NCEA
LEVEL 1
MATHEMATICS
Part 6 - AS90194
Determine Probabilities
QUESTIONS & ANSWERS
MAHOBE
Published by Mahobe Resources (NZ) Ltd
Distributed free at www.mathscentre.co.nz
2
NCEA Level 1 Mathematics, Questions & Answers
Part 6 - AS90194 Determine Probabilities
Contributors: Dr Chris Davidson, Kim Freeman, Dr Sophia Huang, Farisha Khan, Ian O’Connell
This edition is Part 6 of a 6 Part eBook series designed to help you study towards NCEA.
Published in 2009 by:
Mahobe Resources (NZ) Ltd
P.O. Box 109-760
Newmarket, Auckland
New Zealand
www.mahobe.co.nz
www.mathscentre.co.nz
© Mahobe Resources (NZ) Ltd
ISBN(13) 9781877489075
This eBook has been provided by Mahobe Resources (NZ) Ltd to The New Zealand Centre of Mathematics.
School teachers, University lecturers, and their students are able to freely download this book from The New
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out in accordance with Copyright Licensing Ltd guidelines. The content presented within the book represents
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All rights reserved. All the views expressed in this book are those of the author. The questions and
suggested answers are the responsibility of the author and have not been moderated for use in NCEA
examinations.
MAHOBE
YEAR 11 MATHEMATICS
3
NCEA Level 1 Mathematics - Questions & Answers
Contents
Probability
8
Achievement Examples
9
Achievement Exercises
11
Tree Diagrams
16
Merit Example
17
Merit Exercises
18
Excellence Example 1
23
Excellence Exercises
24
Excellence Example 2
27
Sample Exam
29
Answers
36
YEAR 11 MATHEMATICS
MAHOBE
4
MAHOBE
YEAR 11 MATHEMATICS
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About This Book
Q&A eResources are recognised as the leading study guides for NCEA. Each
freely available title has been compiled by a team of experienced educators to
meet the study and revision needs of NCEA students. They are proving to be
valuable resources in the hands of students who want to work ahead of their
regular class programme. They also serve as effective revision programmes in the
run up to the final examinations.
This book carefully explains the mathematical concepts that will be tested in
NCEA then illustrates them with Achievement, Merit and Excellence examplars. It
allows students to practise on NCEA-type questions and provides detailed
solutions. After working through this programme, all students will be well prepared
for their final assessments.
Simplify:
32 =
6
8y + 12
=b + 1
5
=
10
4 × 2y + 3 =
(b + 1)2
1 + 4
3
7
2
The student who wrote the above answer on a recent assessment paper did not
use a Q&A Level 1 Mathematics eResource.
YEAR 11 MATHEMATICS
MAHOBE
6
MAHOBE
YEAR 11 MATHEMATICS
7
MATHEMATICS 1.6 - AS90194
Determine probabilities
Always understand what the examiner wants! A past examination answer is
shown below. The student who wrote this answer on a recent assessment paper
did not use a Q&A Level 1 Mathematics eResource.
Mr and Mrs Jones have three children
all of whom are girls. What is the
probability of their next child being a
boy?
Their next child will actually be
Chinese. This is because one in
every 4 children born in the world is
Chinese.
YEAR 11 MATHEMATICS
MAHOBE
8
Probability
1.
2.
3.
4.
Probabilities can be expressed as either fractions, decimals or percentages.
The notation P(x) = ½ can be read as: “the probability that event x happens
is one half”. P(x) is “probability of x happening”, P(x’) is “of x not happening”.
Probabilities add up to 1. This means the individual events that make up an
overall event will add to 1, e.g P(pass) = 0.35 then P(fail) = 0.65
List all the possible outcomes. e.g. The diagram below shows a die and a
fair equilateral triangle spinner. When the die is tossed and the spinner spun
the two scores are added together. The example below shows a total of 8.
What are all the possible outcomes? Calculate P(x > 8).
In this type of example draw a table showing all possible scores.
Spinner
Die
1
2
3
4
5
6
3
4
5
6
7
8
9
5
6
7
8
9
10
11
7
8
9
10
11
12
13
There are 18 possibilities. There are 9 examples of scores greater
than 8. This means P(x > 8) = 9
18
5.
The “and” Rule: P(A and B) = P(A) × P(B)
The probability of both event A and event B happening is equal to the two
separate probabilities multiplied together. For this to happen the events have
to be independent i.e. one event can not affect the result of the other.
e.g. Find the probability of drawing two Aces from a pack of cards.
P(1st card is an Ace) = 4/52
P(2nd card is an Ace) = 3/51- assume the first card isn’t
replaced. Now apply the and/or rule. Both events must happen
therefore use “and” rule.
4
1
3
52 × 51 = 221
6.
The “or” Rule: P(A or B) = P(A) + P(B)
The probability of either event A or event B happening is equal to the two
separate probabilities added together. For this to happen the events have to
be mutually exclusive i.e. if the first event happens the other can not happen.
MAHOBE
YEAR 11 MATHEMATICS
9
Probability - Achievement Examples
1.
Food City Supermarket is running a promotion to help increase sales. They
have six cards in a box at the checkout. Each card has a different letter from
the word “SAVING” written on it, as shown below:
S
A
V
I
N
G
If a customer spends $20 or more they draw one of these cards from a box at
the checkout. If the letter has not been written on the Result Board, at the
front of the store then they write it up. Then they return the letter to the box.
a. What is the probability that a customer spending at least $20 will draw a
card with a vowel on it?
There are 6 letters altogether. There are two vowels - A and I.
Therefore the probability is 2/6 = 1
3
b. One morning the board showed:
S
V
N
What is the probability that the next letter drawn from the box has not
already been written on the board?
There are 6 letters altogether. There are three possible letters
1
not drawn. Therefore the probability is 3/6 = 2
c. Later in the day the board shows:
A
V
I
N
G
If the next customer draws a card that completes the word “SAVING” on
the board they win a free ticket to the movies. What is the probability that
the next customer that spends $20 will win a free movie ticket?
There are 6 letters altogether. There is only one possible
winning letter. Therefore the probability= 1
6
YEAR 11 MATHEMATICS
MAHOBE
10
2.
The table shows the number of people arriving at New Zealand international
airports in the 5-year period from 2005 to 2009.
Year
Overseas
NZ Residents
Long Term
Visitors
Returning
Arrivals
Total
2005
2006
2007
2008
2009
What is the probability that a person arriving at a NZ international airport:
a.
in 2009 was an overseas visitor?
It is easier to calculate when
Total number of arrivals in 2009 was: you eliminate some zeros!
(1690 + 1250 + 4.1) × 1000 = 2,944,100
There are 1,690,000 overseas visitors in 2007.
1690000
Therefore the probability is: 2944100 = 0.5740
b.
in the 5-year period, would have arrived as a NZ Resident Returning?
Add all the figures in the NZ Resident Returning column.
(980 + 1100 + 1150 + 1180 + 1250) × 1000
= 5,660,000
Add all the figures in the Overseas Visitors column.
(1470 + 1540 + 1460 + 1540 + 1690) × 1000
= 7,700,000
Add all the figures in the Long Term Arrivals column
(54 + 48 + 46.1 + 43 + 41) × 100 = 23,210
Now add the three totals together. This gives us the total
arrivals: 5660000 + 7700000 + 23210 = 13,383,210
5660000
Therefore the probability is: 13383210 = 0.4229
c.
in 2009, was not an overseas visitor?
2009 there are (1690 + 1250 + 4.1) × 1000
= 2,944,100 arrivals
Not overseas visitors (1250000 + 4100) = 1,254,100
MAHOBE
Therefore the probability is:
1254100
2944100 = 0.4260
YEAR 11 MATHEMATICS
11
Exercises
1.
Hilary works for the camper van company Holiday Vans Ltd. The table below
gives the number of different sized camper vans hired during the last year.
Camper vans were hired by both New Zealanders and by overseas visitors.
Holiday Vans Ltd
2-person
4-person
6-person
Hired by a New Zealander
155
250
121
Hired by an overseas visitor
240
96
67
a.
What was the probability that a van was hired by a New Zealander?
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......
b.
What was the probability that a 4 person van was hired?
... . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......
Hilary, George and Obama hire a camper van for a two-week holiday. Each
day they spin to see who will drive. The spinner is shown below. It has equal
sized sectors.
Hilary
Obama
George
c.
What is the probability that Obama drives the camper van on each of
the first two days of the holiday?
... . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......
d.
What is the probability that there will be different drivers on each of
the first two days?
... . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......
YEAR 11 MATHEMATICS
MAHOBE
12
2.
Shania has been keeping records of her wins and losses for the different
types of athletic events that she enters. This allows her to find out in which
event she has the most success. Her tabulated records are below.
Triathalon
Athletics
Swimming
Cycling
Totals
a.
Wins
30
41
5
12
88
Total .
37
47
13
20 .
117
Losses
7
6
8
8
29
What is the probability that Shania wins a triathalon?
.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......
b.
What is the probability that the last event Shania won was a
swimming one?
.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......
$0 $100
$0
$0
$10
$0
0
$5
$0
$10
Helen and John are at a political fundraiser.
Helen has one spin of the prize wheel shown.
a.
Which prize is she likely to win?
(Explain your answer)
$50 $10
$0
3.
.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......
.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......
b.
John has one spin of the prize wheel. By using at least one of the
terms “certain”, “likely”, “possibly”, or unlikely describe his chances of
winning either $100, $50, $10 or a monetary prize.
.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......
.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......
MAHOBE
YEAR 11 MATHEMATICS
13
4.
Below is a table showing student numbers at Mahobe High School.
Year Level
Boys
Girls
Totals
a.
Year 9
120
130
250
Year 10
115
125
240
Year 11
135
155
290
Year 12
80
75
155
Year 13
45
55
100
Totals
495
540
1035
What is the probability that a randomly chosen student is in Year 11?
........................................................
b.
What is the probability that a randomly chosen girl is in Year 11?
........................................................
The canteen staff at Mahobe High School want to encourage students to buy
healthy food. They introduce an incentive scheme. Every time a student
purchases healthy food they are given a card with one of the following
letters. D V I
S K The object is for the students to collect the
cards until they have 7 cards that make up the words DVD DISK. When this
happens then they can exchange the cards for a free DVD.
c.
Rubeun buys two healthy foods and is given two cards.
What is the probability that the two cards are a “V”?
........................................................
d.
Steve buys two healthy foods and is given two cards.
What is the probability that the two cards have different letters?
........................................................
e.
Benson gets a card on Wednesday and another on Friday.
What is the probability that Benson has exactly one D?
........................................................
YEAR 11 MATHEMATICS
MAHOBE
14
5.
The table shows the sports in which New Zealanders have won medals at
the Olympic Games.
Gold Medals
Water Sports
19
Athletics
8
Other
7
Totals
34
Silver Medals
8
1
6
15
Bronze Medals
15
9
7
31
Totals
42
18
20
80
a.
What is the probability that a randomly chosen New Zealand medal is
a gold medal?
........................................................
b.
What is the probabilty that a randomly chosen New Zealand Olympic
gold medal was won in athletics?
........................................................
Paul and Bernadene are playing a game with a spinner. The spinner has
sectors coloured RED, BLUE and GREEN (as shown below). Paul and
Bernadene each spin the arrow once.
RED
BLUE
c.
GREEN
What is the probability that Paul and Bernadene both spin RED?
........................................................
........................................................
d.
What is the probability that Paul and Bernadene both spin the same
colour?
........................................................
................ . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .
MAHOBE
YEAR 11 MATHEMATICS
15
YEAR 11 MATHEMATICS
MAHOBE
16
Tree Diagrams
Tree diagrams are used to illustrate the various possibilities when several events
occur.
Add the end results
4
7
Multiply along
the branches
1
5
4
5
Black
(Black Black) 1 × 4 = 4
5 7
35
White
(Black White)
Black
3
7
4
7
Black
1 3
3
×
=
5 7
35
4 4
16
(White Black) 5 × 7 = 35
White
3
7
White
4 3
12
(White White) 5 × 7 = 35
Total = 1
Event 1
Event 2
4
5
Total is always 1
3
7
= 12
35
1
5
3
7
4
5
4
7
= 19
35
MAHOBE
YEAR 11 MATHEMATICS
17
Probability - Merit Examples
A survey has shown the following reasons for people coming to New Zealand.
25% arrive mainly for business, the rest arrive mainly for a holiday.
20% of the people arriving on business visit friends while they are here.
35% of the people arriving for a holiday visit friends while they are here.
Some of this information is illustrated on the diagram below.
People coming to New Zealand
Visit friends
Holiday
Do not visit friends
Visit friends
0.25
Business
Do not visit friends
What is the probability that a person coming to New Zealand:
a.
will arrive on holiday and visit friends?
0.75 × 0.35 = 0.2625
b.
will not visit friends?
(0.75 × 0.65) + (0.25 × 0.80) = 0.6875
70% of those people who arrive in New Zealand on business and visit friends,
also tour the North and South Islands.
c.
What is the probability that a person comes to NZ on business, and
visits friends but doesn’t visit both the North and South Islands?
For this question add to the tree diagram
(the relevant branches are shown).
0.25 × 0.2 × 0.3 = 0.015
Tours
Visits friends
0.25
0.2
Business
YEAR 11 MATHEMATICS
0.3
No tours
MAHOBE
18
6.
Holiday Vans Ltd hires camper vans to New Zealanders and visitors from
overseas. They collected the data below from last year’s rentals.
i.
4 out of 5 camper vans are hired by overseas visitors.
ii.
Of the overseas visitors, 75% hire a 2 person van, 15% hire a 4
person van and the rest hire a 6 person van.
iii. Of the New Zealanders, 20% hire 2-person van, 45% hire a 4-person
van and the rest hire a 6-person van.
Some of this information is shown on the tree diagram below.
75%
4 out of 5
Overseas visitors
2-person
4 person
6-person
2-person
New Zealanders
4 person
6-person
a.
What is the probability that a 6-person Holiday Vans camper was hired
by an overseas visitor last year.
........................................................
b.
What is the probability that a Holiday Vans camper van hired out last
year was a 2-person van?
........................................................
c.
95% of New Zealanders return the camper van to the town where they
hired it. 55% of overseas visitors return the camper van to the town
where they hired it. Calculate the probability that the next camper van
hired will be returned to the town here it was hired.
........................................................
........................................................
MAHOBE
YEAR 11 MATHEMATICS
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7.
Marina, Onni and Chris like to play tennis. When Marina and Onni play,
Marina wins 2 out of 5 games. When Marina and Chris play, Marina wins 3
out of 4 games.
Marina plays one game of tennis against Onni and one game against Chris.
Some of the information is given on the tree diagram below.
3
4
2
5
Marina wins
Marina wins
Chris wins
Marina wins
Onni wins
Chris wins
a.
What is the probability that Marina wins both games?
........................................................
b.
What is the probability that Marina wins exactly one of the two games?
........................................................
c.
When it is windy, Marina does not play well against Chris.
She only wins 2 out of 5 games when she plays Chris on a windy day.
They are playing tomorrow and the probability of it being windy
tomorrow is 0.55.
What is the probability that Marina will win the game?
........................................................
........................................................
........................................................
........................................................
YEAR 11 MATHEMATICS
MAHOBE
20
8.
The Athletics Sports Day at Mahobe High School is scheduled for either late
February or early March. Over the last few years it has been held 42% of the
time in February and 58% of the time in March.
The probability of fine weather in during February is 0.8
The probability of fine weather in March is 0.5
If the weather on Athletics Sports day is wet then the day is postponed.
a.
What is the probability that the Athletic Sports will be postponed on
any chosen year?
Some relevant information is given on the tree diagram below.
Fine
February
42%
Wet - Athletics postponed
0.5
Fine
March
Wet - Athletics postponed
........................................................
........................................................
b.
When the postponement date is also not fine, the Athletic Sports day is
cancelled and not held at all. Over the last few years if a February
date is postponed then there is a 5% chance that the sports day is
cancelled. If a March date is postponed then there is a 55% chance
that the sports day is cancelled. What is the probability that the
Athletics Sports day will be cancelled in any one year?
........................................................
........................................................
MAHOBE
YEAR 11 MATHEMATICS
21
9.
Willis is going to run a 1500 metre race. He has run personal best times in 8
out of the last 20 races when the temperature was over 28ºC. He has run
personal best times in 4 out of the last 20 races when the temperature was
28ºC or under. During February, when he races, 75% of the days have
recorded a temperature of over 28ºC. Some of the information is given on
the tree diagram below.
Personal best recorded.
Temp > 28º
75%
No Personal best recorded
4
20
Personal best recorded.
Temp < 28°
No Personal best recorded
a.
What is the probability that Willis runs a personal best time and the
temperature is over 28°C?
........................................................
b.
What is the probability that Willis will run a personal best time in his
next race?
........................................................
c.
If the temperature is 28° or below and Willis runs a personal best time
the probability that he wins the race is 7 . What is the probability that,
8
if the temperature is 22°C, Willis will run a personal best time but will
not win the race?
........................................................
........................................................
........................................................
YEAR 11 MATHEMATICS
MAHOBE
22
10.
Cullen travels by train to school each day. He collects data on the arrival
time of the train each morning. He records:
1.
Whether the weather is wet or fine.
2.
If the train is less than 5 minutes late.
3.
If the train is more than 5 minutes late.
The train is never early! Cullen’s results show the weather is wet for 0.3 of
the days. If it is wet, the train is 5 or more minutes late 3 days in 5.
If it is fine then the probability of the train being less than 5 minutes late is
0.45. The probability tree below gives some of his data results.
Timing
Weather
0.45
< 5 minutes late
Fine
5 or more minutes late
< 5 minutes late
0.3
Wet
3
5
a.
5 or more minutes late
What is the probability that on any day:
i.
The weather is wet and the train is less than 5 minutes late.
........................................................
ii.
The train will be 5 or more minutes late?
........................................................
b.
The probability that Cullen gets a seat on a wet day is 0.25.
Given that it is a wet day, what is the probability that the train is 5 or
more minutes late and Cullen will have to stand?
........................................................
........................................................
MAHOBE
YEAR 11 MATHEMATICS
23
Probability - Excellence Example 1
Holiday Vans Ltd hires camper vans to New Zealanders and visitors from
overseas. The most common booking is for a 10-day holiday.
88% of customer requests for a 10 day holiday booking can be confirmed
immediately. The rest of the customers (who require a 10 day holiday booking) go
onto a waiting list. Of those on the waiting list only 20% eventually have their
booking confirmed. Holiday Vans Ltd had to turn away 85 customers last year
because their request for a 10-day holiday booking could not be confirmed.
From the information given, calculate the total number of customer requests for a
10-day holiday booking.
Start by drawing a tree diagram of the situation.
Booking confirmed immediately
0.88
0.20
0.12
Booking confirmed later
Booking not confirmed
Booking unsuccessful
P(successful 10-day booking)
= 0.88 + 0.12 × 0.2
= 0.904
P(unsucessful 10-day booking) = 1 - 0.904
= 0.096
85 customers were unsuccessful in obtaining a 10-day holiday
booking
Expected number = probability × number of trials
Therefore 85 = 0.096 × N
N = 885.417
This means a total of 885 customer requests.
YEAR 11 MATHEMATICS
MAHOBE
24
11.
Ellis has two unusual dice. They both have 6 sides but they have the
following characteristics.
Dice 1: Has 3 sides labeled 1 and 3 sides labeled 2.
Dice 2: Has either a 5 or a 6 on each of its sides.
Ellis rolls the dice 150 times and records the results (shown below).
Total shown on 2 dice
Frequency
6
48
7
77
8
25
Determine the most likely number of sides on Dice 2 that are labeled
with a “5”.
....... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .
....... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .
....... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .
....... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .
....... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .
....... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .
....... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .
....... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .
....... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .
....... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .
....... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .
....... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .
MAHOBE
YEAR 11 MATHEMATICS
25
12.
At the World Athletics Champs, New Zealand has one competitor in each of
the 800m, 1500m and 5000m events.
Our 800m runner has a probability of 0.12 chance of qualifying for the final.
Our 1500m runner has a probability of 0.27 of qualifying for the final.
Our 5000m runner has a probability of 0.15 of qualifying for the final.
What is the probability that there will be at least two New Zealanders in the
Athletics finals?
........................................................
........................................................
........................................................
........................................................
........................................................
........................................................
........................................................
........................................................
........................................................
........................................................
........................................................
........................................................
........................................................
YEAR 11 MATHEMATICS
MAHOBE
26
MAHOBE
YEAR 11 MATHEMATICS
27
Probability - Excellence Example 2
Daisy, Daniel and Celine play a game called “Paper, Scissors, Rock”. A game
consists of each player holding out one hand at the same time to show Paper,
Scissors or Rock in random choices. In the first game, Daisy chooses Paper, and
both Daniel and Celine choose Scissors. They decide to keep playing until they all
have the same outcome - either Paper, Scissors or Rock. They play 20 games,
however, it does not happen. Daisy considers this unusual and decides to carry
out an experiment.
a.
Describe a probability experiment (simulation) that would allow her to
estimate the probability of all players in a game making the same choice.
• Assume that you have access to: coins, cards, spinners, dice, and random
number generators on a calculator or computer.
• Give sufficient detail in your description so someone else could carry out the
experiment successfully.
Example explanations follow:
1.
Use three dice (or one dice three times). Use outcomes 1 and
2 for 'rock', 3 and 4 for 'paper' and 5 and 6 for 'scissors'.
Throw the dice between 20 and 50 times recording how
many times the outcomes are the same.
Or
2.
Using a computer write a program that generates three
numbers between 0 and 4 (1, 2 or 3). Assign rock paper or
scissors to each number. Have the program generate between
20 and 50 sets of 3 numbers and record how many times
the same three numbers are generated (outcomes). Note: you
would need to include a sample program and explain what
each line of code is being used for.
Or
3.
Have a box with 9 cards 3 each with 'scissors', 'paper' and
'rock' written on them. Draw out three cards, 20-50 times.
b.
Calculate the theoretical probability of all three players in a single game
making the same choice.
P(all same) = P(PPP) or P(SSS) or P(RRR)
1× 1× 1= 1
3 3 3 27
YEAR 11 MATHEMATICS
MAHOBE
28
13.
The DVD Emporium has a promotion to sell more DVDs. Every time a
customer purchases a DVD they are given a card with a letter printed on it.
The letters on the cards are:
Z E
P R I
Aishwarya wants to find out how many DVDs she would need to purchase
to get a free DVD prize. Describe a probability experiment to simulate this
situation. Your description must be clear enough for someone else to
duplicate. Make sure you list all the instructions needed to get an answer to
the question “What is the average number of DVDs needed to be
purchased to get a free prize?”. Aishwarya has access to such tools as dice,
coins, scientific calculator (with random key) and random number tables.
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MAHOBE
YEAR 11 MATHEMATICS
29
MAHOBE
Level 1 Mathematics - Sample Exam
AS90194 Determine Probabilities
Published by Mahobe Resources (NZ) Ltd
Distributed free at www.mathscentre.co.nz
YEAR 11 MATHEMATICS
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31
You are advised to spend 30 minutes answering the questions in this section.
QUESTION ONE
Ange has three different species of flowers in her garden. There are also 4
different colours of flower. The table below shows the flowers and colours present
on one particular day.
Flowers
Tulips
Roses
Daisies
Totals
(a)
White
12
8
25
45
Gold
10
5
20
35
Violet
40
0
10
50
Red
27
12
15
54
Totals
89
25
70
184
If Ange picked a flower at random from her garden, what is the probability
that it would be a gold coloured flower?
...........................................................
............................................................
(b)
Brad picks a rose from Ange’s garden. What is the probability that Brad’s
rose is red?
...........................................................
............................................................
QUESTION TWO
Cyrus has a collection of music on his MP3 player from Black Eyed Pods, Tyler
Swift and Lady Gaa. For every one Black Eyed Pods there are three Tyler Swift
tracks. There are four times as many Lady Gaa tracks as there are Tyler Swift
tracks on the player. Cyrus has the MP3 player select a random track. What is
the probability that it selects a Tyler Swift track?
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YEAR 11 MATHEMATICS
MAHOBE
32
QUESTION THREE
Miley has a normal 6 sided die.
(a)
Miley throws the die, records the number, then throws the die again.
What is the probability that she records a 2 on the first throw and a 4 on the
second throw?
............................................................
............................................................
(b)
What is the probability that the total of the two numbers thrown is greater
than 9?
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MAHOBE
YEAR 11 MATHEMATICS
33
QUESTION FOUR
The Auckland Super City Driver’s License Office records details of the name, age
and gender of residents applying for a driver’s license.
Last year the records show that:
60% of applicants were male, 40% of the male applicants were 15 years
old, 30% of the male applicants were 16 years old and the rest of the male
applicants were 17 years or older.
Of the females applying for a driver’s license:
15% were 15 years old, 20% were 16 years old, and the rest of the female
applicants were 17 years or older.
(a)
Complete the tree diagram to show all the relevant information.
Use the information on your tree diagram to answer the following 4
questions.
15 years old
M
16 years old
0.6
F
(b)
Using the information on the tree diagram, find the probability that a person
chosen at random will be a female aged 16 years.
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YEAR 11 MATHEMATICS
MAHOBE
34
(c)
Using the information on the tree diagram, find the probability that a person
chosen at random will be 15 years.
............................................................
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(d)
Last year 285 males who were 16 years old applied for a driver’s license.
Calculate the number of females who were 16 years old who applied for a
driver’s license.
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(e)
Statistics show that 5% of males and 7% of females require spectacles
when driving. If 5000 people apply for a driver’s license how many would
you expect to be 15 year olds who require spectacles when driving?
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MAHOBE
YEAR 11 MATHEMATICS
35
QUESTION FIVE (Extra Challenge)
A coin is tossed n times.
Write an expression to describe the probability of throwing at least 1 tail.
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QUESTION SIX (Extra Challenge)
A survey by the Fresh Fruit Company found that in boxes of their apples there
were on average 6% of the apples that were bad. If you choose 6 apples from a
box at random, what is the probability that at least one of the apples chosen will
be bad?
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YEAR 11 MATHEMATICS
MAHOBE
36
MAHOBE
The Answers
MAHOBE
YEAR 11 MATHEMATICS
37
The Answers
Page 11
1.
e.
Firstly the totals
1 × 4 + 4 × 1 = 8 (0.32)
5
5
25
5
5
2 P
4P
6P
Total
NZ
155
250
121
526
Page 14
Non NZ
240
96
67
403
5.
Totals
395
346
188
929
a.
b.
c.
d.
526
929
346
929
1
1
3 × 3
P(D.D¹) + P(D¹D)
a.
c.
= 0.5662
d.
= 0.3724
34
80 = 0.425
1 × 1 = 0.25
2
2
1
Possibilities:
4 out of 5
(HO)(HG)(OG)(OH)(GO)(GH)
2.
a.
0.15
Overseas visitors
3.
a.
6-person
5
2
1
$10 =12 , $50 = 12 , $100 = 12
Helen has a 50% (equal) chance of
6.
a.
b.
0.45
4 person
0.35
6-person
4 × 1 = 0.08
5 10
4 × 75 + 1 × 20 = 0.64
5 100 5
100
either winning or not winning. If she
0.55
wins, she has more chance of
0.8
winning $10.
b.
0.45
1
0.95
0.2
2
3
0.05
$10 = 0.25 ( 12 ) (unlikely),
6
A prize = 0.5 (
) (possibly).
c.
12
Impossible
0
Certain
Possible
Unlikely
Page 19
0.75
0.4
Likely
b.
c.
d.
0.25
0.75
0.6
Marina wins
Chris wins
Marina wins
Onni wins
0.25
7.
Winning possibilities: DV, DI, DS, DK,
SD, SK, KD, KV, KI, KS
Other possibilities DD, VV, II, SS, KK
i.e. 20 possible outcomes from 25
20
× 20 = 25 (0.8)
YEAR 11 MATHEMATICS
Chris wins
a.
0.4 × 0.75 = 0.3
b.
Marina Chris + Onni Marina
VD, VI, VS, VK, ID, IV, IS, IK, SI, SV,
1 ×1
5
5
Different
Marina wins
Page 13
a.
Same
0.8 × 0.55 + 0.2 × 0.95 = 0.63
1
290
= 0.2802
1035
155
= 0.2870
540
1 × 1 = 0.04
5
5
Different
New Zealanders
$50 = 0.017 ( 12 ) (unlikely),
The scale is based on this diagram.
Same
Overseas
Chances of winning:
$100=0.08 ( 12 ) (unlikely)
4.
2-person
0.2
New Zealanders
b. 88 = 0.8108
Probabilities for each prize are:
3
4 person
0.1
0.2
30
= 0.0568
37
2-person
75%
1
= 9
× 6 = 0.6667
Page 12
1
1
BB + GG + RR = 16 + 16 + 4
= 0.375
= 0.1111
Page 18
8
34 = 0.2353
b.
0.4 × 0.25 + 0.6 × 0.75 = 0.55
c.
W = windy, M = Marina Wins
= P(Weather) + P(Marina V Chris)
= P(WM) + P(W¹M)
= 0.55 × 0.4 + 0.45 × 0.75
= 0.5575
MAHOBE
38
Page 20
Page 24
8.
11.
Below is all the given information
transferred to the tree diagram.
1
Fine
0.8
0.5
6 Total = 7
February
0.42
0.2
Wet (postponed)
Dice 1
6
Total = 8
Dice 2
0.5
Wet (postponed)
Draw up a table of possible results
= 0.42 × 0.2 + 0.58 × 0.5
Number of 5s
Total = 6
Ellis’ Result
1
0.5×0.167×150
= 12.5
48
0.42 × 0.2 × 0.05
2
0.5×0.333×150
= 25
48
+ 0.58 × 0.5 × 0.55
3
0.5×0.5×150
= 37.5
48
4
0.5×0.667×150
= 50.0
48
5
0.5×0.833×150
= 62.5
48
= 0.374
b.
= 0.1637
From the table the closest expected result
Page 21
9.
(50) to Ellis’ experimental result (48)
Personal best
0.4
occurs when there are four 5s on dice 2.
Temp >28
0.75
0.6
4
6
No Personal best
Personal best
0.8
0.25
2
6
0.5
0.2
4
6
No Personal best
0.5
a.
0.75 × 0.4 = 0.3
b.
0.75 × 0.4 + 0.25 × 0.2
Dice 1
0.45
< 5 min late
P (Total = 7) = 0.500
2
6
6
P(Total = 8) = 0.167
Total = 6: 150 × 0.333 = 49.95
(48)
Total = 7: 150 × 0.5 = 75
(77)
5 or more min late
Total = 8: 150 × 0.167 = 25.05
(25)
< 5 min late
These values are very close to the
Fine
0.55
0.4
0.6
experimental totals (in brackets).
i.
5 or more min late
if it is wet
0.3 × 0.4 = 0.12
ii.
0.3 × 0.6 + 0.7 × 0.55
= 0.565
MAHOBE
P(Total = 7) = 0.333
Using 150 trials, calculate the expected
Wet
b.
5
numbers:
0.3
a.
6 P(Total = 7) = 0.167
Dice 2
0.2 × 0.125 = 0.025
Page 22
0.7
P(Total = 6) = 0.333
2
= 0.35
c.
5
1
Temp <28
10.
Total = 7
2
March
a.
5
0.5
Fine
0.5
0.58
Total = 6
5
Page 25
12.
E = 800m runner in final P(E) = 0.12
F = 1500m runner in final P(F) = 0.27
I = 5000m runner in final P(I) = 0.15
P(EFI) + P(EFI¹) + P(EF¹I) + P(E¹FI)
0.6 × 0.75 = 0.45
YEAR 11 MATHEMATICS
39
Page 25 (cont)
12.
Page 32
“at least two” means two or three.
Question Three
= P(EFI) + P(EFI¹) + P(EF¹I) + P(E¹FI)
(a)
P(EFI): 0.12 × 0.27 ×0.15 = 0.00486
(b)
1 × 1 = 1
6
36
6
Combinations are: (4,6), (6,4), (5,5),
P(EFI¹): 0.12 × 0.27 × 0.85 = 0.02754
(5,6), (6,5), (6,6)
P(EF¹I): 0.12 × 0.73 × 0.15 = 0.01314
6
1
36 = 6
P(E¹FI): 0.88 × 0.27 × 0.15 = 0.03564
= 0.00486 + 0.02754 + 0.01314
+ 0.03564
= 0.08118
Page 33
Question Four
(a)
F
0.15
15 yo
0.2
16 yo
0.65
> 17 yo
0.6
Page 28
13.
M
0.4
15 yo
0.3 16 yo
0.3
> 17 yo
There are many possible answers: e.g.
1. Use a calculator random number
button to produce 5 digits. Assign
0.4
outcomes e.g. 0 or 1 = P, 2 or 3 = R,
4 or 5 = I etc or use a dice.
2. Complete a trial and record how many
(b)
P = 0.4 × 0.2
dice rolls or random numbers need to be
= 0.08
generated before 1 complete set of cards
is obtained.
Page 34
3. Repeat the experiment 50 times.
(c)
P = 0.6 × 0.4 + 0.4 × 0.15
Average the results (÷ by 50) to give an
estimate of how many items need to be
= 0.3
(d)
purchased on average to obtain a free
DVD.
285 ÷ (0.6 × 0.3) = 1583 applicants
1583 × 0.4 × 0.2 = 126 applicants
Males
(e)
Females
0.05 × 0.6 × 0.3 + 0.07 × 0.4 × 0.2
Page 31
= 0.0146
Question One
5000 × 0.0146 = 73
(a)
(b)
There are 184 flowers, 35 are gold.
35
= 84
Page 35
There are 25 roses, 12 are red.
Question Five
12
= 25
P(1 Tail) = 1 - (H H H H ...... n times)
= 1 - ( 1 × 1 × 1 × 1 × ...... n times)
Question Two
1 Black Eyed Pods, 3 Tyler Swift
12 Lady Gaa
3
Probability = 16
= 1 -
2
1
2
n
2
2
2
Question Six
Probability of at least one bad apple
means either 1,2,3,4,5 or 6 bad apples
i.e. 1 - (probability of choosing 6 fresh
apples)
1 -
94
93 × 92 × 91 × 90 × 89
× 99
98
97
100
95
96
= 31.7%
YEAR 11 MATHEMATICS
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