MCR g6 what are the chances

Math Concept Reader

What are the
Chances?


DIGITAL FINAL PROOF

W

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DIGITAL FINAL PROOF

Math Concept Reader

What are the Chances?
By Aenea Mickelsen

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.
Requests for permission to make copies of any part of the work should be addressed
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

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DIGITAL FINAL PROOF

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.



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DIGITAL FINAL PROOF

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:



Ahn’s probability of winning:



Tanya’s and Ahn’s probability of winning:

1
100
1
100
2
100

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.



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DIGITAL FINAL PROOF

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 1 . Tanya writes down this fraction in her notebook.
100

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, 2 is a better chance of
100

winning than

1
100

. 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.


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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 1 .
100

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
1

12

probability of winning. 1 is a greater fraction than 1 . If the
12

100

drawing were held right now, their probability of winning would be
much greater.



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DIGITAL FINAL PROOF

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 21
32

and the probability of a boy winning is

11
32

.

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 3 chance of winning. As Tanya and Ahn
100

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.


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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.


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DIGITAL FINAL PROOF

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

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.
The humuhumunukunukuapua’a fish
is smaller than its name!



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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.



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DIGITAL FINAL PROOF

Justin does his homework at the library after turning
in more books for the contest.

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 8 . When there are 100 entries, the probability of
83

each of them winning would be 4 or 1 .
100

25

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 43 while the boys’ probability
83

of winning would be 40 . Justin and Dylan
83

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.

10

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Dylan tells Tanya and Ahn that he will
bring more books to increase the number
of entries he has for the contest.

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DIGITAL FINAL PROOF

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
1
5, 000

probability of winning. Together

they have a 2

5, 000

, or 1

2, 500

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

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 100 , or 1 is more likely than
50 , 000

500

1
2, 500

. They agree that the

probability of winning a contest like that is unlikely.

Contest 1

Contest 2

5,000 Entries

50,000 Entries

Each entry has

1
5, 000

probability of winning.

Each entry has

1
100
50
, 000
5, 000

probability of winning.
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DIGITAL FINAL PROOF

Chapter 3:

What are Our Chances?
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 48 and the probability of a boy winning is 47 .
95

95

12

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DIGITAL FINAL PROOF

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

47

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

95

the probability of an event not happening is 1 minus the probability
that the event will happen.
47

95

47

48



1−



The probability that boys will not win is 48 .

95

=

95

−

95

=

95

95

Tanya and Ahn calculate their own probability of not winning.
Ahn writes an equation. Each girl has a 91 probability of not winning.


95
95

−

4
95

=

95

91
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 8 . Tanya writes one more equation.
95



95



Their probability of not winning is still high at 87 ,

95

−

8
95

=

87
95

95

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.

Ahn figures out probabilities
while Tanya reads about Hawaii.

13

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DIGITAL FINAL PROOF

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 5 , or 1
100

20

probability of winning. The contest is now full and no more entries will
be accepted. Together, the girls’ probability of winning is 10 , or 1 .
100

10

They quickly calculate their probability of not winning. Ahn writes
this equation.


100 10
90
9
−
=
=
100 100 100 10

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 8 , or 2 .
100

25

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


100

The probability that Justin and Dylan will not win is 92 , or 23 .

100

−

8
100

=

92
100

100

25

Probability Justin and Dylan

Probability Justin and Dylan

Will Win

Will Not Win

8
100

, or 2

25

92
100

, or 23
25

14

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DIGITAL FINAL PROOF

Ahn and Tanya are proud of the help they got from family and
friends to collect books for the book sale.

As Tanya and Ahn leave the library, Mrs. Roper tells them that the
drawing will be held on Saturday. They cannot wait to find out who
wins! The girls check out some books about Hawaii and look forward
to reading more about this tropical place. They have already learned a
lot about Hawaii and are eager to share what they know with people
at the festival. That is, if one of them wins the contest! Both girls are
proud that their families, relatives, and neighbors helped them donate
so many books to the annual book sale. They are happy that their

donations will directly help the library. Maybe one day the library will
have a new wing for them to visit Tuesday afternoons while they do
their homework.
As they walk home, the girls talk about the drawing
for the Hawaiian festival tickets. They are thrilled
that they each have a 5

100

probability

of winning. Tanya and Ahn agree that even
if they do not win, it has been fun to learn
about Hawaii. Plus, having the opportunity to
help the library with their fundraising project is
rewarding.

Tanya and Ahn had fun learning
about Hawaii and probability.

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DIGITAL FINAL PROOF

Glossary

certain sure to happen
impossible never able to happen
probability the likelihood that an event will happen
Nene a rare wild goose found on the Hawaiian Islands
and the state bird of Hawaii.
Humuhumunukunukuapua’a a type of triggerfish and
the state fish of Hawaii.

Photo Credits: cover, title page © Corbis; pp. 3, 8 (top), 11, 15 (bottom)
National Biological Information Infrastructure; pp. 4, 6, 10 (bottom), 13, 15
(top) © Michael Newman/Photo Edit; pp. 8 (bottom), 9 (both) NOAA; p. 10
© Myrleen Ferguson Cate/Photo Edit.

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Think and Respond
1. Mikael has three entries in a contest. The contest is
limited to 500 entries. What is the probability that
Mikael will win?
2. What is the probability of getting tails if you flip an
unfair coin with heads on both sides?
3. In a contest with 200 entries, Jack has 10 entries.
What is the probability that he will not win?

4. Create your own contest. Explain how many entries

there will be and describe the prize.