Math Concept Reader What are the Chances? DIGITAL FINAL PROOF W ca67o_lay_070108_cp.indd 1/9/07 1:19:40 AM 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 10 175 16 15 14 13 12 11 10 09 08 07 ca67o_lay_070108_cp.indd 1/9/07 1:19:43 AM 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 their homework and read books Sometimes friends from their class join them to 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  ca67o_lay_070108_cp.indd 1/9/07 1:19:44 AM 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: 100 100 100 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  ca67o_lay_070108_cp.indd 1/9/07 1:19:54 AM 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 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, is a better chance of 100 winning than 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  ca67o_lay_070108_cp.indd 1/9/07 1:20:01 AM 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 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 probability of winning is a greater fraction than If the 12 100 drawing were held right now, their probability of winning would be much greater  ca67o_lay_070108_cp.indd 1/9/07 1:20:02 AM 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 chance of winning As Tanya and Ahn 100 pull out books from their backpacks to 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  ca67o_lay_070108_cp.indd 1/9/07 1:20:07 AM 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 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  ca67o_lay_070108_cp.indd 1/9/07 1:20:13 AM 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 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!  ca67o_lay_070108_cp.indd 1/9/07 1:20:20 AM 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 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 because they know that it is impossible for a mongoose to use a computer  ca67o_lay_070108_cp.indd 1/9/07 1:20:27 AM 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 When there are 100 entries, the probability of 83 each of them winning would be or 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 ca67o_lay_070108_cp.indd 10 Dylan tells Tanya and Ahn that he will bring more books to increase the number of entries he has for the contest 1/9/07 1:20:54 AM 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 5, 000 probability of winning Together they have a 5, 000 , or 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 is more likely than 50 , 000 500 2, 500 They agree that the probability of winning a contest like that is unlikely Contest Contest 5,000 Entries 50,000 Entries Each entry has 5, 000 probability of winning Each entry has 100 50 , 000 5, 000 probability of winning 11 ca67o_lay_070108_cp.indd 11 1/9/07 1:21:18 AM 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 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 ca67o_lay_070108_cp.indd 12 1/9/07 1:21:20 AM 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 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 − 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 Tanya writes one more equation 95 95 Their probability of not winning is still high at 87 , 95 − 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 ca67o_lay_070108_cp.indd 13 1/9/07 1:22:14 AM 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 , or 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 100 10 They quickly calculate their probability of not winning Ahn writes this equation 100 10 90 − = = 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 , or 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 − 100 = 92 100 100 25 Probability Justin and Dylan Probability Justin and Dylan Will Win Will Not Win 100 , or 25 92 100 , or 23 25 14 ca67o_lay_070108_cp.indd 14 1/9/07 1:22:18 AM 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 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 100 probability of winning Tanya and Ahn agree that even if they 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 ca67o_lay_070108_cp.indd 15 15 1/9/07 1:23:21 AM 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, (top), 11, 15 (bottom) National Biological Information Infrastructure; pp 4, 6, 10 (bottom), 13, 15 (top) © Michael Newman/Photo Edit; pp (bottom), (both) NOAA; p 10 © Myrleen Ferguson Cate/Photo Edit 16 ca67o_lay_070108_cp.indd 16 1/9/07 1:23:22 AM Think and Respond Mikael has three entries in a contest The contest is limited to 500 entries What is the probability that Mikael will win? What is the probability of getting tails if you flip an unfair coin with heads on both sides? In a contest with 200 entries, Jack has 10 entries What is the probability that he will not win? Create your own contest Explain how many entries there will be and describe the prize  ... 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. .. 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... 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