MAP 4172 ACTUARIAL SCIENCE PROBLEMS

Academic Coursewarec date August 23, 2014

Contents

Contents i

Preface 1

1 Counting Techniques 3

1.1 Fundamental Principle of Counting . . . . . . . . . . . . . . . . . . . 3

1.2 Counting Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Bibliography 11

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ii CONTENTS

Preface

The lproblem set that accompanies Actuarial Science 1 is designed to increase yourfamiliarity with those problem types which we have found common to exam P/1.The recessitation session is generally run as a strictly question answer session not asa lecture session. The focus is on students working problems to understand wheretheir difficulties lie and seeking solutions that are quick and illuminating. Yourteaching assistant is there to answer questions you have about the problems youreworking on.

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2 CONTENTS

Chapter 1

Counting Techniques

1.1 Fundamental Principle of Counting

Counting is an important skill. The Fundamental Principle of counting will be usedto

1. Count simple events

2. Compute the number of permutations

3. Compute the number of combinations

Let us recall the Fundamental of Counting

Fundamental Theorem of Counting

With m elements a1, a2, . . . , am and n elements b1, b2, . . . , bm, it is possible to formmn pairs containing one element from each group.

Theorem 2

The number of ways of ordering n distinct objects taken r at a time is equal to

n(n 1)(n 2) (n r + 1) = n!(n r)!

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4 CHAPTER 1. COUNTING TECHNIQUES

Theorem 3

The number of ways of partitioning n distinct objects into k distinct groups con-taining n1, n2, . . . , nk objects, respectively, where each object appears in exactly onegroup and

ki=k ni = n is(

n

n1!n2!n3! nk!)

=n!

n1!n2!n3! nk!

Theorem 4

The number of unordered subsets of size r chosen (without replacement) from navailable objects is (

n

r

)=

n!

r!(n r)!

example 1

A chef has 3 meats, 4 vegetables, 2 desserts, and 2 bevarages that he will use tocreate a meal. How many distinct meals can he create if he must use a meat, avegetable, a beverage, and a dessert to create a meal?

Answer: 3 4 2 2 = 48

example 2

In how many ways can n distinct objects be arranged in a row?

Answer: n(n 1)(n 2) . . . (2)(1) = n!

example 3

The names of 3 employees are to be randomly drawn, without replacement, from abowl containing the names of 30 employees of a small company. The person whosename is drawn first receives $100, and the individuals whose names are drawnsecond and third receive $50 and $25, respectively. How many sample points areassociated with this experiment?

1.1. FUNDAMENTAL PRINCIPLE OF COUNTING 5

Answer: 30 29 28 = 30!(303)!

example 4

A librarian wishes to place 7 books on a bookshelf. There are three mathematicsbooks, a college algebra book, a calculus book, and a liberals arts mathematicsbook, two chemistry books, an organic chemistry book, and a general chemistrybook, one general physics book, and a history book. If the mathametics books areindistinguishable from one another (say an identical jacket with math is written oneach and placed on each), the chemistry books are indistinguishable from oneanother (say an identical jacket with chemistry is written on each and placed oneach) then how many distinguishable ways are there to place the 7 books on thebookshelf.

Answer:(

7!3!2!1!1!

)= 420

6 CHAPTER 1. COUNTING TECHNIQUES

1.2 Counting Problems

1. John, Jim, Jay, and Jack have formed a band consiting of 4 instruments. Ifeach of the boys can play all 4 instruments, how many different arrangementsare possible? What if John and Jim can play all 4 instruments, but Jay andJack can each play only piano and drums?

2. A well known nursery rhyme starts as follows: As I was going to St. Ives Imet a man with 7 wives. Each wife had 7 sacks. Each sack had 7 cats. Eachcat had 7 kittens. How many kittens did the traveler meet?

3. In how many ways can 3 boys and 3 girls sit in a row? In how many ways can3 boys and 3 girls sit in a row if the boys and the girls are each to sit together?In how many ways if only the boys must sit together? In how many ways ifno two people of the same sex are allowed together?

4. How many different letter arrangements can be made from the letters MIS-SISSIPPI?

1.2. COUNTING PROBLEMS 7

5. A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is blue.If the child puts the blocks in a line, how many arrangements are possible?

6. In how many ways can 8 people be seated in a row if persons A and B mustsit together? What if there are 4 men and 4 women and no 2 men or 2 womencan sit next to each other? What if there are 5 men and they must sit nextto eah other. What if there are 4 married couples and each couple must staytogether?

7. Five separate awards (best leadership qualities, and so on) are to be presentedto selected students from a class of 30. How many different outcomes arepossible if a student can receive any number of awards. What if each studentcan receive at most 1 award?

8. Consider a group of 20 people. If everyone shakes hands with everyone else,how many handshakes take place?

8 CHAPTER 1. COUNTING TECHNIQUES

9. How many 5 card poker hands are there?

10. A dance class consists of 22 students, 10 women and 12 men. If 5 men and 5women are to be chosen and then paired off, hwo many results are possible?

11. A student has to sell 2 books from a collection of 6 math, 7 science, and 4economics books. How many choices are possible if both books are to be onthe same subject? What if the books are to be on different subjects?

12. A total of 7 different gifts are to be distributed among 10 children. How manydistinct results are possible if no child is to receive more than one gift?

1.2. COUNTING PROBLEMS 9

13. A committee of 7, consisting of 2 Republicans, 2 Democrats, and 3 Indepen-dents, is to be chosen from a group of 5 Republicans, 6 Democrats, and 4Independents. How many committees are possible?

14. From a group of 8 women and 6 men a committee consisting of 3 men and 3women is to be formed. How many different committees are possible if 2 ofthe men refuse to serve together? How many if 2 of the women refuse to servetogether? How many if one man and one woman refuse to serve together?

15. A person has 8 friends of whom 5 will be invited to a party. How many choicesare there if 2 of the friends are feuding and will not attend together? Howmany choices are there if 2 of the friends will only attend together?

16. A psychology laboratory conducting dream research contains 3 rooms, with 2beds in each room. If 3 sets of identical twins are to be assigned to these 6beds so that each set of twins sleeps in different beds in the same room, howmany assignments are possible?

10 CHAPTER 1. COUNTING TECHNIQUES

17. The game of bridge is played by 4 players, each of whom is dealt 13 cards.How many bridge deals are possible?

18. If 12 people are to be divided into 3 committees of respective sizes 3, 4, and5, how many divisions are possible?

Bibliography

[1] S. Ross A First Course In Probability 7th edition Prentice Hall, Upper SaddleRiver, New Jersey 2006

[2] L. Lamport. LATEX A Document Preparation System Addison-Wesley,California 1986.

[3] M.D. Spivak The Joy of TEXA Gourmet Guide to Typesetting with the AMS-TEXmacro package American Mathematical Society, Providence, Rhode Island1986

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ContentsPrefaceCounting TechniquesFundamental Principle of CountingCounting Problems

Bibliography