Students’ Integrated Maths Module for Probability · E: Newspaper Search: Relatively Speaking 29...

Students’

Integrated Maths

Module

for

Probability

Author: Mark R. O’Brien

Editor: Mark R. O’Brien

OTRNet Publications

www.otrnet.com.au

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Copyright © 2003 by OTRNet Publications, www.otrnet.com.au. All rights reserved. Nopart of this publication may be reproduced or transmitted in any form, or by any means,electronic or manual, including photocopying, scanning, recording, or by any informationstorage or retrieval system, without permission in writing from the publisher.

First published: December 2003Revised: January 2019

Design and Editing: Mark R. O’Brien Cover Design: Ali B Design

Ph: 0411 4301 09E-mail: [email protected]

National Library of AustraliaCataloguing-in-Publication data

For secondary studentsISBN 1 876800 65 8

3[SIMM] Probability

Introduction for students:

At the end of this module you should have increased your ability to;

C List equally likely outcomes for experimentsC Assign probabilities from sample spacesC Assign probabilities from relative frequencies C Assign probabilities using counting techniquesC Determine probabilities using probability tree diagramsC Understand the relationship between odds and probabilityC Make predictions based on probabilities

Table of Contents:

Activities: Puzzle:

A: The Money Wheel 4 Card Games 18

B: Secrity Codes 6

C: Life Tables 7 Applications:

D: Greedy Pigs 9 A: DNA 20

E: Investigation: The Birthday Problem 10 B: Brownlow Medal Betting 22

C: Medical Probabilities 24

Student Recording 12 D: Project: Population Control 27

E: Newspaper Search: Relatively Speaking 29

Notes 13

Exercises:

1: Sample Spaces 15 Student Reflection 31

2: Probability 15

3: Expectation 17 Answers to Exercises 32

© OTRNet Publications: www.otrnet.com.au

4 [SIMM] Probability

Activity A:

The Money Wheel

The diagram on the right shows the Money Wheel at aCasino. Players place their bets on a table with areasrepresenting the sectors on the wheel.

The wheel is divided into 52 equal sectors. Each sector hasa monetary value or the word Casino or Winner on it.There are: (a) 24 - $1 sectors - these pay 1 to 1

(b) 12 - $3 sectors - these pay 3 to 1(c) 8 - $5 sectors - these pay 5 to 1(d) 4 - $11 sectors - these pay 11 to 1(e) 2 - $23 sectors - these pay 23 to 1(f) 1 - sector marked Casino - this pays 47 to 1(g) 1 - sector marked Winner - this pays 47 to 1

Task 1: Using the formula:

Find the probability of winning if you bet on each of the monetary values(a), (b), (c), (d), (e), or on a word (f) or (g).

The casino pays winners according to the probabilities as shown in the list above. A$10 winning bet on a $5 sector pays $50 (5 to 1) and hence the player gets a return of$60, the $50 won and the $10 bet.

Task 2: Complete this table of win and loss amounts:

Bet Result Win/Loss Return

$10 on the $5 sector WIN Win $50 $60

$5 on the $23 sector LOSS Lose $5 $0

$10 on the $3 sector WIN


$25 on the $11 sector LOSS

$5 on the Casino sector WIN


$100 on the Winner sector LOSS


5[SIMM] Probability

To simulate the Money Wheel the following set of four digit random numbers hasbeen calculated:

$1 $3 $5 $11 $23 Casino Winner

0.0000to

0.4615

0.4616to

0.6924

0.6925to

0.8463

0.8464to

0.9233

0.9234to

0.9618

0.9619to

0.9809

0.9810to

1.0000

Task 3: Explain how these sets of random numbers, simulate the probabilitiesfrom the Money Wheel.

Task 4: Set your calculator up to generate four digit random numbers from zeroto one.

Task 5: Generate and record 5 of these random numbers alsorecording what sector on the Money Wheel each represents.

Task 6: Run a simulation of 20 spins of the Money Wheel making bets of between$5 and $100 each time and calculating the amount won or lost after eachspin. Record your results in a table like the one shown here. Two exampleshave been shown.

Sectorbet on

Bet ($) RandomNo.

WheelResult

Win orLoss

Amount Won or Lost

$5 sector $10 0.0290 $1 sector Loss Lost $10

$3 sector $5 0.5286 $3 sector Win Won $15

Task 7: Tally up the last column and make a record of the amount you won or lost.

Task 8: Record the amount you won or lost as part of a set of class results.

Task 9: What was the mean result for the class?



Activity B:

Security Codes

Luke has a combination lock on his locker with three tumblers. The combination for opening his lock is 804.

Task 1: If Riley comes along when Luke has already set the8 and the 0, how many different numbers can he tryfor the last digit?

Task 2: What is the probability that if Luke gives Riley one guess he will get theright number for the last digit?

Task 3: How many different possible combinations are there for Luke’s lock?

Task 4: PIN numbers for bank access cards consist of 4 digits. How many differentpossible PIN numbers are there?

Task 5: Automatic teller machines give three attempts at getting the PIN numberright before they ‘keep’ the access card. What is the probability ofsomeone getting access to an account by randomly guessing the PINnumber of a stolen card?

Task 6: The password to login to asecure area of a computersystem consists of eightcharacters. These characterscan be the digits from 0 to 9or any lower case letter of thealphabet. How many different possible passwords are there?


7[SIMM] Probability

Activity C:

Life Tables

Probabilities are often estimated from relative frequencies. An example of this is life insurance companies using pastrecords to predict the probability of people dying.

i.e. Probability (event) = relative frequency

=

Use this formula and the Life Tables below to complete the tasks that follow:Life Tables for Year 2000

Australian Indonesian

Male Female Male Female

AgeActualDeaths

ActualPopulation

ActualDeaths

ActualPopulation

ActualDeaths

ActualPopulation

ActualDeaths

ActualPopulation

<1 760 126 635 486 121589 115 106 2 265 987 82 414 2 158 083

1 - 4 146 521 444 118 493491 29 921 8 827 779 27 225 8 530 054

5 - 9 89 686 626 65 649857 13 148 11 015 331 10 669 10 643 491

10 - 14 102 680 974 84 645961 11 279 11 066 826 8 723 10 723 998

15 - 19 514 694 906 205 661233 19 948 10 924 150 14 069 10 639 975

20 - 24 777 687 850 250 657844 26 427 10 491 127 17 239 10 212 852

25 - 29 968 744 625 293 732733 25 932 9 622 946 18 334 9 386 349

30 - 34 932 705 902 382 707708 27 132 8 825 898 20 457 8 595 340

35 - 39 1 010 747 925 501 747386 29 162 7 357 771 23 195 7 163 030

40 - 44 1 279 722 355 767 723784 35 545 6 463 446 27 369 6 295 924

45 - 49 1 621 669 867 1 057 673489 40 548 5 074 543 30 373 4 956 098

50 - 54 2 527 631 748 1 398 615951 45 043 3 768 203 35 288 4 015 040

55 - 59 3 408 486 876 1 890 472509 58 710 3 280 630 45 884 3 643 377

60 - 64 4 305 391 367 2 335 389123 75 430 2 782 660 60 227 3 140 354

65 - 69 6 245 328 670 3 446 344566 84 271 2 062 393 72 191 2 410 931

70 - 74 9 989 293 793 5 959 331067 86 500 1 366 834 80 377 1 636 589

75 - 79 11 656 215 844 8 841 285198 71 930 735 764 73 241 918 491

80 - 84 10 721 116 528 11 328 185698 48 093 322 630 57 315 447 289

85 - 89 8 349 55 289 12 800 113270 22 856 103 404 31 318 158 274

90 - 94 3 433 16 192 7 752 43067 5 944 18 773 9 828 33 544

95 - 99 1 011 3 316 3 114 11122 742 1 692 1 598 3 841

100+ 212 473 788 1794 45 76 134 237

All ages 70 054 9 529 205 63 859 9 608 440 873 712 106 378 863 747 468 105 713 161

Source: WHO Life Tables

For quick calculations this table of data is also available as an Excel spreadsheet from the TLRpage for this module at: http://www.otrnet.com.au/IntegratedMathsModules/H06/H06TLR.html



Task 1: How many Australian females in the age group 15 - 19 died in 2000?

Task 2: What was the probability of an Australian female in this age group dying?

Task 3: How many Indonesian females in the age group 15 - 19 died in 2000?

Task 4: What was the probability of an Indonesian femalein this age group dying?

Task 5: In this age group and gender, which nationality hadthe greatest chance of dying?

Task 6: Calculate and compare the probabilities of dying, for Australian malesand Australian females.

Task 7: Calculate and compare the probabilities of dying, for Australian malesand Australian females in the 15 - 24 age groups.

Task 8: Use the tables to find whether the probability of an Australian dyingchanges once they are over 4 years of age.

Task 9: What is the probability of not dying in a given year for an Australian agedbetween 15 - 19?

Task 10: Select and describe another pair of probabilitiesfrom the table to compare.

Task 11: Calculate these probabilities and comment onyour findings.


9[SIMM] Probability

Activity D:Materials required: Die

Greedy Pigs

Greedy Pigs is a dice game that has been played in many classrooms.

The game is played with the following rules:1. Each student stands up to start playing a round of greedy pigs.2. The teacher (or a student) rolls a single die.3. If the die comes up 3 all of the students standing score zero and that round is

completed. Go back to step 1 and restart.4. If the die comes up 1, 2, 4, 5 or 6 all of the students standing score this value.

This is added to the score they already have accumulated in the round. 5. Students decide if they wish to sit down and collect their score or if they wish to

remain standing up and continue to try to increase their score. Once decided, goback to step 2.

Task 1: Play one or two rounds of Greedy Pigs to alloweveryone to become familiar with how it operates andto start considering strategies.

Task 2: Play five rounds of Greedy Pigs keeping a record ofyour score in each round.

Task 3: What was your total score for the five rounds?

Task 4: What was your mean score per round?

Task 5: Collect together the results for all students in the class over the five rounds.

Task 6: What is the mean score per round for the students in your class.

Task 7: What would you expect the mean score to be in this game?

Task 8: Write down what you think would be a good strategy for playing this game. Discuss strategies within your group and within your class.



Activity E:

Investigation :

The Birthday Problem

Task 1: Estimate the probability that if there weretwo people in a class they would have the same birthday.

Task 2: Estimate the probability that if there were 365 people in a class, two ofthem would have the same birthday.

Task 3: Estimate the number of people there would need to be in a class for theprobability of two people having the same birthday, to be greater than0.5.

Task 4: How many people are in your class?

Task 5: Are there two (or more) people in your class with the same birthday?

Task 6: Explain how we could use the idea from tasks 4 and 5 to investigate thisproblem further. What would be the problems of using this method?

To calculate the probabilities for tasks 1 and 2 the following method can be used:

For one person there are 365 different birth dates (assume a non leap year).

The probability of a second person having their birthday on the same day as thefirst person when there are only two people is:

1 - probability it was on a different day

= 1 - (as there are 364 different days left out of 365)

= 1 - 0.9973

= 0.0027 (This is the exact answer for task 1).


11[SIMM] Probability

The probability of a second person having their birthday on the same day whenthere are three people is:

1 - probability other peoples birthdays are on a different day

= 1 - x (for the third person there are 363 different days left)

= 1 - 0.9918

= 0.0082

Task 7: Show that the probability of two people having their birthdays on thesame day when there are four people is 0.0164.

Task 8: Calculate the probabilities for two people having the same birthday whenthere are 5, 6, 7 and 8 people.

Task 9: Work out the correct response for task 3.

Task 10: How many people are necessary before theprobability of two of them having the samebirthday is equal to 1.0000 (rounded to 4 d.p.)?

Task 11: How many people are necessary before the probability of two of themhaving the same birthday is equal to exactly 1?



Student Recording:

Write at least two pieces of information about each of theseconcepts that you have explored in earlier lessons. Then try to givean example relating to each. Use diagrams where it helps.

probability:

relative frequency:

simulation:

random:

random number:



Notes

Sample Space

A sample space is a list of all of the possible outcomes of anexperiment or chance process.

The sample space for flipping two coins is: HH, HT, TH, TT

Some of the possible ways of finding a sample space are: Listing the outcomes, drawing up a grid, making a table, drawing a tree diagram

An event is a group of outcomes from a sample space. Getting an even number is oneevent from rolling a die.

Probability

Probability is a measure of how likely an event is. Probabilities range from zero -where an event is impossible, to one - where an event is certain.

The formula for calculating probability from a sample space is:

The probability of getting an even number on a die roll is

i.e. three favourable outcomes (2,4,6 are even) and six possible outcomes (1,2,3,4,5,6).

Probability can also be estimated from relative frequencies using this formula:

Probability .

When using relative frequencies to estimate probabilities, a greater number of trialswill improve the accuracy of the estimate.

Probabilities are usually written as fractions or decimals.



Complementary events

Complementary events are events which add to give the complete sample space. For example; getting an even number on a die and getting an odd number on a die. Thesum of the probabilities of complementary events equals one.

Independent events

Two events are independent if the occurrence of one does not affect the probabilityof the other occurring. If a coin is tossed six times the result of the sixth toss isindependent of the results in the first five tosses.

An example of two events which are not independent are when rolling a red and agreen die: Event A, getting a six on the red die, and Event B, getting a sum of 12. Theoccurrence of Event B is dependent on the result of Event A.

Expected Values

Each chance process has an expected result. This is the outcome with the greatestprobability of occurring. In thirty rolls of a die the expected number of 2's would be five.

The expected value of a chance process is the value that the mean result would be closestto over many repeated trials. This is why long run relative frequency (especiallysimulation) can be used to determine probabilities of events. In the game GreedyPigs the expected result for each round is 18. This means that over many games, theaverage score per round would approach 18.

Simulations

Simulations can be useful for the following purposes:C to trial an event that may take too long in real life C to trial an event that may be too expensive to trial in real lifeC to trial an event that may be unsafe in real lifeC to enable variables to be easily changed



Exercise 1: Sample Spaces

1. Draw tree diagram to show that there are sixteen different waysof having boys and/or girls in a family of four children.

2. List the 24 possible four letter “word” arrangements of the letters that make up theword RATE.

3. Draw up the sample space of the sum of the values on twodice.

4. Draw up the sample space of equally likely outcomes when selecting two coins atrandom from this set of coins:

5. A pair of players are chosen from the school tennis team of fiveplayers to play a doubles match. If the players in the team are Alison,Bree, Courtney, Danielle and Emma, write down all of the possiblepairs that could be chosen.

Exercise 2: Probability

1. A bag contains four coloured blocks, two green, one blue, one red. What is theprobability that if one block is drawn from the bag it will be;(a) green (b) red(c) not blue (d) red or green?

2. Use your sample space from Ex 1, Q3 to calculate these probabilities for thesum of two dice. The probability of rolling;(a) a sum of 6 (b) a sum of 12(c) a sum of 1 (d) a sum less than 10(e) a sum other than 8 (f) a double (both dice the same)(g) a 2 on one of the dice (h) a 2 on either of the dice.



3. Use your sample space from Ex 1, Q2 to calculate these probabilities for the“words” making up the sample space. The probability that the “word”;(a) starts with R (b) ends with R(c) starts with R or ends with R (d) starts with R and ends with R(e) starts with R or ends with E (f) starts with R and ends with E(g) has the letter A following the letter R (h) has the letters A and E adjacent.

4. Use your sample space from Ex 1, Q5 to calculate these probabilities for thedoubles team. The probability that the team;(a) is Courtney and Emma (b) contains Alison(c) does not contain Bree (d) contains Bree or Emma.

5. What is the probability that a four digit randomly allocated PIN number;(a) starts with a 4 (b) has all four digits the same(c) starts and ends with the same digit?

6. Use your sample space from Ex 1, Q4 to calculate these probabilities for thevalue of the two coins. The probability the coins are worth;(a) 10c (b) 70c(c) 40c (d) less than 50c(e) more than 20c (f) an amount other than 15c(g) 60c (h) 60c if one of them is a 50c coin.

7. Based on your answers to parts (g) and (h) are “the value of the two coins” and“one of the coins being a 50c” independent events? (See the note on page14).

8. Explain why ‘it makes sense’ that these two events are not independent.

9. A computer company has a variety of different CPU’s in stock. The quantities ofeach type are shown in this table. Find the probability that if a CPU is sold (atrandom) from this stock it;(a) has 3.2 GHz Chip Speed(b) has 800 MHz BUS Speed(c) is not 3.0 GHz Chip Speed(d) has 2.8 GHz Chip Speed with 800

MHz BUS Speed(e) has 3.2 GHz Chip Speed given that it

has an 800MHz BUS Speed.(f) Based on your answers to (a) and (e),

are the Chip Speed and BUS speedindependent?

BUS Speed

Chip Speed 533 MHz 800 MHz

2.4 GHz 8 5

2.67 GHz 15 12

2.8 GHz 12 24

3.0 GHz 0 10

3.2 GHz 0 5



10. Eighty four high school students who had their driving license were asked “Do youdrive your own car, your parents car or both?” The results are shown in this Venndiagram.

Use the diagram to find the probability that arandomly chosen student from the survey;

(a) drives their own car only

(b) drives their parents’ car at some time

(c) answered “both” to the survey question.

(d) What answer would have been given tothe survey question by the five studentsrepresented in the bottom right corner of the Venn diagram?

Exercise 3: Expectation

1. In a game of Monopoly the two dice are rolled 300 times. How many times wouldyou expect a double (both dice the same) to come up?

2. In the game Two Up “spinning tails” is when both coins show a tail. Over a nightsplay if the coins are thrown up 320 times how many times would you expect to“spin tails”?

3. In 80 families of four children, how many of the families would you expect to have;

(a) all boys (b) no girls

(c) at least one girl (d) two boys and two girls

(e) children born in the order boy, girl, boy, girl?

(You drew up the sample space for this situation in Ex 1, Q1.)



Puzzle:

Card Games

The diagram below shows the 52 cards in a standarddeck of cards. There are 2 red suits, Hearts Ì, andDiamonds Ë, and 2 black suits, Clubs Ê, Spades Í.Each suit contains 13 cards; Ace, King, Queen, Jack,10, 9, 8, 7, 6, 5, 4, 3, 2.

Instructions:By answering the twenty one probability questions that follow you can find the names often card games. Just look for your answers in the code boxes and match the questionnumber with the letter. (You may need to find the smallest equivalent fraction for yourprobability.)

1. If a card is drawn from a deck what is the probability of it being the 4Í?2. If you are dealt a hand of 4 cards how many hearts would you expect to get?3. If a card is drawn from a deck what is the probability of it being a spade?4. In sixteen cards how many would you expect to be diamonds?5. If a card is drawn from a deck what is the probability of it being an Ace?6. If you split the deck between two people how many kings would you expect in each

half of the deck?7. If a card is drawn from a deck what is the probability of it being red?8. In a hand of ten cards, how many would you expect to be black cards? 9. If a card is drawn from a deck what is the probability of it being a picture card

(i.e. a King, Queen or Jack)?10. If you are dealt a hand of 13 cards how many picture cards would you expect to

get?11. If a card is drawn from a deck what is the probability of it being a 3 or a 4?12. If you are dealt a hand of 13 cards how many cards would you expect not to be

picture cards?13. If a card is drawn from a deck what is the probability of it not being a 10Ì?



14. If a card is drawn from a deck what is the probability of it being King or a Heart? 15. If a card is drawn from a deck what is the probability of it not being a 3?16. If an Ace is drawn from a deck what is the probability of it not being the AÌ?17. If two Jokers are added to a standard deck of cards what is the probability of

then drawing a Joker?18. If a 3Í is drawn from a deck and then another card is drawn without replacing the

3Í what is the probability it will be a 3?19. If you already have Ace, King, Queen, Jack what is the probability the next card

you will get from a deck is a ten?20. If you have a pair of sevens and you get one more card from the deck without

replacing the pair what is the probability it is also a seven?21. In a game of Blackjack you are dealt an Ace as your first card, what is the

probability of the next card you are dealt being either a King, Queen, Jack orTen to give you Blackjack?

Code Box:

A B C D E G H I J K L M N O P R S T U W Y

1 10 2 3 5 4

Ten Card Games:

13 6 21 20 14 7 21 20 14 16 20 21 19

13 21 20 20 21 18 21 19 11 4 14 17 18

20 21 8 21 16 19 21 17 9 20 12 18 17

13 18 15 1 2 17 16 11 15 19

5 12 15 16 19 18 9 10 10 3



Application A:

DNA

Deoxyribonucleic acid (DNA) is a molecule foundin all life forms that determines the inheritedcharacteristics of that life form. How the DNA isput together dictates the colour of a persons hairor the height of a tree.

As the diagram here shows, a DNA molecule is aspiralling structure consisting of two strands(made up of sugar and phosphate) connected by what are known as nucleotides.

Each nucleotide is made up oftwo matching bases. There arefour bases; thymine (T), adenine(A), cytosine (C) and guanine (G). The bases can only be matched Awith T (or T with A) and C with G(or G with C) as shown in thisdiagram.

The order that the nucleotides are arranged determines the inherited characteristics.

This diagram shows onesequence of 5 nucleotides:

Task 1: Complete this table showing how many possible sequences of nucleotidesthere can be:

No. of nucleotides 1 2 3 4 5 6

Possible sequences

(Note that AT is different from TA and that GC is different from CG).



Task 2: Use the pattern in this table to predict how many possible sequences therecould be for 8 nucleotides.

Task 3: What type of function would model the values in this table?

Task 4: Develop a function that gives the number of sequences from the number ofnucleotides.

Task 5: What is the least number of nucleotides in sequence to give more than onemillion different possible structures?

Task 6: What is the minimum number of nucleotidesnecessary to give 7 billion sequences (7 x 109) sothat every human on the planet could have adifferent one?

Dr Fraiser from the Stargate team discovers that the Jaffa (an alien race) have an extratwo bases which she names, Jackine (J) and Samine (S) which can only go together. Task 7: How many different sequences of 10 nucleotides are now possible?

Task 8: What is the minimum number of nucleotides necessary to give 12 trillionsequences (1.2 x 1013) so that every Jaffa in the universe could have adifferent one?



Application B:

Brownlow Medal Betting

At the end of the AFL season voting for the fairest and best player of theyear is counted and the Brownlow Medal is awarded to the winner.

The table here shows the betting price (odds for winning) quoted by abookmaker on the day of the count for the ten most favoured players.

Player Name ClubPrice

(decimal)Price

(fractional)Win for $1 bet

Odds as aprobability

N. Buckley Collingwood $3.25 9/4 $2.25

M. Voss Brisbane $4.50 7/2

G. Wanganeen Port $5.50 $4.50

M. Riccuito Adelaide $9.00

R. Harvey St Kilda $11.00 $10.00

A. Goodes Sydney 10/1

P. Bell Fremantle $13.00 12/1

S. Crawford Hawthorn $14.00 $13.00

W. Tredrea Port $21.00 20/1

J. Hird Essendon 30/1 $30.00

The price is shown in two different but equal formats, the modern decimal form and thetraditional fractional form. C The price in decimal form is the return on a bet of $1. For example $3.25

indicates that for betting $1 you get a return of $3.25, winning $2.25. C The price in fractional form shows the amount won per amount bet. For example

9/4 says you will win $9 if you bet $4, getting a return of $13. C These two prices work out the same for the same bet.

Task 1: Use this information to complete the missing amounts in the “Price”columns and the “Win for $1 bet” column of the table.



The probability of each player winning has been calculated from the price (odds). For example, N. Buckley’s odds of 9/4 indicates 9 chances of losing to 4 chances ofwinning. Hence, his probability of winning is 4 chances out of a total of 13 chances

(4+9) or .

Task 2: Use this information to complete the missing values in the “Odds as aprobability” column.

Task 3: Total up the probabilities for the ten players.

As well as these ten players there were about 550 other players eligible for the BrownlowMedal. These other players were quoted as having very small probabilities of winning

ranging from to about .

Task 4: Taking these other players into account estimate the total probability.

Task 5: Why is this total unusual for probabilities?

Task 6: To make the probability equal to one what would need to happen to theprices?

Task 7: Comment on why the prices offered by the bookmaker might be smallerthan they should be according to probability theory.

Task 8: Calculate the return and how much would be won or lost on the followingbets if the result was as shown:

Player Result Bet Return Amount Won or Lost

N. Buckley Win $40

M. Voss Loss $500

A. Goodes Win $50

P. Bell Loss $200

G. Wanganeen Win $25

If you want to know who won this Brownlow Medal for 2003 you can find a listing of all of themedal winners over the years via the TLR page for this module at:http://www.otrnet.com.au/IntegratedMathsModules/H06/H06TLR.html(You might like to discuss the bets from task 8 with reference to the actual result.)



Application C: Materials required: Method of simulating probabilities of 0.2 and 0.9

Medical Probabilities

In current medicine, testing for a bacteria known as Heliobacter Pyloriis a part of the diagnosis of stomach ulcers. The bacteria has beenfound to be present in approximately 20% of people under 40 in our community. If present, it is often the cause of a stomach ulcer.

One of the tests for this bacteria is known as the CLO test. Research has shown that thistest is about 90% accurate. That is, it correctly diagnoses the presence of the bacteria90% of the time.

Task 1: Explain how you can simulate a probability of 20% representing thepercentage of people in our community with Heliobacter Pylori present.

Task 2: Explain how you can simulate a probability of 90% representing theaccuracy of the CLO test.

Task 3: Work in a group to simulate the testing of 40 people. First decide if theyhave the bacteria (20%) and then decide if the test result is correct (90%). Tally your results in this table:

Bacteria Present Bacteria Absent

Test correct

Test incorrect

For example, if your first person came up as having the bacteria absent andtested correctly they would go in the top right cell.

Task 4: Which two cells contain people that the test showed as having the bacteria?

Task 5: Which two cells contain people that the test showed as not having thebacteria?



Task 6: Combine your results with the rest of the class.

Task 7: What is the purpose of combining your results withthe rest of the class?

Task 8: From the class results, how many people did the test show as having thebacteria?

Task 9: How many of the people who the test showed as having the bacteria,actually did not have it?

Task 10: Of the people who the test showed had the bacteria, what percentageactually did not have it? (Use your results from tasks 8 and 9.)

Task 11: Comment on the usefulness of the test given your result from task 10.

The theoretical results for this simulation can be calculated using a probability treediagram. The diagram for this simulation is shown below with the probabilities.



Outcome A in this diagram represents people who had the bacteria present and got a testresult showing the presence of bacteria. This is 18% of the total.

Task 12: Explain each of the other three outcomes of the tree diagram (B, C andD), in terms of whether they actually had the disease and what the testshowed.

The ability of a test to correctly identify the presence of a disease indiseased people is known as the test’s Sensitivity. Its ability tocorrectly identify the absence of the disease in non-diseased peopleis known as its Specificity.

In rarer diseases even very accurate tests can have the problem of asignificant number of “false positives”. These are the peoplelabelled D in the tree diagram.

Task 13: Draw up a tree diagram to show the results for ascreening test that is 95% accurate and is used to test for a rare diseaseaffecting only 1% of the community.

If 10 000 people were tested, 100 of them (1%), would be expected to have the raredisease .

Task 14: Use the tree diagram to find how many of the 10 000 people would beexpected to show up under the testing as having the diseases but who don’treally have it (the false positives).

As you can see from your result from task 14 the problem of false positives is a significantconcern in testing for rare diseases. The usual response from this is to give a second testif the first test gives a positive result.

You can find out more about this aspect of medical testing and probabilities by usingthe search box on a page linked to the TLR page for this module at: http://www.otrnet.com.au/IntegratedMathsModules/H06/H06TLR.html



Application D:

Project: Population Control

Various methods of controlling the earth’s population have beensuggested over the years. One possible method might be a lawthat states:

“ Families may not produce any more children once they have a boy”.

This means that we could have families like;

Girl, girl, boy or Girl, girl, girl, girl, boy or maybe just Boy

� Assuming that families under this law would have as many children as legally possible:

Task 1: Draw a tree diagram to show the families that would result with five orless children.

Task 2: The tree diagram partly completed below shows some of the outcomeswith probabilities. If the probability of a boy or a girl remains at 0.5,complete your tree diagram showing all of the outcomes andprobabilities.

Task 3: What is the probability of a family with girl, girl, girl, girl, boy?



Task 4: Complete the second row in this table showing the probabilities ofvarious family types occurring.

Family type B GB GGB GGGB GGGGB

Probability 0.125

Out of 64 8

Task 5: If 64 random families followed this law, complete the 3rd row of the tableto show many of each of these types of families you would expect.

Task 6: Total up the families in the 3rd row of the table.

Task 7: Explain why only sixty two families are shown in the table.

Task 8: For these 62 families, how many childrenwould you expect there to be, using thismethod of population control?

Task 9: If we consider that an average of two children per family would keep thepopulation steady, how well does the population law seem to work?

Task 10: Reinvestigate the results here without restricting families to 5 or lesschildren. A spreadsheet available via the TLR page for this module at:http://www.otrnet.com.au/IntegratedMathsModules/H06/H06TLR.html can be used to help.

Task 11: Investigate the population control law: “Families may not produce anymore children once they have a boy and a girl”.



Application E:Materials required: Newspapers

Newspaper Search:

Relatively Speaking

Relative frequencies are generally used to estimate probabilities when it is notpossible to calculate the probability of an event occurring.

To do this the following formula is used:

Task 1: Get a copy of the “Homes For Sale” from a newspaper or from a RealEstate Agent.

Task 2: Set up a frequency table similar to the one below and record the prices of asample of 200 homes. Use the following guidelines when choosing yoursample:C Choose properties at random.C Include houses, units, villas or duplexes but not blocks of land.C Where the exact price is not given disregard the advertisement. For

example: $170 -190,000 or $140,000'sC Disregard advertisements for multiple sales. For example: Choice of

two from $134,750

PRICE TALLY FREQUENCY

Less than $100,000

$100,000 - $149,999

$150,000 - $199,999

etc.

etc.

$500,000 or over



Task 3: Using the table you have constructed and therelative frequency formula for probabilitygiven earlier answer these questions:

1. What is the relative frequency of homeprices in the category $200,000 to$249,999?

2. What is the probability that if a home ischosen at random it will be between$300,000 - $349,999?

3. What is the probability that a home will be below $300,000?

4. What is the probability that a home will be between $400,000and $499,999?

5. What is the probability that a home will not be $350,000 ormore?

6. What is the probability that a home will be over $399,999?

7. If there are 4000 homes for sale in this city what number would youexpect to be priced less than $250,000?

8. If you were told that a home price is less than $350,000, what is theprobability of it being more than $200,000?

Properties for sale:

Rivervale: Fantastic969sqm with 3brm housefor $259,900.Rockingham: Investmentpotential 4 x 2 with oceanglimpses a steal at$328,000.Roleystone: Countrygrace in this 4 x 2mansion near golf courseat $595,000.



Student Reflection

What have I learned in this module?

What new words did I learn during this module?

Look at the outcomes at the start of the module (page 3). Have I progressed on each ofthese outcomes?

What do I need to improve on?

Write about one thing in this module I found interesting.

What do I think was the most important concept in this module?

Where could the maths in this module be used in our society?

One area I would like to look more at is:

Write something about how the bits in this module connect to each other.

Write something about how the bits in this module connect to other modules.



ANSWERS TO EXERCISES:

Exercise 1:1. BBBB, BBBG, BBGB, BGBB, GBBB, BBGG, BGBG, GBBG,

BGGB, GBGB, GGBB, BGGG, GBGG, GGBG, GGGB, GGGG2. RATE, RAET, RTAE, RTEA, REAT, RETA, ARET, ARTE, ATRE, ATER, AERT, AETR,

TRAE, TREA, TARE, TAER, TERA, TEAR, ERAT, ERTA, EART, EATR, ETRA, ETAR3.

2 3 4 5 6 7

3 4 5 6 7 8

4 5 6 7 8 9

5 6 7 8 9 10

6 7 8 9 10 11

7 8 9 10 11 12

4. 5,5: 5,10: 5,10: 5,10: 5,20: 5,50: 5,5: 5,10: 5,10: 5,10: 5,20: 5,50: 10,5: 10,5: 10,10: 10,10:10,20: 10,50: 10,5: 10,5: 10,10: 10,10: 10,20: 10,50: 10,5: 10,5: 10,10: 10,10: 10,20: 10,50:20,5: 20,5: 20,10: 20,10: 20,10: 20,50: 50,5: 50,5: 50,10: 50,10: 50,10: 50,20

5. AB, AC, AD, AE, BC, BD, BE, CD, CE, DE (order does not matter here).

Exercise 2:[Probabilities from sample spaces are unsimplified to aid checking.]

1. (a) (b) (c) (d)

2. (a) (b) (c) 0 (d)

(e) (f) (g) (h)

3. (a) (b) (c) (d) 0

(e) (f) (g) (h)

4. (a) (b) (c) (d)

5. (a) (b) (c)

6. (a) (b) (c) 0 (d)

(e) (f) (g) (h)

7. They are NOT independent.8. Because if the 50c coin is one of the two coins then the chance of the total being 60c is increased.

9. (a) (b) (c) (d) (e) (f) NOT independent

10. (a) (b) (c) (d) I don’t drive either.

Exercise 3:1. 50 times2. 80 times3. (a) 5 (b) 5 (c) 75 (d) 30 (e) 5


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