
that more observations will provide a more accurate
estimate of the probability.

12. Law of Large Numbers
Suppose we toss a fair coin. The result of each toss is either a head
or a tail. If we toss the coin a great number of times, the probability
of the outcome of heads will approach .5. The following table
reports the results of an experiment of flipping a fair coin 1, 10, 50,
100, 500, 1,000 and 10,000 times and then computing the relative
Frequency of heads

13. Let n be the number of events, m be the favorable
event to event E then empirical probability can be
written as
The probability of an event Eis approximately
the number of times event Eis observed
divided by the number of repetitions of the
experiment.n
m
ltEP
n)(

14. III. Subjective Probability
•This approach is entirely based on the beliefs,
experience and judgment of individual to a certain
activity.
•Due to the variation in attitude and opinion of
different person, it is almost impossible to reach at
n objective decision.
•It may be useful for the decision making for the
quality related issues.
•For example, there is a 40% chance of raining
tomorrow

15. Properties of probability
1.The probability of any event E, P(E), must
be between 0 and 1 inclusive. That is,
0 <P(E) <1
2. If an event is impossible, the probability of
the event is 0.
3. If an event is a certainty, the probability of
the event is 1.
4. If S= {e
1, e
2, …, e
n}, then
P(e
1) + P(e
2) + … + P(e
n) = 1.

16. Addition theorem
Case I: If the cases are not mutually exclusive
Let A and B are two independent (not mutually
exclusive) events then the probability of happening
either A or B is
P(A or B) = P(A) + P(B) –P(A and B)
This can further be written as
P (A U B)= P(A) + P(B)-P (A and B)
Similarly for three events A, B and C
P(AUBUC)=P(A)+P(B)+P(C)-P(A and B)-P(B and
C)-P(A and C)+ P(A and B and C)

17. Case II: If the events are mutually exclusive
If A and B are two mutually exclusive events then
the probability of happening either A or B is
P(A or B) = P(A) + P(B)
This can further be written as
P (A U B)= P(A) + P(B)
In mutually exclusive case no common part exist
between the events.
Similarly for three events A, B and C
P(AUBUC)=P(A)+P(B)+P(C)

18. Multiplicative law of probability
Let A and B are two independent events then the
probability of occurring of two events is
P (A and B) = P(A). P(B)
Similarly for three events A, B and C
P (A and B and C) = P(A). P(B). P(C)
However, if the events A and B are dependent then the
probability can be computed as
P(A and B) = P(A). P(B/A) where P(B/A) is the probability
of occurring event B when event A has already occurred.
Likewise,
P (A and B) = P (B). P(A/B)Interpret P(A/B)?????

19. Conditional Probability
Let A and B are the events of random experiment.
The probability of happening event A when event
B has already occurred is known as conditional
Probability. The conditional probabilities for events A and B
can be written as
Where P(A/B) is the conditional probability of A when B has
already occurred
Similarly P(B/A) is the conditional probability of B when A
has already occurred. 0)(
)(
)(
)/(
0)(
)(
)(
)/(


AP
AP
BAP
ABP
BP
BP
BAP
BAP



20. Example: The probability that a manufacturer will produce ‘brand X’
product is 0.13, the probability that he will produce ‘brand Y’ product
is 0.28 and the probability that he will produce both brand is 0.06.
What is the probability that the manufacturer who has produced
‘brand Y’ will also have produced ‘brand X’?

21. Age (Years)
Bachelor degree
only (B.E.)
Master degree
only (M.E.)
Total
Under 25
60 8 68
25 to 30
7 26 33
over 30
9 10 19
Total
76 44 120
•Thepersonneldepartmentofa
companyhasrecordswhichshowthe
followingofits120engineersIfone
engineerisselectedatrandom,findthe
probability that
hehasonlyabachelor’sdegree
he isunder 25 years.
hehasamaster’sdegree,giventhatheis
over 30.
heis25to30yearsgiventhathehas
bachelor’s degree.
heisover30andhasamaster’sdegree.
Solution:
Total number of engineers = 120
Let event A = an engineer chosen is
bachelor’s degree , B = an engineer
chosen is Masters' degree, C = an
engineer chosen is under age 25
years, D = an engineer chosen is 25
to 30 years and E = an engineer
chosen is over 30 years
If one engineer is selected at
random, then

22. Example: An urn X contains 3 white balls and 5 black balls. Another urn Y contains 6 white
balls and 4 black balls. A ball is transferred from the urn X to urn Y and then a ball is taken
from urn Y. Find the probability that it will be a white ball.3 White
5 Black
6 White
4 Black
Transferred a ball
from urn X to Y
A ball
P (White ball) = ?
Urn YUrn X

23. Baye’s Theorem
Baye’s theorem is the extended form of conditional
Probability in which prior information is used to
compute the existing probability.
Conditional probability provides an information
about the happening of one event given that
specific event has already occurred.
But in Baye’s theorem, we compute the probability
of particular effect due to the specific cause
based on the concept that specific event has
already occurred.

24. Example
Let’s suppose there are three urns A, B and C
containing 8 balls each with white and blue balls.
Suppose an urn is chosen at random and a ball is drawn
from the chosen urn. It is easier to compute the
probability of drawing a white or blue from the selected
urn.
But if a ball is chosen randomly from these three urns and
found to be white, what is the probability that the
chosen ball is from A or B or C. In this condition, the
probability of selection of urn depends on the color of
ball.
To solve such problem Bayes' Theorem is extensively used
in management science.

26. Baye’s formula
Let the events E
1, E
2, , E
npartition the finite discrete
sample space Sfor an experiment and let Abe an event
defined on S., then Baye’s formula can be written as;
Now if n=2 then there are two
mutually exclusive events E1
and E3 then
S
E
1 E
2 E
3
A)
3
()
3
|()
2
()
2
|()
1
()
1
|()( EPEAPEPEAPEPEAPAP  

)/().(
)().(
)/(
i
EAP
i
EP
i
E
AP
i
EP
A
i
EP )()
2
/().
2
()
1
/().
1
(
)
1
/().
1
(
)/
1
(
APEAPEPEAPEP
EAPEP
AEP



27. For (E2/A), this can be written as
Now if there are three events (n=3) E1, E2and E3
then)(
)
2
/().
2
(
)/
2
(
AP
EAPEP
AEP  )/().()/().()/().(
)
1
/().
1
(
)/
1
(
332211 EAPEPEAPEPEAPEP
EAPEP
AEP

 )/().()/().()/().(
)/().
2
(
)/
2
(
332211
2
EAPEPEAPEPEAPEP
EAPEP
AEP

 )/().()/().()/().(
)/().
3
(
)/
3
(
332211
3
EAPEPEAPEPEAPEP
EAPEP
AEP



28. Tree Diagram of Baye’s Law
Let there are two events E1and E2 where A be an event
that occur within these two events then the tree diagram
can be depicted as follows.
E1A P(E1). P(A/E1)
E2A P(E2).P(A/E2)
P(E1)
P(E2)
P (A/E1)
P (A/E2 )
Probability
Conditional
Probability
Event
Probability

29. Similarly, the tree diagram for three events E1, E2
and E3can be shown as
Joint probability
P(E1)
P(E2)
P(E3)
P(A/E1)
P(A/E2)
P(A/E3)
P(E1). P(A/E1)
P(E2). P(A/E2)
P(E3). P(A/E3)
Conditional probability
Probability
Total probability P(A)= P(E1). P(A/E1) +P(E2). P(A/E2)+ P(E3). P(A/E3)

31. =P(A)
P(E1/A)=0.0715 P(E2/A)=0.286
Therefore, machine C has higher chance of getting defective item as compared to Machines A and B

32. E
3
1/6
2/6
3/6
E
1
E
2
0.01
A
A
A
0.02
0.03
Events Probability
E 
1
E 
2
E 
3
1/6×0.01 = 0.00167
2/6×0.02 = 0.0067
3/6×0.03 = 0.015
P(A) = 0.0233
Total Tree Diagram Method

33. Example of Cards
2, 3, 4, 5, 6, 7,
8, 9, 10
Pack /Deck =52
Color
26 Black and 26 Red
Black
26
Red
26
Spade =13Club =13
Heart =13Diamond =13