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# Graded Homework PowerPoint PPT Presentation

Graded Homework. P. 163, #29 P. 170, #35, 37. Graded Homework, cont. P. 163, #29 U. Of Pennsylvania 1,033 admitted early (E) 854 rejected outright (R) 964 deferred (D) Typically 18% of deferred early admission are admitted in the regular admission process (173.5)

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#### Presentation Transcript

P. 163, #29

P. 170, #35, 37

• P. 163, #29

• U. Of Pennsylvania

• 854 rejected outright (R)

• 964 deferred (D)

• Total number of students admitted = 2,375

P(E) = 1033/2851 = 0.362

P(R) = 854/2851 = 0.300

P(D) = 964/2851 = 0.338

Yes, a student cannot be both admitted and deferred, so P(E∩D)=0

1033/2375 = 0.435

1033/2851 + (964/2851)(.18) = 0.423

P. 170, 35

P. 170, 35

P(Manager|Female) = 0.17/0.46 = 0.37

P(Precision production|Male) = 0.11/0.54 = 0.20

No, P(Manager|Female) = 0.37, P(Manager) = 0.34

P. 170, 37

P(PC) = .37P(Y) = .14

P(Y|PC) = .19P(O|PC) = .81

P(PC|Y) = P(Y ∩ PC)/P(Y) = [P(PC)P(Y|PC)]/P(Y) = [(.37)(.19)]/(.14) = 0.5

P(PC|O) = P(O ∩ PC)/P(O) = [P(PC)P(O|PC)]/P(O) = [(.37)(.81)]/(.86) = 0.35

People under 24 years old are more likely to use credit cards.

Yes, otherwise they can’t establish a credit history and the companies want customers who will make heavy use of the cards. They could put strict limits on the maximum balance for the card.

### Multiplication Law

P(A ∩ B) = P(B)P(A|B)

or

P(A ∩ B) = P(A)P(B|A)

### Multiplication Rule, cont.

If events A and B are independent then P(A|B) = P(A)P(B). In this special case the multiplication rule reduces from:

P(A ∩ B) = P(B)P(A|B)

to:

P(A ∩ B) = P(B)P(A)

### Tree Diagram

Assume we take the four aces out of a deck of cards and we draw twice with replacement:

Are A and B statistically independent in this case?

### Sampling and Statistical Independence

If we sample without replacement the outcomes will not be statistically independent.

However, if we are drawing from a large population the change in probability will be so small we can treat the draws as being statistically independent.

### Bayes’ Theorem

A technique used to modify a probability given additional information.

### Bayes’ Theorem, cont.

Assume that 10% of the population has a disease. Assume there is a test to see if someone has the disease but it is not very accurate.

### Bayes’ Theorem, cont.

Assume we want to calculate the probability that someone has the disease if the test says they have the disease.

### Bayes’ Theorem, cont.

.08

.02

.27

.63

A1 = Has the disease

B = Test says the patient has the disease

P(A1) = .1P(A2) = .9

P(B|A1) = .8P(B|A2) = .3

### Practice

Assume that 40% of a company’s parts are produced in Boston and 60% are produced in Chicago. Also assume that 20% of the parts produced in Boston are defective, and 10% of the parts produced in Chicago are bad.

A randomly chosen part is defective. Use Bayes Theorem to find the probability the part came from Boston.

### Bayes’ Theorem, cont.

.08

.32

.06

.54

A1 = Boston

B = Part is defective

P(A1) = .4P(A2) = .6

P(B|A1) = .2P(B|A2) = .1

### Counting Rules

• Number of possible outcomes

• Combinations

• Permutations

### Number of Possible Outcomes

Given k steps (or rounds) in an experiment and ni possible outcomes at step i, the total number of possible outcomes is:

(n1) (n2)…(nk)

### Number of Possible Outcomes, cont.

• Assume a diner can choose:

• One of three main dishes (beef, chicken or vegetarian)

• Either potatoes or beans

• How many possible meals are there?

• (2)(3)(2) = 12

### Number of Possible Outcomes, cont.

• Assume car buyer can choose:

• automatic or standard transmission

• 6 different colors

• 3 body styles

• 4 different accessory packages

• How many possible different outcomes are there?

• (2)(6)(3)(4) = 144

### Factorials

N! = (N)(N-1)(N-2)…(2)(1)

5! = (5)(4)(3)(2)(1) = 120

What is the value of 4! ?

(4)(3)(2)(1) = 24

What is the value of 5!/3! ?

[(5)(4)(3)(2)(1)]/[(3)(2)(1)] = (5)(4) = 20

By definition 0! = 1

### Permutations

The number of permutations of N objects taken n at a time:

### Permutations, cont.

Assume a broker is going to pick 3 stocks from a pool of 10 stocks. Also assume he will invest 60% of his money in one stock, 30% in another, and 10% in another. How many portfolios can be constructed?

### Combinations

The number of combinations of N objects taken n at a time:

### Combinations, cont.

A bank is constructing a bond based on mortgages. It is going to base the bond from four mortgages, it has ten mortgages to choose from. How many ways can the bond be structured?

### Practice

Three employees will be chosen from an office of 8 workers for a committee to evaluate a new production technique. How many possible committees could be formed?

Assume a club has 5 members and they are going to elect a president, treasurer, and secretary. How many ways can the offices be filled?

A magazine is going to recommend two of ten products to its readers. It will identify the rankings of the two products that are selected. How many potential rankings are there?