Basic Terminology:-

• Trial & Event

• Exhaustive number of cases

• Favorable number of cases

• Mutually Exclusive events

• Independent events

• Classical Method:-

Assigning probabilities based on the assumption of equally likely outcomes.

cases of no. Exhaustive

cases of no. favorableeventan ofy probabilit

• Relative Frequency Method:- Assigning probabilities based on experimentation or historical data.

Probability rules:-

1. The case where one event or another event occur.

2. The case where two or more events will both occur.

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 IStrongly support 9 10Mildly support 11 3Undecided 2 2Mildly oppose 4 8Strongly 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?

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)

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

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 Age1. Male 302. Male 323. Female 454. Female 205. 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 02 0.09 0.12 0.06

Which station has the higher pbt of being shut down?

Pbts under conditions of Statistical independence:

1. Marginal / Unconditional pbts

2. Joint pbts

3. Conditional pbts

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)

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?

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?

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

Similarly the conditional pbt of A given that already the event B happened is given by

P(A)

P(AB)P(B/A)

P(B)

P(AB)P(A/B)

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)

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?

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?

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

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?

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.

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.

Patron Reference Periodicals Books

Undergraduate 44 26 72

Graduate 24 61 20

Faculty 16 69 18

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?

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.

NewNewInformationInformation

NewNewInformationInformation

ApplicationApplicationof Bayes’of Bayes’TheoremTheorem

ApplicationApplicationof Bayes’of Bayes’TheoremTheorem

PosteriorPosteriorProbabilitiesProbabilities

PosteriorPosteriorProbabilitiesProbabilities

PriorPriorProbabilitiesProbabilities

PriorPriorProbabilitiesProbabilities

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?

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

Let X be the event of its being defective

Prior pbts:

P(E1)=25%

P(E2)=35%

P(E3)=40%

Likelihood pbts:

P(X/E1)=5%

P(X/E2)=4%

P(X/E3)=2%

Additional information:

A defective bolt was selected from the output.

To find the chance that it was produced by machine B we apply the Bayes’ theotem and is given by

)|(P)(P...)|(P)(P)|(P)(P

)|(P)(P)|(

2211 nn

iii EXEEXEEXE

EXEXEP

)E|)P(XP(E)E|)P(XP(E)E|)P(XP(E)E|)P(XP(E

X)|P(E332211

22

2

0.40586928

345140

2)(0.40)(0.04)(0.35)(0.05)(0.25)(0.04)(0.35)(0.0

Events

Ei

Prior Probabilities

P(Ei)Conditional Probabilities P(X|Ei)

Joint Probabilities P(Ei ∩ X)

Posterior Probabilities

P(Ei |X)

E1 0.25 0.05 0.0125 0.3623

E2 0.35 0.04 0.014 0.4058

E3 0.40 0.02 0.008 0.2319

Ex: A proposed shopping center will provide strong competition for downtown businesses like John Clothiers. If the shopping center is built, the owner of John Clothiers feels it would be best to relocate.

The shopping center cannot be built unless a zoning change is approved by the town council. The planning board must first make a recommendation, for or against the zoning change, to the council.

Let:

A1 = town council approves the zoning change

A2 = town council disapproves the Zoning change

• Prior Probabilities

Using subjective judgment: P(A1) = 0.7, P(A2) = 0.3

• New Information The planning board has recommended against the zoning change. Let B denote the event of a negative recommendation by the planning board. Given that B has occurred, should John Clothiers revise the probabilities that the town council will approve or disapprove the zoning change?

• Conditional Probabilities Past history with the planning board and the town council indicates the following:

P(B|A1) = 0.2 P(B|A2) = 0.9

• To find the posterior probability that event Ai will occur given that event B has occurred we apply Bayes’ theorem.

• Bayes’ theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space.

P A BA B A

A B A A B A A B Aii i

n n

( | )( ) ( | )

( ) ( | ) ( ) ( | ) ... ( ) ( | )

P P

P P P P P P1 1 2 2

P A BA B A

A B A A B A A B Aii i

n n

( | )( ) ( | )

( ) ( | ) ( ) ( | ) ... ( ) ( | )

P P

P P P P P P1 1 2 2

• Posterior ProbabilitiesGiven the planning board’s recommendation not to approve the zoning change, we revise the prior probabilities as follows.

= 0.34• Conclusion

The planning board’s recommendation is good news for John Clothiers. The posterior probability of the town council approving the zoning change is 0.34 versus a prior probability of 0.70

P A BA B A

A B A A B A( | )

( ) ( | )( ) ( | ) ( ) ( | )1

1 1

1 1 2 2

P PP P P P

P A BA B A

A B A A B A( | )

( ) ( | )( ) ( | ) ( ) ( | )1

1 1

1 1 2 2

P PP P P P

(. )(. )(. )(. ) (. )(. )

7 27 2 3 9

(. )(. )(. )(. ) (. )(. )

7 27 2 3 9

Tabular Approach:

Events

Ai

Prior Probabilities

P(Ai)

Conditional Probabilities P(B|

Ai)Joint Probabilities

P(Ai ∩ B)

Posterior Probabilities

P(Ai |B)

A1 0.7 0.2 0.14 0.3415

A2 0.3 0.9 0.27 0.6585

1. A doctor has decided to prescribe two new drugs to 200 heart patients as follows: 50 get drug A, 50 get drug B, 100 get both. The 200 patients were chosen so that each had an 80% chance of having a heart attack if given neither drug. Drug A reduces the chance of a heart attack by 35%, drug B reduces the same by 20%, and the two drugs, when taken together, work independently. If a randomly selected patient in the program has a heart attack, what is the pbt that the patient was given both drugs?

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%?