Math 13 Chapter 7 Handout Helene Payne …hpayne/Math 13/Handouts/Math13Handout7x...Math 13 Chapter 7 Handout Helene Payne Name: Probability Sets A set is a well-de ned collection

Math 13 Chapter 7 Handout Helene Payne

Name:

ProbabilitySets A set is a well-defined collection of distinct objects. Well-defined means that we candetermine whether an object is an element of a set or not. Distinct means that we can tellthe objects apart.

Basic Set Definitions

Well-defined Set The set of natural numbers between 3 and 7, inclusive: {1, 2, 3, 4, 5, 6, 7}.

The set of rational numbers:{ab|a, b are integers, b 6= 0

}.

Not Well-defined Set The set of large numbers.

Set with Non-distinct Objects {A,B,C,C}

Universal set - U In a given problem, the single larger set from which all the elements ofthe sets under discussion are drawn.

Empty set or Null set - ∅ or {} A set containing no elements.

Venn diagram A tool used in visualizing sets invented by John Venn (1834-1923).

U

BA


Set definitionsDefinition Example

∈ is an element of 2 ∈ {0, 1, 2, 3, 4, 5, 6}/∈ is not an element of 2 /∈ {1, 3, 5}⊆ is a subset of {1, 3, 5} ⊆ {0, 1, 2, 3, 4, 5, 6, 7, 8}

{1, 3, 5} ⊆ {1, 3, 5}* is not a subset of {0, 1, 2, 3, 4, 5, 6} * {1, 3, 5}⊂ is a proper subset of {3, 5} ⊂ {1, 3, 5}6⊂ is not a proper subset of {1, 3, 5} 6⊂ {1, 3, 5}= set equality {1, 3, 5} = {3, 1, 5}

The order of elements does not matter

1. Determine whether A is a subset of B. Draw a Venn diagram for each case.

(a) A = {2, 3, 4} and B = {1, 2, 3, 4, 5}

(b) A = {0, 2, 3, 4} and B = {1, 2, 3, 4, 5}

The Complement and Set Definitions involving Two Sets

For the examples in the table below, suppose set A = {0, 2, 4, 6, 8}, set B = {1, 3, 5, 7, 9},set C = {1, 2, 3, 4} and the universal set, U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Set definitions

Definition Example

∩ The intersection of two sets A ∩B = ∅A and B = the set of A ∩ C = {2, 4}

all elements in both A and B

∪ The union of two sets A and B = A ∪ C = {0, 1, 2, 3, 4, 6, 8}the set of all elements A ∪B = U

in either A or B or both− The complement of a set A, C = {0, 5, 6, 7, 8, 9}

A = all elements in the A = {1, 3, 5, 7, 9} = B

Universal set, U , not in set A

Two sets, A and B are disjoint, A and B are disjoint,if A ∩B = ∅. since A ∩B = ∅

B and C are not disjoint,since B ∩ C = {1, 3}

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2. For the sets A = {a, b, c}, B = {c, e, f}, C = {g, h} and the universal set,U = {a, b, c, d, e, f, g, h, i, j}, find

(a) A

(b) B

(c) A ∪B

(d) A ∩B

(e) ∅

(f) B ∩ C

3. Let the universal set, U be the set of all Foothill College students, let Ebe the set of all Foothill College students enrolled in an English class, letG be the set of all Foothill College students enrolled in a geology class,and let M be the set of all Foothill College students enrolled in a mathclass.

(a) Use the given set symbols to construct a Venn diagram identifyingthe various sets of students.

(b) Use the given set symbols along with ∩, ∪ or −, to identify the set ofstudents enrolled in English, but not in geology or math.

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4. By labeling regions in a Venn diagram, represent the following set:A ∩B ∪ C.

U

BA

C

Set Region Labels

A ∩BA ∩BC

A ∩B ∪ C

5. By labeling regions in a Venn diagram, show that: A ∪B = A ∩B.

U

BA

Set Region Set RegionLabels Labels

A A

B B

A ∪BA ∪B A ∩B

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Properties of Set AlgebraCommutative Laws1. A ∩B = B ∩ A 2. A ∪B = B ∪ A

Associative Laws1. A∩(B∩C) = (A∩B)∩C 2. A∪(B∪C) = (A∪B)∪C

Distributive Laws1. A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)2. A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)

Set Difference Law: A−B = A ∩BDouble Complement Law: (A) = A

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The Number of Elements in a Setn(A) denotes the number of elements in A, if A is a finite set.

6. Calculate n(A) for each set A below.

(a) A = {1,√

2, π, e,√

11, 4√

123}, n(A) =

(b) A = the letters of the alphabet, n(A) =

(c) A = ∅, n(A) =

Important Counting FormulasThe Inclusion-Exclusion Principle1. n(A∪B) = n(A) + n(B)− n(A ∩B), orn(A or B) = n(A) + n(B)− n(A and B)

The Complement Principle

2. n(A) = n(U)− n(A)

7. Verify the counting formulas above using the Venn diagram below. Eachelement is represented by a diamond.

U

BA

(a) n(A ∪B) = n(A) = n(B) = n(A ∩B) =

(b) n(A) = n(U) = n(A) =

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8. A Chevrolet dealer has 160 new cars for sale on her lot. Among thesecars,

• Fifty have four-cylinder engines.

• Eighty have a tilted steering wheel.

• Thirty have power windows.

• Forty-two have four-cylinder engines and a tilted steering wheel.

• Eighteen have four-cylinder engines and power windows.

• Fifteen have all three features.

• Sixty-five have none of the features.

Let F be the set of cars with four-cylinder engines, T be the set of carswith a tilted steering wheel, and let P be the set of cars with powerwindows. Use the Venn diagram to answer the questions below.

U

TF

P

How many of these cars have

(a) A tilted steering wheel and power windows?

(b) At least one of these features?

(c) Exactly two of these features

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9. Suppose 250 domestic CEO’s were surveyed about their company’s in-dustry type and geographic location in the United States. The CEO’swere allowed to choose only one industry type (manufacturing, communi-cations, or finance) and one location (Northeast, Southeast, Midwest, orWest). The results are given below:

North- South- Mid-east east west West

Manufacturing 37 8 27 15

Communications 35 23 15 20

Finance 30 12 10 18

Find

(a) The number of CEO’s whose response was not Southeast.

(b) The number of CEO’s whose response was Communications or West.

(c) The number of CEO’s whose response was Northeast but not Manu-facturing.

(d) The number of CEO’s whose response was Manufacturing or Com-munications or Midwest or West.

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The Multiplication Principle

cb

a

10. A box contains three slips of paper with the letters a written on one, theletter b written on another and the letter c written on the third slip. Aslip is drawn without replacement (=without putting it back), and theletter is recorded. Then a second slip is drawn and its letter is recorded.A typical outcome for this experiment is (b, a) if the first letter drawn wasa b and the second one was a c.

(a) List the set of outcomes, the sample space, S. How many differentoutcomes are there?

(b) Make a tree diagram of this experiment

Based on our findings in the tree diagram, suggest a formula for calcu-lating the number of outcomes for this experiment. The outcomes in theprevious experiment are called ordered arrangements since the order isimportant. (a, b) is a different outcome from (b, a).Number of outcomes:

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11. Repeat the previous experiment, but now, do it with replacement (=puttingthe slip back), after the letter is recorded.

(a) List the set of outcomes, i.e. the sample space, S. How many differentoutcomes are there now?

(b) Make a tree diagram of this experiment

(c) Use the formula found in the previous exercise to calculate the num-ber of outcomes when drawing two slips with replacement.

The Multiplication Principle of Counting.An experiment is performed where there are p selections for the firstchoice, q selections for the second choice, r selections for the third choiceand so on,

p =the number of selections for the first choiceq =the number of selections for the second choicer =the number of selections for the third choice...

Then the number of distinct ordered outcomes, n(S) is:

n(S) = p · q · r · · · · (1)

12. A coin is tossed 3 times and each time H (head) or T (tail) is recorded. Atypical outcome for this experiment is HTT which represents the outcomewhere the first toss was a head and the second and third tosses were tails.

(a) List the sample space for this experiment.

(b) Use the multiplication principle to determine the total number ofoutcomes for this experiment.

(c) Use the multiplication principle to determine how many outcomeshave a head on the first toss?

13. Tom is getting dressed for work. He will wear a shirt, a pair of pantsand a jacket. He owns 4 shirts, 3 pairs of pants and 2 jackets. How manydifferent outfits could he wear?

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14. How many different license plates could be made using three letters fol-lowed by three digits if

(a) there are no restrictions?

(b) no letter can be repeated?

(c) no letter can be repeated and zero cannot be used as the first digit?

15. In how many ways can five of nine people be seated in five chairs?

Permutations Factorials Counting problems often involve the productof consecutive numbers. To save on the amount of writing, we use thefactorial notation. For example, 3! is read ”three factorial” and is de-fined by:3! = 1 · 2 · 3 = 6, and6! = 1 · 2 · 3 · 4 · 5 · 6 = 720.

n FactorialLet n be a positive integer. Then the product of integers from 1 to n, n!,read ”n factorial” is:

n! = 1 · 2 · 3 · · · · · n. (2)

By definition, 0! = 1, to make all calculations work out properly.

16. Use your calculator to find:

(a) 10!

(b) 20!

(c) 50!

(d) 100!

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Permutations - ORDER IS IMPORTANTA permutation is an ordered arrangement of objects for which:

• All objects are selected from the same set, S.

• All objects are considered distinguishable, i.e. we can tell them apart.

• Successive selections from S are made without replacement.

The result is called an ordered arrangement.

The Number of Permutations of n Distinguishable Objects Takenr at a Time where 0 ≤ r ≤ n.

P (n, r) = n · (n− 1) · (n− 2) · · · · · (n− r + 1) (3)

=n!

(n− r)!

On your TI-83 or TI-84 calculator, you can find factorial (!) and P (n, r)(nPr on your calculator) on the MATH menu by pressing MATH→PRB.

To find 35!, press: 35→MATH→PRB→!To find P (7, 4), press: 7→ MATH→PRB→ nPr → 4

17. In how many ways can three out of seven executives be seated in a rowfor a corporate picture?

18. In how many ways can three people be elected president, treasurer andsecretary, in a chess club with 22 members?

19. In how many ways can we arrange 3 red books, 1 blue book and 1 greenbook on a shelf?

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Number of Distinguishable Arrangements with Indistinguish-able ObjectsLet S be a set of n elements, and let

k1 = the number of elements of type 1k2 = the number of elements of type 2k3 = the number of elements of type 3...km =the number of elements of type m

Then the number of distinguishable permutations of the n elements takenn at a time is:

n!

k1!k2!k3! · · · · · km!(4)

20. How many permutations are there of the letters in the word INTELLI-GIBLE?

21. In how many ways can three people be elected president, treasurer andsecretary, in a chess club with 22 members (9 female and 13 male) if atleast one of the positions needs to be filled by a female?

22. A firm has 750 employees. Explain why at least 2 of the employees wouldhave the same pair of initials for their first and last name.

23. For an experiment, 12 sociology students are to be divided into twogroups, one containing 7 students and the other containing 5 students.In how many ways can this grouping be done?

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Combinations - ORDER IS NOT IMPORTANTA combination is a group of objects for which:

• All objects are selected from the same set, S.

• All objects are considered distinguishable, i.e. we can tell them apart.

• Successive selections from S are made without replacement.

• The order in which they are chosen does not matter.

The result is called a combination, subset or group.

The Number of Combinations of n Distinguishable Objects Takenr at a Time where 0 ≤ r ≤ n.

C(n, r) =P (n, r)

r!=

n!

r!(n− r)!(5)

24. Use the formulas above to find following number of combinations:

(a) C(8, 3)

(b) C(7, 4)

On your TI-83 or TI-84 calculator, you can find factorial C(n, r) (nCr onyour calculator) on the MATH menu by pressing MATH→PRB.

Redo the exercise above using the nCr function on your calculator.

25. How many doubles tennis teams can be formed from 12 players?

26. Among 18 computers, 12 are in working order. How many samples of 4are possible, wherein

(a) all are in working order?

(b) exactly 2 are in working order?

(c) at least 1 is in working order?

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27. In how many ways can a 4-card hand be dealt if

(a) all if the cards in the hand are to be red cards?

(b) all are to be nines?

(c) all are to be from the same suit?

28. A committee of four is to be selected from among eight graduate studentsand a professor. The committee is to meet with the dean about newclassroom equipment. In how many ways can the committee be selectedif

(a) there are no restrictions?

(b) the professor must be in the committee?

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Sample Spaces and the Assignment of Probabilities.The foundations of modern probability theory was laid by the Frenchmathematicians, Blaise Pascal (1623 - 1665) and Pierre de Fermat (1601-1665). The need for probability theory came from games of chance (=gam-bling) as well as mortality tables and insurance rates.

Probability The chance or likelihood of an event occurring.

Experiment An activity or procedure that produces distinct, well-definedpossibilities called outcomes.

Outcomes Observations or measurements from a probability experiment,which cannot be predicted with certainty.

Sample Space, S The set of all possible outcomes, i.e. the universal setof outcomes.

Event An set of outcomes from a probability experiment. An event is asubset of S, the sample space.

Simple Event An event consisting of a single outcome in sample space.

29. A balanced coin is to be tossed, and the result of each toss is recorded.Find the probability of each outcome in the sample space.

Sample space, S = {

Assign a probability for each outcome in this experiment. List the prob-ability distribution in the table below:

Outcome, en

Probabilityof Outcome, P (en)

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30. Toss a coin 50 times and record number of heads and tails in the tablebelow.

Number Number Empirical Empiricalof Heads of Tails Probability Probability

of Heads of Tails

As you can see, the empirical probabilities are not usually equal to 12 ,

but close. When we say that the probability of getting a head is 12 we

mean that in the long run if the experiment is repeated over and over, onaverage, we will get a head 1

2 the time.

31. A pair of dice, one red and one green is to be rolled and the number ofpips on the upper face recorded. A typical outcome forthis experimentis (2, 3) meaning the red die showed a 2 and the green die showed a 3.Complete the table below listing the sample space for this experiment.Red Die 1 2 3 4 5 6Green Die

1

2

3

4

5

6

(a) How many outcomes are there? n(S) =

(b) What is the probability for each outcome in sample space?

(c) Let E be the event: E: A sum of 5 is rolled.List the outcomes in E:E = {

(d) What is the probability of rolling a sum of 5, i.e. find P (E):

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32. Complete the table below, listing the sum of the red and green dice:Red Die 1 2 3 4 5 6Green Die

1

2

3

4

5

6

(a) What is the probability of rolling a sum of 7?

(b) What is the probability of rolling a sum less than 7?

Probability of an OutcomeSuppose the sample space of an experiment has n outcomes given by:S = {e1, e2, . . . , en}The following must hold for the probabilities of these events:1. 0 ≤ P (e1) ≤ 1,

0 ≤ P (e2) ≤ 1,...0 ≤ P (en) ≤ 1- The probability of each outcome must be a number between 0 and 1.

2. P (e1 + P (e2) + . . .+ P (en) = 1- The sum of the probabilities of all outcomes of sample space must equal 1

The Probability of an EventLet S be a sample space with n = n(S) equally likely outcomes. The theprobability of each outcome is

1

n(S). (6)

If E is an event in this sample space with m = n(E) outcomes, then theprobability of E, P (E) is

P (E) =n(E)

n(S)=m

n. (7)

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33. A ball is picked at random from a box containing three red, five greenballs, seven blue and four white balls. What is the probability that

(a) a white ball is picked?

(b) a green ball is picked?

(c) a blue or red ball is picked?

34. Dr. Sun, a mathematics professor, gives a two-question, multiple-choicequiz in which each question has four possible answers: A,B,C, and D.Assume that the student guesses at the answers on this quiz.

(a) Write the outcomes of the sample space for this experiment.

(b) Assign a probability to each outcome.

(c) What is the probability that the student will answer both questionscorrectly?

(d) What is the probability that the student will answer one questioncorrectly?

35. Three girls and three boys will be randomly seated in a row. What is theprobability that

(a) boys will be seated on both ends?

(b) boys and girls will be alternately seated?

(c) a particular girl, Tonya, will be seated on the left end?

36. Among 30 microwave ovens, 10 are known to have defects. A sample of 5ovens is to be selected, without replacement and without regard to order.What is the probability that the sample will

(a) contain no defective ovens?

(b) contain exactly 2 defective ovens?

(c) contain all defective ovens?

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37. The number of new home sales (in thousands) in the United States as ofMarch, 24, 2010, is given below.

Northeast Midwest South West

Feb-Mar 2009 47 94 402 143

Apr-Jun 2009 82 607 607 278

Jul-Sep 2009 114 179 623 302

Oct-Dec 2009 102 174 578 253

Jan-Feb 2009 67 91 299 170

(a) What is the probability a home selected at random was sold in thesouth?

(b) What is the probability a home selected at random was sold fromApr-Jun 2009?

(c) What is the probability a home selected at random was sold in thesouth from Apr-Jun 2009?

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Properties of the Probability of an Event

Event; Simple EventAn event is any subset of a sample space. If an event has exactly oneelement, that is, if it consists of only one outcome, it is called a simpleevent.

38. Consider the experiment of selecting one family from the set of all possiblethree-child families. The simple event GGB is a family whose childrenare girl, girl, then boy, in that order.

(a) List the sample space for this experiment:S = {

(b) List the event that the family had a girl first, then two boys.E = {

(c) List the event, F that the family had exactly one boy.F = {

General Rules for Probability.

• The sum of the probabilities of the simple events of sample space, en,is always 1, i.e.,P (S) = P (e1) + · · ·+ P (en) = 1.

• The probability of any event, E, is always a number between 0 and1, i.e. 0 ≤ P (E) ≤ 1.

• The probability of an empty event, ∅, is always zero, i.e. P (∅) = 0.

• The probability of an event, E is found by adding the probabilities ofall simple events in sample space making up E, i.e. P (E) = P (e1) +P (e2) + · · ·+ P (er).

39. Suppose a particular experiment results in the sample space S = {a, b, c, d, e}.(a) If all outcomes in S are equally likely, what is the probability of a

single outcome?

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(b) If not all outcomes are equally likely, P (a) = 0.1, P (b) = 0.3, P (c) =0.2, and P (d) = 0.1, what is P (e)?

(c) If E = {a, b, d, e} is an event in S with probability 0.8, what is P (c)?

(d) Explain why the probability assignment P (a) = 0.4, P (b) = 0.3,P (c) = 0.1, P (d) = 0.4, and P (e) = −0.2 is not valid.

Mutually Exclusive Events.Two events, E and F are mutually exclusive if they cannot occur si-multaneously (they share no outcomes), i.e. E ∩ F = ∅.

Probability of E or F for Mutually Exclusive Events.If two events, E and F are mutually exclusive, i.e.E∩F = ∅, then the probability of E or F is the sum of their probabilities:P (E or F ) = P (E) + P (F ), orP (E ∪ F ) = P (E) + P (F ).

40. Let A and B be two mutually exclusive events of a sample space. IfP (A) = 0.35, and P (B) = 0.60, find each of the following probabilities.

(a) P (A ∩B)

(b) P (A ∪B)

(c) P (A)

(d) P (B)

(e) P (A ∪B)

All events are not mutually exclusive. Recall the ”Inclusion-ExclusionPrinciple” from section 7.2, we used to calculate the number elements inthe union of two sets. It is:

n(A ∪B) = n(A) + n(B)− n(A ∩B)

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The formula for calculating the probability of the union of two events isanalogous to the inclusion-exclusion principle and it is listed next:

Additive Rule for Probability: The Union of Two EventsFor any two events E and F in a sample space

P (E∪F ) = P (E) + P (F )− P (E ∩ F ), orP (E or F )= P (A) + P (B)− P (A and B)

41. If P (E) = 0.25, P (F ) = 0.35, and the probability of E or F is 0.55, whatis the probability of E and F?

42. Let S be the sample space in which the events A and B are such thatP (A) = 0.4, P (B) = 0.5, and P (A ∩B) = 0.2.

(a) Illustrate these probabilities with a Venn diagram.

(b) Use the Venn diagram to find the probability of the event A or B,i.e. A ∪ B, and compare the answer with P (A) + P (B). Are eventsA and B mutually exclusive?

(c) Use the Venn diagram to find the probability of A but not B, i.e.P (A ∩B).

43. An auditor of personal income tax forms submitted to a certain stategathered the following information from a randomly selected sample of300 forms:

Income Level Correct UnderstatedTax

OverstatedTax

$30, 000 −$70, 000

125 20 10

$70, 000 −$120, 000

110 30 5

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(a) Find the probability that a person will submit a correct form?

(b) Find the probability that a person will be in the $30, 000− $70, 000income level and will submit a taxform that understates his or hertax?

(c) Find the probability that a person will be in the $30, 000− $70, 000income level or will submit a taxform that understates his or hertax?

P (E ∪ E) = P (E) + P (E) = 1. From this we derive the formula for theprobability of the complement of an event E, P (E):

Probability of the Complement of an Event

P (E) = 1− P (E) (8)

44. For a certain weighted die, the probability of getting a six is 0.1. Whatis the probability of not getting a six?

45. What is the probability that a seven digit telephone number has one ormore repeated digits? Note that the telephone number cannot begin witha zero.

46. What is the probability that in a group of 20 people, at least 2 peoplehave the same birthday?

Odds Formula.If the odds for an event E is stated as a to b, then probability that eventE will occur, P (E) is:

P (E) =a

a+ b(9)

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and the probability of E, P (E) is:

P (E ′) =b

a+ b(10)

On the other hand, the odds for an event E, a to b are found bydividing P (E) by P (E ′) and reducing to its lowest terms, a

b :

The odds for event E,P (E)

P (E)=a

b, (11)

and similarly,

the odds against event E,P (E)

P (E)=b

a. (12)

47. A fast-food chain is conducting a game in which the odds of winning adouble cheeseburger are reported to be 1 : 100. Find the probability ofwinning a double cheeseburger.

48. A pair of dice, one red and one green, is to be rolled. Find the odds

(a) for the sum of the pips on the top sides to be 6.

(b) for the sum of the pips on the top sides to be 11.

(c) for the red die to show three pips on the top side.

(d) against the pips’ sum on the top sides being 3.

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Expected ValueWhenever a probability distribution has numerical outcomes, i.e. out-comes that are numbers. We can calculate average or expected valuefor the outcomes.

49. Suppose you are playing the following card game. You draw a card atrandom from a deck of cards. If it is a face card, you win $2. If it is anon-face card, you lose $1. What are your expected earnings from thisgame?

Outcome Payoff Assigned= Earnings Probability

face cardnon-face card

Expected Value.Let S be a sample space, and let A1, A2, · · · , An be n events of S thatform a partition of S, that is the union of the events is S and they arepairwise disjoint:A1∪A2∪ . . .∪An = S and Ai∩Aj = ∅ for all i 6= j. Let p1, p2, p3, ..., pn, bethe probabilities of the eventsA1, A2, · · · , An. If each event, A1, A2, · · · , An

is assigned the payoffs, m1,m2, · · · ,mn the expected value correspond-ing to these payoffs is:

E = m1p1 +m2p2 + . . .+mnpn. (13)

50. To move from your current position on a game board, you draw one ball,with replacement, from a container that has five red and six green balls.If the ball you draw is red, you advance four places; if the ball is green,you move back three places.

(a) What is your expected movement in this game?

(b) In 44 turns at drawing a ball, where would you expect to be locatedon the game board relative to your present position?

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51. Assume that a boy is just as likely as a girl at each birth. In a three-childfamily, what is the expected number of boys?

52. Suppose you pay $5 to play this game: From a box containing six redand eight green marbles, you are allowed to select one marble after beingblindfolded. If the marble is red, you win $10, but if the marble is green,you win nothing. What are your expected earnings from this game?

53. A department store wants to sell 11 purses that cost the store $40 each,and 32 purses that cost the store $10 each. If all purses are wrapped in43 identical boxes, and if each customer picks a box randomly, find

(a) Each customer’s expectation (=average value of purse picked).

(b) The department store’s expected profit per purse, if it charges $20for each box.

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Math 13 Chapter 7 Handout Helene Payne …hpayne/Math 13/Handouts/Math13Handout7x...Math 13 Chapter 7 Handout Helene Payne Name: Probability Sets A set is a well-de ned collection

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