1. Basic Terminology:-
• Trial & Event
• Exhaustive number of cases
• Favorable number of cases
• Mutually Exclusive events
• Independent events
2. • Classical Method:-
Assigning probabilities based on the assumption of
equally likely outcomes.
favorable no. of cases
probabilit y of an event
Exhaustive no. of cases
3. • Relative Frequency Method:-
Assigning probabilities based on
experimentation or historical data.
4. Probability rules:-
1. The case where one event or another
event occur.
2. The case where two or more events will
both occur.
5. Ex:-Union shop steward Peter has drafted a set of wage and benefit
demands to be presented to management. To get an idea of
worker support for the package, he randomly polls the two largest
groups of workers at his plant, The machinists (M) and the
inspectors (I). He polls 30 of each group with the following results:
Opinion of package M I
Strongly support 9 10
Mildly support 11 3
Undecided 2 2
Mildly oppose 4 8
Strongly oppose 4 7
i. What is the pbt that a machinist randomly selected from the polled
group mildly supports the package?
ii. What is the pbt that an inspector randomly selected from the
polled group is undecided about the package?
iii. What is the pbt that a worker (machinist or inspector) randomly
selected from the polled group strongly or mildly supports the
package?
iv. What types of pbts are these?
6. Addition rule:
If A and B are any two events and are
disjoint then
P(A or B)= The Pbt of either A or B
happening
= P(A)+P(B)
7. If A and B are any two events and are not
disjoint then
P(A or B)= The Pbt of either A or B
happening
= P(A)+P(B)-P(AB)
P(A) is that Pbt of A happening
P(B) is that Pbt of B happening
P(AB) is that Pbt of A and B happening
together
8. Ex: A leading marketing research firm in
India, wants to collect information about
households with computers and Internet
access in urban Mumbai. After conducting
an intensive survey, it was revealed that
60% of the households have computers
with Internet access; 70% of the
households have two or more computer
sets. Suppose 50% of the households
have computers with Internet connection
and two or more computers. A household
with computer is randomly selected.
9. 1. What is the pbt that the household has
computers with Internet access or two or
more computers?
2. What is the pbt that the household has
computers with Internet access or two or
more computers but not both?
3. What is the pbt that the household has
neither computers with Internet access
nor two or more computers?
10. Ex: A company is interested in understanding
the consumer behavior of the capital of the
newly formed state Chattisgarh, Raipur. For
this purpose, a company has selected a
sample of 300 consumers and asked a
question, “Do you enjoy shopping?” Out of
300 respondents 200 were males, 100 were
females. Out of 200 males, 120 responded
“yes” and out of 100 females, 70 responded
“yes”.
A respondent is selected randomly. Construct
a pbt matrix and find the following pbts:
11. • The respondent is a male
• Enjoy shopping
• Is a female and enjoys shopping
• Is a male and does not enjoy shopping
• Is a female or enjoys shopping
• Is a male or does not enjoy shopping
• Is a male or female
12. 1. The employees of a certain company have elected 5 of their number to
represent them on the employee-management productivity council. Profiles
of the 5 are as follows:
Gender Age
1. Male 30
2. Male 32
3. Female 45
4. Female 20
5. Male 40
This group decides to elect a spokesperson by drawing a name from a chit.
What is the Pbt the spokesperson will be either female or over 35?
2. An inspector of the Alaska pipeline has the task of comparing the reliability
of two pumping stations. Each station is susceptible to two kinds of failure:
pump failure and leakage. When either (or both) occur, the station must be
shut down. The data at hand indicate that the following pbts prevail:
Station P (Pump failure) P (Leakage) P (Both)
1 0.07 0.10 0
2 0.09 0.12 0.06
Which station has the higher pbt of being shut down?
13. Pbts under conditions of Statistical
independence:
1. Marginal / Unconditional pbts
2. Joint pbts
3. Conditional pbts
14. If A & B are two independent events then
the Joint pbt of A and B is given by
P(A and B)=P(AB)=P(A)P(B)
If A & B are two independent events then
the conditional pbt of B given that A is
given by
P(B/A)=P(B)
Similarly the conditional pbt of A given that
B is given by P(A/B)=P(A)
15. 1.A bag contains 32 marbles:4 are red, 9 are
black, 12 are blue, 6 are yellow, and 1 is
purple. Marbles are drawn one at a time
with replacement. What is the pbt that
i. The second marble is yellow given
the first was yellow?
ii. The second marble is yellow given
the first was black?
iii. The third marble is purple given both
the first and second were purple?
16. 2. The health dept. routinely conducts two independent
inspections of each restaurant, with the restaurant
passing only if both inspectors pass it. Inspector A is
very experienced, and hence passes only 2% of
restaurants that actually do have health code violations.
Inspector B is less experienced and passes 7% of
restaurants with violations.
What is the pbt that
a). Inspector A passes a restaurant with a violation,
given that inspector B has found a violation?
b). Inspector B passes a restaurant with a violation,
given that inspector A passes it?
c). A restaurant with a violation is passed by the
health department?
17. 3. Unique Pvt. Ltd is a company involved in
the production of small bearings. One day
an important machine stops working. The
company has three operators. Their
chances of repairing machine are:
½,1/3,1/4 respectively. What is the
probability that the machine will be
repaired when they try independently?
18. Pbts under conditions of Statistical dependence:
1. Conditional pbts:
If A and B are any two events then the
conditional pbt of B given that already the
event A happened is given by
P(AB)
P(B/A)
P(A)
Similarly the conditional pbt of A given that
already the event B happened is given by
P(AB)
P(A/B)
P(B)
19. 2. Joint pbts: (Multiplication law of pbt)
If A and B are any two events then
P(A and B)=P(AB)=P(A)P(B/A)
=P(B)P(A/B)
20. Let a box contains 10 balls distributed as below
• Three are colored and dotted
• One is colored and striped
• Two are gray and dotted
• Four are gray and striped
a) Suppose a ball drawn from the box and found to
be colored. What is the pbt that it is dotted?
b) Suppose a ball drawn from the box and found to
be colored. What is the pbt that it is striped?
c) Suppose a ball drawn from the box and found to
be gray. What is the pbt that it is dotted?
d) Suppose a ball drawn from the box and found to
be gray. What is the pbt that it is striped?
21. d) Suppose a ball is drawn from the
box, find the pbt that it is colored and
sriped?
e) Suppose a ball is drawn from the
box, find the pbt that it is colored?
22. 1.Two events A and B are statistically
dependent. If P(A)=0.39, P(B)=0.21and
P(A or B)=0.47, find the pbt that
a). Neither A nor B will occur
b). Both A & B will occur
c). B will occur, given that A has
occurred.
d). A will occur, given that B has
occurred
23. 2. During a study of auto accidents, the
Highway safety found that 60% of all
accidents occur at night. 52% are alcohol-
related, and 37% occur at night and are
alcohol- related.
a) What is the pbt that an accident was
alcohol-related, given that it occurred at
night?
b) What is the pbt that an accident
occurred at night, given that it was alcohol-
related?
24. 3. The university’s library has been randomly surveying patrons over the
last month to see who is using the library and what services they
have been using. Patrons are classified as undergraduate, graduate,
or faculty. Services are classified as reference, periodicals, or books.
The data for 350 people are given below. Assume a patron uses only
one service per visit.
Patron Reference Periodicals Books
Undergraduate 44 26 72
Graduate 24 61 20
Faculty 16 69 18
Find the pbt that a randomly selected chosen patron
a) Is a graduate student
b) Visited the periodicals section, given that the patron is a
graduate
c) Is a faculty member, given a reference section visit.
d) Is an undergraduate who visited the book section.
25. 4. The southeast regional manager of General
Express, a private parcel – delivery firm, is
worried about the likelihood of strikes by some of
his employees. He has learned that the chance
of a strike by his pilots is 0.75 and the chance of
a strike by his drivers is 0.65. Further, he knows
that if the drivers strike , there is a 90% chance
that the pilots will strike in sympathy.
a) What is the pbt of both group’s striking?
b) If the pilots strike, what is the pbt that the
drivers will strike in sympathy?
26. Posterior probabilities or Bayes’ theorem:-
• Often we begin probability analysis with initial or prior
probabilities.
• Then, from a sample, special report, or a product test we
obtain some additional information.
• Given this information, we calculate revised or posterior
probabilities.
• Bayes’ theorem provides the means for revising the prior
probabilities.
Application
Prior New Posterior
of Bayes’
Probabilities Information Probabilities
Theorem
27. 1.In a bolt factory machines A, B and C
manufactures respectively 25%,35% and
40% of the total output. Of their output 5%,
4%,2% are defective bolts. A bolt is drawn
from the output and is found to be
defective. What is the chance that it was
produced by machine B?
28. Let
E1 be the event of drawing a bolt at
random manufactured by the machine A
E2 be the event of drawing a bolt at
random manufactured by the machine B
E3 be the event of drawing a bolt at
random manufactured by the machine C
32. To find the chance that it was produced by
machine B we apply the Bayes’ theotem and is
given by
P( Ei )P( X | Ei )
P( Ei | X )
P( E1 )P( X | E1 ) P( E2 )P( X | E2 ) ... P( En )P( X | En )
35. 1. T.C.Fox, marketing director for Metro-Goldmine Motion Pictures,
believe that the studio’s upcoming release has a 60% chance of
being a hit, a 25% chance of being a moderate success, and a 15%
chance of being a flop. To test the accuracy of his opinion, T.C.Fox
has scheduled two test screenings. After each screening, the
audience rates the film on scale of 1 to 10, 10 being best. From his
long experience in the industry, T.C. Fox knows that 60% of the time
a hit picture will receive a rating of 7 or higher; 30% of the time, it will
receive a rating of 4,5, or6; and 10% of the time, it will receive a
rating of 3 or lower. For moderately successful picture, the
respective pbts are 0.30, 0.45, and 0.25; for a flop film, respective
pbts are 0.15, 0.35, and 0.50.
a) If the first test screening produces a score of 6, what is the pbt that
the film will be a hit?
b) If the first test screening produces a score of 6 and the second test
screening yields a score of 2, what is the pbt that the film will be a
flop assuming that the screening results are independent of each
other?
36. Case Let:
A state Democratic official has decided that changes in the state
unemployment rate will have a major effect on his party’s chance
of gaining or losing seats in the state senate. He has determined
that if unemployment rises by 2% or more, the respective pbts of
losing more than 10 seats, losing 6 to 10 seats, gaining or losing
5 or less seats, gaining 6 to 10 seats , and gaining more than 10
seats are 0.25, 0.35, 0.15, 0.15, and 0.10 . If unemployment
changes by less than 2% , the respective pbts of losing more
than 10 seats, losing 6 to 10 seats, gaining or losing 5 or less
seats, gaining 6 to 10 seats , and gaining more than 10 seats are
0.10, 0.10, 0.15, 0.35, and 0.30. If unemployment falls by 2% or
more, the respective pbts of losing more than 10 seats, losing 6
to 10 seats, gaining or losing 5 or less seats, gaining 6 to 10
seats , and gaining more than 10 seats are 0.05, 0.10, 0.10,
0.40, and 0.35. Currently this official believes that unemployment
will by 2% or more with pbt 0.25, change by less than 2% with
pbt 0.45, and fall by 2% or more with pbt 0.30.
a) If the Democrats gained seven seats, what is the pbt that
unemployment fell by 2% or more?
b) If the Democrats lost one lost seat, what is the pbt that
unemployment changed by less than 2%?
