MCR g6 what are the chances

The classmates will collect books to enter a contest for tickets to the festival.. The contest is to help the library collect books for its annual book sale.. Each time a person donate

What are the

Chances?

What are the Chances?

DIGITAL FINAL PROOF

What are the Chances?

By Aenea Mickelsen Math Concept Reader

Copyright © Gareth Stevens, Inc All rights reserved.

Developed for Harcourt, Inc., by Gareth Stevens, Inc This edition published by Harcourt, Inc., by agreement with Gareth Stevens, Inc No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the copyright holder.

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to Permissions Department, Gareth Stevens, Inc., 330 West Olive Street, Suite 100, Milwaukee, Wisconsin 53212 Fax: 414-332-3567.

HARCOURT and the Harcourt Logo are trademarks of Harcourt, Inc., registered

in the United States of America and/or other jurisdictions.

Printed in the United States of America ISBN 13: 978-0-15-360209-2

ISBN 10: 0-15-360209-0

1 2 3 4 5 6 7 8 9 10 175 16 15 14 13 12 11 10 09 08 07

Chapter 1:

The Contest



Tanya and her friend Ahn go to the library nearly every Tuesday

after school The library is right down the street and they walk there

when school gets out They take their schoolwork and spread it out

across one of the large library tables They enjoy being there in the

comfortable, quiet building, where they can do their homework and

read books Sometimes friends from their class join them to do their

homework as well

As they enter the library, their attention is drawn to a beautiful

poster on the wall which features a photograph of a peaceful tropical

sunset The sunset covers most of the bright poster There are palm

trees along a white, sandy beach, and the sun is setting in the

distance The photo is very inviting

The girls walk over to the poster and read the words printed on it

The poster announces a new contest

DIGITAL FINAL PROOF

The library near Tanya and Ahn will hold a Hawaiian

festival to raise funds for a new library wing The

classmates will collect books

to enter a contest for tickets

to the festival.

The contest is to help the library collect books for its annual book

sale The funds raised at the book sale will help the library add a new

wing to the building Adding a new library wing means the library

would have many more books available to the community This year

the book sale is part of a Hawaiian festival the library is hosting as part

of their fundraising effort The festival will include booths featuring

Hawaiian food, Hawaiian music and dancing, and tropical-themed

games It will also feature booths with information about each of the

Hawaiian Islands Each time a person donates books to the library, they

are entered in a drawing for two tickets to the Hawaiian festival The

contest is limited to 100 entries

The girls continue to read and learn that they can enter the

contest as many times as they like Tanya and Ahn see this and

excitedly look at each other They would both love to help a good

cause, and if they were rewarded for their donations, that would be a

great bonus

Number of Opportunities for the Festival Tickets: 100

Tanya’s probability of winning:

100 1

Ahn’s probability of winning:

100 1

Tanya’s and Ahn’s probability of winning:

100 2

Tanya and Ahn talk about their chances of winning the library’s contest



The girls wonder what their probability of winning tickets to the

Hawaiian festival is as they find a table in the library Both girls are

determined to get at least one entry into the ticket drawing If they

do, that means their probability of winning will not be zero If there

are 100 entries, and they each have one entry, the probability one of

them will win is

100

1 Tanya writes down this fraction in her notebook

They both know that the top part of a probability fraction shows how

many successful possibilities there are, while the bottom part reflects

the total possibilities

Tanya and Ahn begin to think of ways they will collect used books

because they want their chances of winning to be as high as possible

Even if they each have just two entries,

100

2 is a better chance of winning than

100

1 The girls remember from studying probability in math class that the greater the fraction, the more likely a successful

outcome The lesser the fraction, the less likely the outcome will be

successful This means that in order to win the contest, they need to

enter it as many times as possible

DIGITAL FINAL PROOF

Chapter 2:

The Possibilities

Tanya and Ahn understand that the more contest entries they

have, the greater their chances are of winning tickets for the

Hawaiian festival fundraiser Every entry they submit also helps the

library gain more books for its fundraiser The next day, they each

bring a used book to the library Mrs Roper, the head librarian, is

pleased to accept their donations and asks the girls if they know what

the probability is of their winning the tickets

Tanya and Ahn tell Mrs Roper that since the ticket drawing is

limited to 100 entries, they know they each have a probability of

100

1 Mrs Roper smiles and tells them that right now, there are 12 entries

Tanya and Ahn have two of those entries, which means that if the

library selected the winning entry today, each person would have a

12

1 probability of winning

12

1 is a greater fraction than

100

1 If the drawing were held right now, their probability of winning would be

much greater

Mrs Roper counts some of the donated books submitted as contest entries before she puts them away for the book sale.



Over the course of one week, Tanya and Ahn have each collected

and donated more books to the library for the book sale They

gathered used books from relatives and neighbors whom they told

about the library’s fundraising event That means Ahn and Tanya,

now, each have three entries into the drawing

When they go to the library to turn in the additional books, they

ask Mrs Roper how many entries there are in the drawing now She

tells the girls that she now has 32 entries There are 21 entries from

girls and 11 entries from boys The probability of a girl winning is

32 21

and the probability of a boy winning is

32

11 The friends are certain that more boys and girls will enter the

drawing over the next few days Once there are 100 entries, Tanya

and Ahn will each have a

100

3 chance of winning As Tanya and Ahn pull out books from their backpacks to do homework, their

classmates, Justin and Dylan sit down next to them They inform

Tanya and Ahn they also have three entries each in the drawing The

girls are determined to collect more books

DIGITAL FINAL PROOF

Hawaii is made up of several islands It is America’s 50th state.

While they are at the library, Tanya and Ahn decide to do some

research on Hawaii because they are interested in learning more about

this exotic location The upcoming Hawaiian Festival has inspired them

to find out all they can about the place The girls sign up to use a

computer and begin searching for information about Hawaii

Tanya and Ahn learn that Hawaii is America’s 50th state, it is the

only state completely formed from volcanic material, and it is the only

state that grows coffee Hawaii is made up of many islands, some of

them barely above the surface of the water The eight main islands of

Hawaii are Hawaii, Maui, Oahu, Kauai, Molokai, Lanai, Niihau, and

Kahoolawe

The biggest island, which is the island of Hawaii, is also the

youngest The island of Hawaii is also known as the Big Island and is

where the volcano Kilauea is located This volcano is still highly active

The lava that flows from Kilauea eventually hardens and becomes new

landmass for the Big Island The Big Island is also home to the world’s

tallest mountain, called Mauna Kea

The Nene, or Hawaiian Goose, is Hawaii’s state bird.

The humuhumunukunukuapua’a fish

is smaller than its name!



Tanya and Ahn read that sugar is Hawaii’s major crop In the year

2006, Hawaii produced about 1,762,000 tons of sugar In the late

1800s, sugar farmers were concerned because rats were destroying

some of their crops, so they decided to import mongooses from the

Caribbean island of Jamaica Since mongooses eat rats, the farmers

hoped to control the rat population with these predatory animals In

1883, 72 mongooses arrived in Hawaii

As it turns out, while mongooses do eat rats, they also prey on

ground-nesting birds and their eggs Hawaii’s population of its state

bird, the Nene, or Hawaiian Goose, was almost completely wiped

out The mongoose has no natural predators in the Hawaiian Islands,

so several other species of birds actually did become

extinct because of the animal

As Ahn and Tanya read more about native Hawaiian creatures, they come across one of the longest words they have ever seen: humuhumunukunukuapua’a (humu humu nuku nuku apua’a) This is a fish that

is also called the Hawaiian Triggerfish

DIGITAL FINAL PROOF

Thousands of acres of pineapple fields used to cover the Hawaiian islands.

One of the fruits grown in Hawaii is the pineapple, and the

pineapple happens to be Ahn’s favorite fruit Tanya and Ahn discover

that Hawaii used to be the world’s largest producer of pineapple

The industry began in 1899 when James Drummond Dole purchased

60 acres of fertile land to grow pineapple Soon, pineapple fields

covered thousands of acres Today, however, Thailand is the world’s

largest producer of pineapple, and Hawaii no longer even ranks in a

list of the top ten producers

The girls consider what their chances of eating pineapple would

be if they go to the festival They decide that eating pineapple while

attending the Hawaiian festival is likely That means the probability is

close to 1 because the closer an event’s probability is to 1, the more

likely it is to occur Ahn hopes that it will be certain that they eat

pineapple at the festival

Tanya and Ahn wonder what event might have a probability of

zero Ahn suggests that the chance of her seeing a mongoose using

a computer at the library cannot happen This means the probability

is 0 because they know that it is impossible for a mongoose to use a

computer

Justin does his homework at the library after turning

in more books for the contest.

Dylan tells Tanya and Ahn that he will bring more books to increase the number

of entries he has for the contest.

10

Later that week, the girls go to the library again Many people have

donated books for the book sale and there are now 83 entries in the

contest Tanya and Ahn each have four entries now and are still hoping to

get more If the drawing for the tickets were held today, the probability of

either girl winning is

83

8 When there are 100 entries, the probability of each of them winning would be

100

4 or

25

1 Mrs Roper tells them that there are still more entries from girls than

from boys, though the difference in the numbers is less Now there are

40 entries from boys and 43 from girls If the drawing were held today, the

probability that a girl would win would be

83

43 while the boys’ probability

of winning would be

83

40 Justin and Dylan tell the girls that they each have three entries and are planning on bringing in more books

Tanya and Ahn know the boys will be fierce competition They must collect many more books and gain more contest entries

DIGITAL FINAL PROOF

Ahn has Hawaii on her mind when she figures out her probability of winning the contest.

Tanya wonders how to figure out the

probability of winning the tickets with more

than 100 entries She and Ahn sit down

together and get out a sheet of paper

What if they entered a contest that was

limited to 5,000 entries? If they could each

have one entry, they would each have a

000

,

5

1 probability of winning Together

they have a

000 , 5

2 , or

500 , 2

1 probability

of winning Compared to their chances to

win the Hawaiian festival tickets, this probability of winning a contest

with 5,000 entries seems less likely but not impossible

The girls then imagine a contest with 50,000 entries Even if they

had 100 entries in a contest like that, their chances would only be

100 out of 50,000 Their probability of winning is still low, even

though

000 , 50

100 , or

500

1 is more likely than

500 , 2

1 They agree that the probability of winning a contest like that is unlikely

Contest 1

5,000 Entries Each entry has

000 , 5

1 probability of winning

Contest 2

50,000 Entries

Each entry has

000 , 50 100

probability of winning.5,000

1

Chapter 3:

What are Our Chances?

1

The deadline for entries in the contest is next week When they

visit Mrs Roper at the beginning of the week, Tanya and Ahn learn

that there are now 95 entries for the tickets to the Hawaiian festival

The contest is nearly full Both girls have four entries each and they

are curious to find out if girls still have a better chance of winning the

prize than boys If more girls have entered the contest, that means

girls do have a better chance than boys Mrs Roper tells them that

it is closer than before Now there are 47 entries from boys and 48

entries from girls

Tanya and Ahn are happy to know that girls still have a better

probability of winning As the contest stands that day, the probability

of a girl winning is

95

48 and the probability of a boy winning is

95

47

DIGITAL FINAL PROOF

Ahn figures out probabilities while Tanya reads about Hawaii.

Tanya and Ahn are curious If the probability of a boy winning is

95

47, what is the probability a boy will not win? They remember that

the probability of an event not happening is 1 minus the probability

that the event will happen

95

48 95

47 95

95 95

47

The probability that boys will not win is

95

48

Tanya and Ahn calculate their own probability of not winning

Ahn writes an equation Each girl has a

95

91 probability of not winning

95

91 95

4 95

95

=

At that moment, the girls decide that they want to share their

entries If one of them wins, the other friend will come along to the

Hawaiian festival That means that together they have eight entries

Their probability of winning is

95

8 Tanya writes one more equation

95

87 95

8 95

95

=

−

Their probability of not winning is still high at

95

87, but their probability of winning the contest has

improved They have found an easy way to

increase their chance of going to the

Hawaiian festival

On Friday afternoon, Tanya and Ahn make one more trip to the

library They are each carrying a bag of books to donate to the library

Thanks to donations from neighbors, their final donations are the 99th

and 100th entries in the drawing and each girl has a

100

5 , or

20

1 probability of winning The contest is now full and no more entries will

be accepted Together, the girls’ probability of winning is

100

10 , or

10

1 They quickly calculate their probability of not winning Ahn writes

this equation

10

9 100

90 100

10 100

They agree that the probability of not winning seems high

Justin and Dylan are sitting at the table over by the windows The

boys are also excited about the contest They have a total of eight

entries in the drawing The probability that they will win is

100

8 , or

25

2 Justin writes an equation to calculate the boys’ probability of not

winning

100

92 100

8 100

100 − =

The probability that Justin and Dylan will not win is

100

92 , or

25

23

Probability Justin and Dylan

Will Win

100

8 , or

25

2

Probability Justin and Dylan

Will Not Win

100

92 , or

25

23

DIGITAL FINAL PROOF