Contents MODULE 2 1 Scatter graphs 1

Transcription

00 Higher-Prelims ii-vi.qxd 5/2/07 10:45 PM Page ii
Contents
MODULE 2
1 Scatter graphs
1
1.1
Scatter graphs and relationships
1.2
Lines of best fit and correlation
1.3
Using lines of best fit
Chapter summary
Chapter review questions
1
5
6
10
10
2 Collecting and recording data 14
2.1
Introduction to statistics
2.2
Data by observation and by experiment
2.3
Grouping data
2.4
Questionnaires
2.5
Sampling
2.6
Databases
Chapter summary
3 Averages and range
3.1
Mean, mode and median
3.2
Using frequency tables to find averages
3.3
Range and interquartile range
3.4
Stem and leaf diagrams
3.5
Estimating the mean of grouped data
3.6
Moving averages
Chapter summary
14
14
16
18
20
23
27
28
31
31
34
36
38
41
44
47
47
4 Processing, representing and
interpreting data
4.1
Frequency polygons
4.2
Cumulative frequency
4.3
Box plots
4.4
Comparing distributions
4.5
Frequency density and histograms
Chapter summary
5 Probability
5.1
5.2
5.3
Writing probabilities as numbers
Sample space diagrams
Mutually exclusive outcomes and the
probability that the outcome of an event
will not happen
5.4
Estimating probability from relative
frequency
5.5
Independent events
5.6
Probability tree diagrams
5.7
Conditional probability
Chapter summary
51
51
56
64
65
68
73
73
77
77
79
81
84
86
88
89
92
93
MODULE 3
6 Number
6.1
6.2
Properties of whole numbers
Multiplication and division of directed
numbers M4
6.3
Squares, cubes
6.4
Index laws
6.5
Order of operations
6.6
Using a calculator
6.7
Prime factors, HCF and LCM
Chapter summary
7 Angles (1)
7.1
7.2
7.3
7.4
7.5
ii
Triangles
Equilateral triangles and isosceles triangles
Corresponding angles and alternate angles
Proofs
Bearings
97
97
98
100
101
102
104
106
110
110
113
113
114
116
119
120
Chapter summary
8 Expressions and sequences
8.1
8.2
Expressions and collecting like terms
Working with numbers and letters and
using index notation M4
8.3
Index laws M4
8.4
Sequences
Chapter summary
9 Measure (1)
9.1
9.2
Compound measures – speed and density
Converting between metric and imperial
units
Chapter summary
124
124
127
127
129
131
134
138
139
141
141
144
145
145
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10 Decimals and fractions
10.1
10.2
10.3
10.4
Fractions revision
Arithmetic of decimals
Manipulation of decimals
Conversion between decimals and
fractions M4
10.5 Converting recurring decimals to
fractions
10.6 Rounding to significant figures
Chapter summary
11 Expanding brackets and
factorising
11.1
11.2
11.3
11.4
11.5
Expanding brackets
Factorising by taking out common factors
Expanding the product of two brackets
Factorising by grouping
Factorising expressions of the form
x2 bx c
11.6 Factorising the difference of two squares
Chapter summary
147
147
149
151
13 Graphs (1)
13.1 Coordinates and line segments
13.2 Straight line graphs
Chapter summary
192
14.1 Significant figures
14.2 Accuracy of measurements
Chapter summary
192
194
196
196
154
157
159
161
161
15 Three-dimensional shapes (1) 197
15.1 Volume of three-dimensional shapes
15.2 Surface area of three-dimensional shapes
15.3 Coordinates in three dimensions
Chapter summary
197
202
204
205
206
164
164
165
167
168
170
171
174
174
12 Two-dimensional shapes (1) 176
12.1 Special quadrilaterals
12.2 Perimeter and area of rectangles
12.3 Area of a parallelogram
12.4 Area of a triangle
12.5 Area of a trapezium
12.6 Problems involving areas
Chapter summary
14 Estimating and accuracy
176
177
178
178
179
181
184
184
186
186
187
191
191
16 Indices and standard form
16.1 Zero and negative powers
16.2 Standard form M4
16.3 Fractional indices
Chapter summary
207
M4
17 Further factorising,
simplifying and algebraic
proof
207
208
215
217
217
220
17.1 Further factorising
17.2 Simplifying rational expressions
17.3 Adding and subtracting rational expressions
17.4 Algebraic proof
Chapter summary
18 Circle geometry (1)
220
222
225
228
230
230
232
18.1 Parts of a circle
18.2 Isosceles triangles
18.3 Tangents and chords
Chapter summary
232
232
233
236
237
MODULE 4
19 Angles (2)
19.1 Quadrilaterals
19.2 Polygons
19.3 Exterior angles
Chapter summary
238
238
240
244
246
247
20 Fractions
20.1
20.2
20.3
20.4
248
Addition and subtraction of fractions
Addition and subtraction of mixed numbers
Multiplication of fractions and mixed
numbers
Division of fractions and mixed numbers
248
249
251
253
iii
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CONTENTS
20.5 Fractions of quantities
20.6 Fraction problems
Chapter summary
21 Scale drawings and
dimensions
21.1 Scale drawings and maps
21.2 Dimensions
Chapter summary
255
256
258
258
260
260
262
263
264
22 Two-dimensional shapes (2) 266
22.1 Drawing shapes
22.2 Circumference of a circle
22.3 Area of a circle
22.4 Circumferences and areas in terms of 22.5 Arc length and sector area
22.6 Segment area
22.7 Units of area
Chapter summary
23 Linear equations
23.1 The balance method for solving equations
23.2 Setting up equations
23.3 Solving equations with fractional terms
23.4 Simultaneous linear equations
23.5 Setting up simultaneous linear equations
Chapter summary
24 Percentages
24.1 Percentages M3
24.2 Increases and decreases
24.3 Use of multipliers
24.4 Reverse percentages
Chapter summary
25 Graphs (2)
25.1 Real life graphs
25.2 Solving simultaneous equations graphically
25.3 The equation y mx c
25.4 Further uses of y mx c
Chapter summary
26 Transformations
26.1 Introduction
26.2 Translations
26.3 Rotations
26.4 Reflections
26.5 Enlargements
26.6 Centre of enlargement
26.7 Combinations of transformations
Chapter summary
iv
266
268
270
272
273
274
276
277
277
280
280
284
287
289
292
293
294
296
296
299
306
309
311
312
314
314
319
321
324
328
329
332
332
332
336
338
343
346
351
354
354
27 Inequalities
27.1 Inequalities on a number line
27.2 Solving inequalities
27.3 Integer solutions to inequalities
27.4 Problems involving inequalities
27.5 Solving inequalities graphically
Chapter summary
28 Formulae
28.1
28.2
28.3
28.4
Using an algebraic formula
Writing an algebraic formula
Changing the subject of a formula
Expressions, identities, equations and
formulae
28.5 Further changing the subject of a formula
Chapter summary
29 Pythagoras’ theorem and
trigonometry (1)
29.1 Pythagoras’ theorem
29.2 Finding lengths
29.3 Applying Pythagoras’ theorem
29.4 Line segments and Pythagoras’ theorem
29.5 Trigonometry – introduction
29.6 Finding lengths using trigonometry
29.7 Finding angles using trigonometry
29.8 Trigonometry problems
Chapter summary
30 Ratio and proportion
30.1 Introduction to ratio
30.2 Problems
30.3 Sharing a quantity in a given ratio
30.4 Direct proportion
30.5 Inverse proportion
Chapter summary
358
358
359
361
362
363
368
368
372
372
374
376
378
379
381
382
384
384
385
388
390
392
393
396
398
401
401
405
405
408
409
411
413
415
415
31 Three-dimensional shapes (2) 418
31.1 Planes of symmetry
31.2 Plans and elevations
31.3 Volume of three-dimensional shapes
31.4 Surface area of three-dimensional shapes
Chapter summary
32 Graphs (2)
32.1
32.2
Graphs of quadratic functions
Using graphs of quadratic functions to
solve equations
32.3 Using graphs of quadratic and linear
functions to solve quadratic equations
Chapter summary
418
420
422
427
430
431
433
433
436
439
442
442
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CONTENTS
33 Further graphs and trial
and improvement
Graphs of cubic, reciprocal and exponential
functions
33.2 Trial and improvement
Chapter summary
445
33.1
34 Constructions, loci and
congruence
34.1 Constructions
34.2 Loci
34.3 Regions
34.4 Drawing triangles
34.5 Congruent triangles
34.6 Proofs of standard constructions
Chapter summary
35 Bounds and surds
35.1 Lower bounds and upper bounds
35.2 Surds
Chapter summary
36 Circle geometry
36.1 Circle theorems
Chapter summary
37 Completing the square
37.1 Completing the square
Chapter summary
38 Quadratic equations
38.1 Introduction to solving quadratic equations
38.2 Solving by factorisation
38.3 Solving by completing the square
38.4 Solving using the quadratic formula
38.5 Solving equations with algebraic fractions
38.6 Problems that involve quadratic equations
Chapter summary
39 Pythagoras’ theorem and
trigonometry (2)
39.1
39.2
39.3
39.4
39.5
39.6
Problems in three dimensions
Trigonometric ratios for any angle
Area of a triangle
The sine rule
The cosine rule
Solving problems using the sine rule,
the cosine rule and 12 ab sin C
Chapter summary
445
449
453
453
40 Simultaneous linear and
quadratic equations and
loci
530
40.1
40.2
40.3
Solving simultaneous equations
Loci and equations
Intersection of lines and circles – algebraic
solutions
Chapter summary
530
532
536
538
538
458
458
461
464
467
468
469
470
471
41 Similar shapes
540
41.1 Similar triangles
41.2 Similar polygons
41.3 Areas of similar shapes
41.4 Volumes of similar solids
41.5 Lengths, areas and volumes of similar solids
Chapter summary
540
544
547
550
552
554
555
474
474
476
478
479
481
481
487
488
491
491
494
494
496
496
496
498
499
501
502
505
505
507
507
512
516
519
522
525
527
527
42 Direct and inverse
proportion
42.1 Direct proportion
42.2 Further direct proportion
42.3 Inverse proportion
42.4 Proportion and square roots
Chapter summary
43 Vectors
43.1
43.2
43.3
43.4
43.5
43.6
Vectors and vector notation
Equal vectors
The magnitude of a vector
Addition of vectors
Parallel vectors
Solving geometric problems in two
dimensions
Chapter summary
44 Transformations of
functions
44.1
44.2
44.3
44.4
44.5
44.6
559
559
561
563
566
568
568
571
571
573
574
575
578
583
587
587
591
Function notation
Applying vertical translations
Applying horizontal translations
Applying reflections
Applying stretches
Transformations applied to the graphs of
sin x and cos x
Chapter summary
591
592
596
599
602
Index
Licence
611
618
605
608
608
v
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Chapter 5 077-096.qxd 5/2/07 11:07 PM Page 77
5
Probability
CHAPTER
Favourites to seize the
Olympic flame
London defy all
the odds
As the day for decision approaches it
seems unlikely that London will win
the battle to host the 2012 Olympic
Games. The probability that Paris will
win this race has always been high. It is
felt that Madrid, Moscow and New
York have little chance of success as
the final presentations are made.
London won with their bid to host
the 2012 Summer Olympic Games.
Yesterday’s vote saw likely winners
Paris stumble at the final hurdle. A
spokesperson said ‘Everyone thought
that Paris was certain to win the vote
but I always felt that we had a greater
than even chance of success.’
5.1 Writing probabilities as numbers
The diagram shows a three-sided spinner.
The spinner can land on red or blue or yellow. If it is equally likely to land on
each of the three colours the spinner is said to be fair.
This spinner, which is fair, is spun once. This is called a single event.
The colour it lands on is called the outcome.
The outcome can be red or blue or yellow. There are three possible
outcomes and each possible outcome is equally likely.
The probability of an outcome to an event is a measure of
how likely it is that the outcome will happen.
1 successful outcome
1
The probability that the spinner will land on blue 3 possible outcomes
3
1
Similarly the probability that the spinner will land on red 3
1
and the probability that the spinner will land on yellow 3
When all the possible outcomes are equally likely to happen
number of successful outcomes
probability total number of possible outcomes
Probability can be written as a fraction or a decimal.
For an event:
● the probability of an outcome which is certain to happen is 1 For example the probability that
the spinner will land on red or blue or yellow is 1 since the spinner is certain to land on one of
these three colours
● the probability of an outcome which is impossible is 0 For example the probability that the
spinner will land on green is 0 since green is not a colour on the spinner
● all other probabilities lie between 0 and 1
77
CHAPTER 5
Probability
Example 1
A fair five-sided spinner is numbered 1 to 5
Jane spins the spinner once.
a Find the probability that the spinner will land on the number 4
b Find the probability that the spinner will land on an even number.
Solution 1
a The possible outcomes are the numbers 1, 2, 3, 4 and 5 So the total
number of possible outcomes is 5 and they are all equally likely.
The 1 successful outcome is the number 4
Probability total number of possible outcomes
Probability that the spinner will land on the number 4 15 or 0.2
b 2 and 4 are the even numbers on the spinner.
The number of successful outcomes is 2
The total number of possible outcomes is 5
So the probability that the spinner will land on an even number = 25 or 0.4
In the following example the term at random is used. This means that each possible outcome is
equally likely.
Example 2
Six coloured counters are in a bag.
3 counters are red, 2 counters are green
and 1 counter is blue.
One counter is taken at random from the bag.
a Write down the colour of the counter which is
i most likely to be taken
ii least likely to be taken
b Find the probability that the counter taken will be
i red
ii green
iii blue
Solution 2
a i Red is the most likely colour to be taken since the number of red counters is greater than the
number of counters of any other colour.
ii Blue is least likely to be taken since the number of blue counters is less than the number of
counters of any other colour.
b There are 6 counters so there are 6 possible outcomes.
Outcome
1
2
3
4
5
i Number of successful outcomes 3 (3 red counters)
Probability that the counter taken will be red 36 12
ii Number of successful outcomes 2 (2 green counters)
Probability that the counter taken will be green 26 13
iii Number of successful outcomes 1 (1 blue counter)
Probability that the counter taken will be blue 16
78
6
Final answers for probabilities
written as fractions should be
given in their simplest form.
CHAPTER 5
5.2 Sample space diagrams
2
6
2 John spins the fair spinner. Write down the
probability that the spinner will land on
a 2
b a number greater than 5
c an even number
d a number greater than 10
8
4
3 Samantha Smith has 8 cards which spell ‘Samantha’.
She puts the cards in a bag and chooses
one of the cards at random.
Find the probability that she will
choose a card showing a
a letter S
b letter A
S A M A N T H A
c letter which is also in her surname ‘SMITH’
4 Ben has 15 ties in a drawer. 7 of the ties are plain, 3 of the ties are striped and the rest are
patterned. Ben chooses a tie at random from the drawer. What is the probability that he chooses
a tie which is
a plain
b striped
c patterned?
5 Peter has a bag of 8 coins. In the bag he has one 10p coin, five 20p coins and the rest are
50p coins. Peter chooses one coin at random. What is the probability that Peter will choose a
a 10p coin
b 20p coin
c 50p coin
d £1 coin
e coin worth more than 5p?
6 Rob has a drawer of 20 socks. 4 of the socks are blue, 6 of the socks are brown and the rest
of the socks are black. Rob chooses a sock at random from the drawer. Find the probability
that he chooses
a a blue sock
b a brown sock
c a black sock
d a white sock
7 Verity has a box of pens. Half of the pens are blue, 11 of the pens are green, 10 of the pens are
red and the remaining 4 pens are black. Verity chooses a pen at random from the box.
Find the probability that she chooses
a a blue pen
b a green pen
c a red pen
d a black pen
5.2 Sample space diagrams
A sample space is all the possible outcomes of one or more events.
For the three-sided spinner, the sample space when the spinner is spun once is
1 2 3
2
3
A sample space diagram is a diagram which shows the sample space.
1
Exercise 5A
1 Nicky spins the spinner. The spinner is fair. Write down
the probability that the spinner will land on a side coloured
a blue
b red
c green
79
CHAPTER 5
Probability
Example 3
The three-sided spinner is spun and a coin is tossed at the same time.
Write down the sample space of all possible outcomes.
Solution 3
There are 6 possible outcomes. For example the spinner landing on 1 and the coin showing heads is
written as (1, head). The sample space is
(1, head)
(1, tail)
(2, head)
(2, tail)
(3, head)
(3, tail)
Example 4
Two fair dice are thrown.
a Write down the sample space showing all the possible outcomes.
b Find the probability that the numbers on the two dice will be
i both the same
ii both even numbers
iii both less than 3
Solution 4
a (1, 1) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1)
(1, 2) (2, 2) (3, 2) (4, 2) (5, 2) (6, 2)
(1, 3) (2, 3) (3, 3) (4, 3) (5, 3) (6, 3)
(1, 4) (2, 4) (3, 4) (4, 4) (5, 4) (6, 4)
(1, 5) (2, 5) (3, 5) (4, 5) (5, 5) (6, 5)
(1, 6) (2, 6) (3, 6) (4, 6) (5, 6) (6, 6)
The total number of possible outcomes is 36
b i (1, 1) (2, 2) (3, 3) (4, 4) (5, 5) and (6, 6) are the successful outcomes with both numbers
the same.
Probability that the numbers on the two dice will be both the same 366 16
ii (2, 2) (2, 4) (2, 6) (4, 2) (4, 4) (4, 6) (6, 2) (6, 4) (6, 6) are the successful outcomes with both
numbers even.
Probability that the numbers on the two dice will both be even numbers 396 14
iii (1, 1) (2, 1) (1, 2) (2, 2) are the successful outcomes with both numbers less than 3
Probability that the numbers on the two dice will both be less than 3 is 346 19
Exercise 5B
In each of the questions in this exercise give all probabilities as fractions in their simplest forms.
1 Two coins are spun at the same time.
a Write down a sample space to show all possible outcomes.
b Find the probability that both coins will come down heads.
c Find the probability that one coin will come down heads and the other coin will come down tails.
2 A bag contains 1 blue brick, 1 yellow brick, 1 green brick and 1 red brick.
A brick is taken at random from the bag and its colour noted.
The brick is then replaced in the bag.
A brick is again taken at random from the bag and its colour noted.
a Write down a sample space to show all the possible outcomes.
b Find the probability that
i the two bricks will be the same colour
ii one brick will be red and the other brick will be green
80
5.3 Mutually exclusive outcomes
CHAPTER 5
3 Two fair dice are thrown. The sample space is shown in Example 4
The numbers on the two dice are added together.
a Find the probability that the sum of the numbers on the two dice will be
i greater than 10
ii less than 6
iii a square number.
b i Which sum of the numbers on the two dice is most likely to occur?
ii Find the probability of this sum.
4 Daniel has four cards, the ace of hearts,
the ace of diamonds, the ace of spades
and the ace of clubs.
Daniel also has a fair dice.
He rolls the dice and takes a card at random.
a Write down the sample space showing all
possible outcomes.
One possible outcome, ace of Diamonds and 4 has been done for you, (D, 4).
b Find the probability that a red ace will be taken.
c Find the probability that he will take the ace of spades and roll an even number on the dice.
5 Three fair coins are spun.
a Draw a sample space showing all eight possible outcomes.
b Find the probability that the three coins will show the same.
c Find the probability that the coins will show two heads and a tail.
d Write down the total number of possible outcomes when
i four coins are spun
ii five coins are spun
5.3 Mutually exclusive outcomes and the probability that the
outcome of an event will not happen
Nine coloured counters are in a bag.
3 counters are red, 2 counters are green and 4 counters are yellow.
One counter is chosen at random from the bag.
Notation: ‘P(red)’ means the probability of red.
The probability that the counter will be red, P(red) 39
P(green) 29 P(yellow) 49
Mutually exclusive outcomes are outcomes which cannot happen at the same time.
For example when one counter is chosen at random from the bag the outcome ‘red’ cannot happen
at the same time as the outcome ‘green’ or the outcome ‘yellow’. So the three outcomes are
mutually exclusive.
P(red) P(green) P(yellow) 39 29 49 99 1
The sum of the probabilities of all the possible mutually exclusive outcomes of an event is 1
There are 9 possible outcomes, 2 of which are green.
The probability that the counter will be green is 29
Out of the 9 possible outcomes 9 2 7 outcomes are NOT green.
The probability that the counter will NOT be green is 1 29 79
If the probability of an outcome of an event happening is p then the probability of it NOT
happening is 1 p
81
CHAPTER 5
Probability
If the counter is not green it must be either red or yellow.
So, P(not green) P(either red or yellow)
The probability that the counter will be either red or yellow is 79
7
9
39 49 P(red) P(yellow).
So P(either red or yellow) P(red) P(yellow)
In general when two outcomes A and B, of an event are mutually exclusive
P(A or B) P(A) P(B)
This can be used as a quicker way of solving some problems.
Example 5
David buys one newspaper each day. He buys the Times or the Telegraph or the Independent. The
probability that he will buy the Times is 0.6 The probability that he will buy the Telegraph is 0.25
a Work out the probability that David will buy the Independent.
b Work out the probability that David will buy either the Times or the Telegraph.
Solution 5
a P(Times) 0.6
P(Times) means the probability
that David will buy the Times.
P(Telegraph) 0.25
P(Independent) ?
As David buys only one newspaper each day, the three outcomes are mutually exclusive.
P(Independent) 0.6 0.25 1
P(Independent) 0.85 1
P(Independent) 1 0.85
The probability that David will buy the Independent 0.15
b P(Times or Telegraph) P(Times) P(Telegraph)
0.6 0.25
0.85
The probability that David will buy either the Times or the Telegraph 0.85
Example 6
The probability that Julie will pass her driving test next week is 0.6
Work out the probability that Julie will not pass her driving test next week.
Solution 6
The probability that Julie will not pass 1 0.6 0.4
Exercise 5C
1 Nosheen travels from home to school. She travels by bus or by car or by tram. The probability
that she travels by bus is 0.4 The probability that she travels by car is 0.5
a Work out the probability that she travels by tram.
b Work out the probability that she travels by car or by bus.
2 Roger’s train can be on time or late or early. The probability that his train will be on time is 0.15
The probability that his train will be early is 0.6
a Work out the probability that Roger’s train will be late.
b Work out the probability that Roger’s train will be either on time or early.
82
CHAPTER 5
5.3 Mutually exclusive outcomes
3 The probability that Lisa will pass her Maths exam is 0.8
Work out the probability that Lisa will not pass her Maths exam.
4 A company makes batteries. A battery is chosen at random. The probability that the battery will
not be faulty is 0.97
Work out the probability that the battery will be faulty.
5 Four athletes Aaron, Ben, Carl and Des take part in a race.
The table shows the probabilities that Aaron or Ben or Carl will win the race.
a
b
c
d
e
Aaron
Ben
Carl
0.2
0.14
0.3
Des
Work out the probability that Aaron will not win the race.
Work out the probability that Ben will not win the race.
Work out the probability that Des will win the race.
Work out the probability that either Aaron or Ben will win the race.
Work out the probability that either Aaron or Carl or Des will win the race.
6 The table shows the probabilities of a dice landing
on each of the numbers 1 to 6 when thrown.
The dice is thrown once.
a Work out the probability that the dice will land on
either 1 or 3
b Work out the probability that the dice will land on
either 2 or 4
c Work out the probability that the dice will land on
i an even number
ii an odd number
7 A roundabout has four roads leading from it. Michael is driving
round the roundabout. The roads lead to Liverpool or Trafford Park
or Eccles or Bolton. The table shows the probabilities that
Michael will take the road to Liverpool or Trafford Park or Bolton.
Liverpool
Trafford Park
Eccles
Bolton
0.49
0.18
x
0.23
a Work out the probability that Michael will not take the road to
Liverpool.
b Work out the value of x.
c Work out the probability that Michael will take either the
road to Trafford Park or the road to Bolton.
Number
Probability
1
0.2
2
0.15
3
0.25
4
0.18
5
0.05
6
0.17
Trafford
Park
Liverpool
Eccles
Bolton
8 Sam has red, white, yellow and green coloured T-shirts only. She chooses a T-shirt at random.
The probabilities that Sam will choose a red T-shirt or a white T-shirt are given in the table.
Sam is twice as likely to choose a green T-shirt as she is to choose a yellow T-shirt.
Work out the value of x.
Red
White
Yellow
Green
0.5
0.14
x
2x
83
CHAPTER 5
Probability
5.4 Estimating probability from relative frequency
The diagram shows two three-sided spinners.
One spinner is fair and one is biased.
A spinner is biased if it is not equally likely to land on each of the
2
numbers. This can be tested by experiment.
If a spinner is spun 300 times it is fair if it lands on each of the
numbers approximately 100 times.
John spins one spinner 300 times and Mary spins the other spinner 300 times.
3
1
3
1
John’s spinner
2
Mary’s spinner
3
1
3
1
2
2
300 spins
300 spins
1
2
3
1
2
3
97
104
99
147
96
57
FAIR spinner
BIASED spinner
John’s spinner is fair because it lands on each of the three numbers approximately the same number
of times.
Mary’s spinner is biased because it is more likely to land on the number 1
It is least likely to land on the number 3
To estimate the probability that Mary’s spinner will land on each number, the relative frequency of
each number is found using
number of times the spinner lands on the number
relative frequency total number of spins
Relative frequency that Mary’s spinner will land on the number 1 134070 0.49
9
6
5
7
An estimate of the probability that the spinner will land on the number 1 is 0.49
If Mary spins the spinner a further 500 times, an estimate for the number of times the spinner lands
on the number 2 is 0.32 500 160
In general
if the probability that an experiment will be successful is p and the experiment is carried
out N times, then an estimate for the number of successful experiments is p N.
Example 7
In a statistical experiment Brendan
Number on dice
1
2
3
throws a dice 600 times.
Frequency
48
120 180
The table shows the results.
Brendan throws the dice again.
a Find an estimate of the probability that he will throw a 2
b Find an estimate of the probability that he will throw an even number.
Zoe now throws the same dice 200 times.
c Find an estimate of the number of times she will throw a 6
84
4
5
6
96
54
102
CHAPTER 5
5.4 Estimating probability from relative frequency
Solution 7
2
0
a Estimate of probability of a 2 is 1
600 0.2
b The number of times an even number is thrown 120 96 102 318
1
8
Estimate of probability of an even number 3
600 0.53
10
2
c Estimate of probability of a 6 is 600 0.17
An estimate for the number of times Zoe will throw a 6 in 200 throws 0.17 200 34
Exercise 5D
1 A coin is biased. The coin is tossed 200 times.
It lands on heads 140 times and it lands on tails 60 times.
a Write down the relative frequency of the coin landing on tails.
b The coin is to be tossed again. Estimate the probability that the coin will land on
i tails
ii heads.
2 A bag contains a red counter, a blue counter, a
Red
Blue
White
white counter and a green counter. Asif takes a
81
110
136
counter at random. He does this 400 times.
The table shows the number of times each of the coloured counters is taken.
a Write down the relative frequency of Asif taking the red counter.
b Write down the relative frequency of Asif taking the white counter.
Asif takes a counter one more time.
c Estimate the probability that this counter will be
i blue
ii green.
Green
73
3 Tyler carries out a survey about the words in a newspaper. He chooses an article at random.
He counts the number of letters in each of the first 150 words of the article.
The table shows Tyler’s results.
Number of letters in a word
1
2
3
4
5
6
7
8
9
10
Frequency
7
14
42
31
21
13
10
6
4
2
A word is chosen at random from the 150 words.
a Write down the most likely number of letters in the word.
b Estimate the probability that the word will have
i 1 letter
ii 7 letters
iii more than 5 letters.
c The whole article has 1000 words. Estimate the total number of 3-letter words in this article.
4 A bag contains 10 coloured bricks. Each brick is white or
White
Red
red or blue. Alan chooses a brick at random from the 10
290
50
bricks in the bag and then replaces it in the bag.
He does this 500 times. The table shows the numbers of each coloured brick chosen.
a Estimate the number of red bricks in the bag.
b Estimate the number of white bricks in the bag.
Blue
160
5 The probability that someone will pass their driving test at the first attempt is 0.45
On a particular day, 1000 people will take the test for the first time.
Work out an estimate for the number of these 1000 people who will pass.
6 Gwen has a biased coin. When she spins the coin the probability that it will come down tails is 35
Work out an estimate for the number of tails she gets when she spins her coin 400 times.
7 The probability that a biased dice will land on a 1 is 0.09 Andy is going to roll the dice 300 times.
Work out an estimate for the number of times the dice will land on a 1
85
CHAPTER 5
Probability
5.5 Independent events
In Example 3 a fair three-sided spinner is spun and a fair coin is tossed at the same time. The
outcomes from spinning the spinner do not affect the outcomes from tossing the coin. The outcomes
from tossing the coin do not affect the outcomes from spinning the spinner.
These are independent events since an outcome of one event does not affect the outcome of the
other event.
What is the probability that the spinner will land on 3 and the coin will land on tails? This is written
as P(3, tail).
P(3) 13 P(tail) 12
To work out P(3, tail) the sample space could be used. The sample space is
(1, head)
(1, tail)
(2, head)
(2, tail)
(3, head)
(3, tail)
1
6
P(3, tail) since this is 1 out of 6 possible outcomes.
But P(3, tail) 13 12 16
so
P(3, tail) P(3) P(tail)
In general when the outcomes, A and B, of two events are independent
P(A and B) P(A) P(B)
Example 8
A bag contains 4 green counters and 5 red counters. A counter is chosen at random and then replaced in
the bag. A second counter is then chosen at random.Work out the probability that for the counters chosen
a both will be green
b both will be red
c one will be green and one will be red
Solution 8
a
P(G) 49
4
9
Find the probability that a green counter will be chosen.
4
9
P(G and G) 1
6
81
The probability that both counters chosen
will be green 1861
b
P(R) 59
5
9
Find the probability of choosing a red counter.
5
9
P(R and R) 2
5
81
The probability that both counters chosen
will be red 2851
c P(one G and one R) P(first G and second R or
first R and second G)
P(first G and second R)
P(first R and second G)
49 59 59 49
2801 2801
P(one G and one R) 4801
The probability that one of the counters chosen
will be green and one will be red 4801
86
The choosing of the two counters are two independent
events so use P(A and B) P(A) P(B)
Use P(A and B) P(A) P(B)
Use P(A or B) P(A) P(B)
Hint: A or B Add probabilities
A and B Multiply probabilities
Note: (G, G) (R, R) (G, R) (R, G) is
the full sample space so answers to
parts a, b and c must add up to 1
CHAPTER 5
5.5 Independent events
Exercise 5E
1 A biased coin and a biased dice are thrown. The probability that the coin will land on heads is 0.6
The probability that the dice will land on an even number is 0.7
a Write down the probability that the coin will not land on heads.
b Find the probability that the coin will land on heads and that the dice will land on an
even number.
c Find the probability that the coin will not land on heads and that the dice will not land on an
even number.
2 A basket of fruit contains 3 apples and 4 oranges. A piece of fruit is picked at random and then
returned to the basket. A second piece of fruit is then picked at random.
Work out the probability that for the fruit picked
a both will be apples
b both will be oranges
c one will be an apple and one will be an orange.
3 Eric and Frank each try to hit the bulls-eye.
They each have one attempt.
The events are independent.
The probability that Eric will hit the
bulls-eye is 23
The probability that Frank will hit the
bulls-eye is 34
a Find the probability that both Eric and
Frank will hit the bulls-eye.
b Find the probability that just one of
them will hit the bulls-eye.
c Find the probability that neither of
them will hit the bulls-eye.
4 When Edna rings the health centre the probability that the phone is engaged is 0.35
Edna needs to ring the health centre at 9 am on both Monday and Tuesday.
Find the probability that at 9 am the phone will
a be engaged on both Monday and Tuesday
b not be engaged on Monday but will be engaged on Tuesday
c be engaged on at least one day
5 Mrs Rashid buys a car.
Fault
Engine
Brakes
Probability
0.05
0.1
The table shows the probability of different mechanical faults.
Find the probability that the car will have
a a faulty engine and faulty brakes
b no faults
c exactly one fault
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CHAPTER 5
Probability
5.6 Probability tree diagrams
It is often helpful to use probability tree diagrams to solve probability problems.
A probability tree diagram shows all of the possible outcomes of more than one event by following
all of the possible paths along the branches of the tree.
Example 9
Mumtaz and Barry are going for an interview.
The probability that Mumtaz will arrive early is 0.7
The probability that Barry will arrive early is 0.4
The two events are independent.
a Complete the probability tree diagram.
b Work out the probability that Mumtaz and Barry will
both arrive early.
c Work out the probability that just one person will arrive early.
Early
Early
Not early
Early
Not early
Not early
Barry not early:
1 0.4 0.6
Must be either early or not early.
Barry early:
0.4
The two events are independent.
0.3
Barry
0.4
Early
P(Early, Early)
0.6
Not early
P(Early, Not early)
0.4
Early
P(Not early, Early)
0.6
Not early
P(Not early, Not early)
Early
Not early
b P(Mumtaz early and Barry early)
P(Early, Early) 0.7 0.4 0.28
c P(just one person early)
P(Mumtaz early and Barry not early OR
Mumtaz not early and Barry early)
P(Early, Not early) P(Not early, Early)
(0.7 0.6) (0.3 0.4)
0.42 0.12 0.54
P(just one person early) 0.54
88
0.7
Must be either early or not early.
0.7
Barry
0.4
Solution 9
a Mumtaz not early: 1 0.7 0.3
Mumtaz
Mumtaz
This probability is found in part b
These probabilities
are added in part c
Use P(A and B) P(A) P(B) When moving along a
path multiply the probabilities on each of the branches.
Possible ways for just one person to be early.
Use P(A or B) P(A) P(B)
Exercise 5F
1 Amy and Beth are going to take a driving test tomorrow.
The probability Amy will pass the test is 0.7
The probability Beth will pass the test is 0.8
The probability tree diagram shows this information.
Use the probability tree diagram to work out the
probability that
a both women will pass the test
b only Amy will pass the test
c neither woman will pass the test.
Amy
0.7
0.3
Beth
0.8
Pass
0.2
Not pass
0.8
Pass
0.2
Not pass
Pass
Not pass
CHAPTER 5
5.7 Conditional probability
2 A bag contains 10 coloured counters,
Bag
Box
7
4 of which are yellow.
10
Yellow
A box also contains 10 coloured counters,
4
Yellow
10
Not yellow
7 of which are yellow.
One counter is chosen at random from the bag
Yellow
and one counter is chosen at random from the box.
Not yellow
Not yellow
a Copy and complete the probability tree diagram.
b Find the probability that
i both counters will be yellow
ii the counter from the bag will be yellow and the counter from the box will not be yellow
iii at least one counter will be yellow.
3 The probability that a biased coin will show heads when thrown is 0.4
Tina throws the coin twice and records her results.
a Draw a probability tree diagram.
b Use your diagram to work out the probability that the coin will show
i heads on both throws
ii heads on exactly one throw.
4 The probability that Jason will receive one DVD for his birthday is 45
The probability that he will receive one DVD for Christmas is 38
These two events are independent.
Find the probability that Jason will receive at least one DVD.
5 Stuart and Chris each try to score a goal in a penalty shoot-out.
The probability that Stuart will score a goal is 0.75
The probability that Chris will score a goal is 0.64
a Work out the probability that both Stuart and Chris will score a goal.
b Work out the probability that exactly one of them will fail to score a goal.
The probability of an outcome of an event that is dependent on the outcome of a previous event is
called conditional probability. For example when choosing two pieces of fruit without replacing the
first one, the choice of the second piece of fruit is dependent on the choice of the first.
Example 10
A bowl of fruit contains 3 apples and 4 bananas. A piece of fruit is chosen at random and eaten. A second
piece of fruit is then chosen at random. Work out the probability that for the two pieces of fruit chosen
a both will be apples
b the first will be an apple and the second will be a banana
c at least one apple will be chosen.
Solution 10
a 1st choice:
P(A) 37
Find the probability that the first piece of fruit will be an
apple. There is a total of 7 pieces, 3 of which are apples.
2nd choice:
P(A) 26
1st choice was apple so there are now only 6 pieces of fruit,
2 of which are apples. Find the probability that the second
piece of fruit will be an apple.
P(A and A) 37 26 17
Probability both will be apples 17
Multiply the probabilities.
89
CHAPTER 5
Probability
P(A) 37
b 1st choice:
P(B) 2nd choice:
Find the probability that the first piece of fruit will be an
apple. There is a total of 7 pieces, 3 of which are apples.
4
6
P(A and B) 37 46 27
Probability the first will be an apple and
the second will be a banana 27
c P(At least one apple) 1 P(B, B)
1st choice:
1st choice was apple so there are now only 6 pieces
of fruit, 4 of which are bananas. Find the probability
that the second piece of fruit will be a banana.
(A, A) (A, B) (B, A) (B, B) are all the possible
outcomes so P(At least one apple) P(B, B) 1
Find the probability that the first piece of fruit will be a
banana. There is a total of 7 pieces, 4 of which are bananas.
P(B) 47
1st choice was banana so there are now only 6 pieces
of fruit, 3 of which are bananas. Find the probability
that the second piece of fruit will be a banana.
P(B) 36
P(B, B) 47 36 27
P(At least one apple) 1 27 57
2nd choice:
Example 11
There are 4 red crayons, 3 blue crayons and 1 green crayon in a box.
A crayon is taken at random and not replaced. A second crayon is then taken at random.
a Draw and complete a probability tree diagram.
b Find the probability that both crayons taken will be
i blue
ii the same colour.
c Find the probability that exactly one of the crayons will be red.
Solution 11
a First crayon: total of 8 crayons out of which 4 are red, 3 are blue, 1 is green.
Second crayon: total of 7 crayons (since 1st not replaced)
When 1st crayon is red, 3 red, 3 blue, 1 green remain. When 1st crayon is blue, 4 red, 2 blue,
1 green remain. When 1st crayon is green, 4 red, 3 blue, 0 green remain
First crayon
Second crayon
3
7
3
7
Red
1
7
4
8
4
7
3
8
2
7
Blue
1
7
1
8
4
7
3
7
Green
0
7
90
Red
4
3
P(R, R) 8 7
Blue
4
3
P(R, B) 8 7
Green
P(R, G) Red
3
4
P(B, R) 8 7
Blue
3
2
P(B, B) 8 7
Green
3
1
P(B, G) 8 7
Red
1
4
P(G, R) 8 7
Blue
1
3
P(G, B) 8 7
Green
P(G, G) 4
1
8
7
1
0
8
7
These results are
used later in the
question.
CHAPTER 5
b
i P(B and B) 38 27 238
Probability that both colours will be blue 238
ii Probability that colours will be the same
P(R and R or B and B or G and G)
P(R, R) P(B, B) P(G, G)
48 37 38 27 18 07
1526 566 506 1586
Probability that colours will be the same 298
c Probability of exactly one red
P(R, B) P(R, G) P(B, R) P(G, R)
48 37 48 17 38 47 18 47
1526 546 1526 546 3526
Probability of exactly one red 47
Follow the branches ‘blue to blue’
and multiply the probabilities.
Colours are either both red or
both blue or both green.
Follow the pairs of branches
which have just one red.
Example 12
When driving to the shops Rose passes through
two sets of traffic lights.
If she stops at the first set of lights the probability that she stops at the
second set of lights is 0.25
If she does not stop at the first set of lights the probability that she
stops at the second set is 0.35
The probability that Rose stops at the first set of lights is 0.4
a Draw and complete a probability tree diagram.
b Find the probability that when Rose next drives to the shops she will not
stop at the second set of traffic lights.
Solution 12
1st set
a
0.4
0.6
2nd set
0.25
Stop
0.75
Not stop
0.35
Stop
0.65
Not stop
Stop
Not stop
b P(Not stopping at 2nd set) P(S, NS or NS, NS)
P(S, NS) P(NS, NS)
0.4 0.75 0.6 0.65
0.3 0.39
Probability that Rose will not stop at the 2nd set 0.69
Exercise 5G
1 A box of chocolates contains 10 milk chocolates and 12 plain chocolates. Two chocolates are
taken at random without replacement.
Work out the probability that
a both chocolates will be milk chocolates
b at least one chocolate will be a milk chocolate
91
CHAPTER 5
Probability
2 Anil has 13 coins in his pocket, 6 pound coins, 3 twenty-pence coins and 4 two-pence coins.
He picks two coins at random from his pocket. Work out the probability that the two coins each
have the same value.
3 Mandy has these five cards
Each card has a number on it.
1 2
She chooses two cards at random
without replacement and records
the number on each card.
b Find the probability that both numbers are even.
c Find the probability that the sum of the two
numbers is an even number.
2
3
4
1st card
2nd card
Even
3
5
Even
Odd
Even
Odd
Odd
4 Michael returns to school tomorrow.
If it is raining the probability that Michael walks to school is 0.3
If it is not raining the probability that Michael walks to school is 0.8
The probability that it will rain tomorrow is 0.1
a Draw a probability tree diagram.
b Find the probability that Michael will walk to school tomorrow.
5 A box contains 3 tins of soup. 2 of the tins are chicken
soup and 1 is tomato soup.
Betty wants tomato soup. She picks a tin at random from the
box. If it is not tomato she gives the tin to her son and then
picks another tin at random from the box.
b Find the probability that Betty does not pick the tin of
tomato soup.
1st tin
2nd tin
Tomato
Tomato
Chicken
Chicken
6 The probability that a biased dice when thrown will land on 6 is 14 In a game Patrick throws the
biased dice until it lands on 6 Patrick wins the game if he takes no more than three throws.
a Find the probability that Patrick throws a 6 with his second throw of the dice.
b Find the probability that Patrick wins the game.
Chapter summary
You should now know:
that probability is a measure of how likely the outcome of an event is to happen
that probabilities are written as fractions or decimals between 0 and 1
that an outcome which is impossible has a probability of 0
that an outcome which is certain to happen has a probability of 1
for an event, outcomes which are equally likely have equal probabilities
that when calculating probabilities you can use
probability total number of possible outcomes
when all outcomes of an event are equally likely to happen
For example the probability of throwing a six on a normal fair dice is 16
92
CHAPTER 5
Chapter 5 review questions
that a sample space is all the possible outcomes of one or more events and a sample space
diagram is a diagram which shows the sample space
how to list all outcomes in an ordered way using sample space diagrams. For example when
two coins are tossed the outcomes are
(H, H) (T, H)
(H, T) (T, T)
that for an event mutually exclusive outcomes are outcomes which cannot happen at the
same time
that the sum of the probabilities of all the possible mutually exclusive outcomes is 1
that if the probability of something happening is p,then the probability of it NOT happening
is 1 p
that when two outcomes, A and B, of an event are mutually exclusive
P(A or B) P(A) P(B)
that from a statistical experiment for each outcome
number of times the outcome happens
relative frequency total number of trials of the event
that relative frequencies give good estimates to probabilities when the number of trials is large
that if the probability that an experiment will be successful is p and the experiment is carried
out a number of times, then an estimate for the number of successful experiments is
p number of experiments
For example if the probability that a biased coin will come down heads is 0.7 and the coin is
spun 200 times, then an estimate for the number of times it will come down heads is
0.7 200 140
that for independent events an outcome from one event does not affect the outcome of
the other event
that when the outcomes, A and B, of two events are independent
P(A and B) P(A) P(B)
that a probability tree diagram shows all of the possible outcomes of more than one event
by following all of the possible paths along the branches of the tree. When moving along a
path multiply the probabilities on each of the branches
that conditional probability is the probability of an outcome of an event that is dependent
on the outcome of a previous event. For example choosing two pieces of fruit without
replacing the first one where the choice of the second piece of fruit is dependent on the
choice of the first.
1 Shreena has a bag of 20 sweets.
10 of the sweets are red.
3 of the sweets are black.
The rest of the sweets are white.
Shreena chooses one sweet at random.
What is the probability that Shreena will choose a
a red sweet
b white sweet?
(1385 June 1999)
93
CHAPTER 5
Probability
2 80 students each study one of three languages.
The two-way table shows some information about these students.
French
German
Spanish
Total
15
Female
39
Male
31
Total
17
41
28
80
a Copy and complete the two-way table.
One of these students is to be picked at random.
b Write down the probability that the student picked studies French.
3 Here are two sets of cards.
Each card has a number on it as shown.
A card is selected at random from set A and a card is
selected at random from set B.
The difference between the number on the card
selected from set A and the number on the card
selected from set B is worked out.
a Copy and complete the table started below to
show all the possible differences.
Set A
1
1
Set B
2
0
3
(1387 June 2005)
1
1
2
2
3
3
4
4
Set A
Set B
4
2
2
0
3
1
4
b Find the probability that the difference will be zero.
c Find the probability that the difference will not be 2
4 There are 20 coins in a bag.
7 of the coins are pound coins.
Gordon is going to take a coin at random from the bag.
a Write down the probability that he will take a pound coin.
b Find the probability that he will take a coin which is NOT a pound coin.
5 Mr Brown chooses one book from the library each week.
He chooses a crime novel or a horror story or a non-fiction book.
The probability that he chooses a horror story is 0.4
The probability that he chooses a non-fiction book is 0.15
Work out the probability that Mr Brown chooses a crime novel.
4
3
94
2
6 Here is a 4-sided spinner. The sides of the spinner are labelled 1, 2, 3 and 4
The spinner is biased.
The probability that the spinner will land on each of
the numbers 2 and 3 is given in the table.
The probability that the spinner will land on 1 is
Number
1
equal to the probability that it will land on 4
Probability
x
a Work out the value of x.
Sarah is going to spin the spinner 200 times.
b Work out an estimate for the number of times it will land on 2
(1387 June 2005)
1
2
3
4
0.3
0.2
x
(1387 June 2005)
CHAPTER 5
7 Meg has a biased coin.
When she spins the coin the probability that it will come down heads is 0.4
Meg is going to spin the coin 350 times.
Work out an estimate for the number of times it will come down heads.
8 A dice has one red face and the other faces coloured white.
The dice is biased.
Sophie rolled the dice 200 times.
The dice landed on the red face 46 times.
The dice landed on a white face the other times.
Sophie rolls the dice again.
a Estimate the probability that the dice will land on a white face.
Each face of a different dice is either rectangular or hexagonal.
When this dice is rolled the probability that it will land on a rectangular face is 0.85
Billy rolls this dice 1000 times.
b Estimate the number of times it will land on a rectangular face.
9 Julie does a statistical experiment.
She throws a dice 600 times.
She scores six 200 times.
a Is the dice fair? Explain your answer.
Julie then throws a fair red dice once and a fair blue dice once.
b Copy and complete the probability tree diagram to show
the outcomes.
Label clearly the branches of the probability tree diagram.
The probability tree diagram has been started.
c i Julie throws a fair red dice once and a fair blue dice once.
Calculate the probability that Julie gets a six on both the
red dice and the blue dice.
ii Calculate the probability that Julie gets at least one six.
Red dice
1
6
Blue dice
Six
Not six
(1387 June 2003)
10 Lauren and Yasmina each try to score a goal.
The probability that Lauren will score a goal is 0.85
The probability that Yasmina will score a goal is 0.6
a Work out the probability that both Lauren and Yasmina will score a goal.
b Work out the probability that Lauren will score a goal and Yasmina will not score a goal.
(1385 June 1998)
11 Amy has 10 CDs in a CD holder.
Amy’s favourite group is Edex.
She has 6 Edex CDs in the CD holder.
Amy takes one of these 10 CDs at random.
She writes down whether or not it is an Edex CD.
She puts the CD back in the holder.
Amy again takes one of these 10 CDs at random.
1st choice
Amy had 30 CDs.
Edex CD
The mean playing time of these 30 CDs
0.6
was 42 minutes.
Amy sold 5 of her CDs.
The mean playing time of the 25 CDs left
Not-Edex CD
was 42.8 minutes.
b Calculate the mean playing time of the 5 CDs that Amy sold.
2nd choice
Edex CD
Not-Edex CD
Edex CD
Not-Edex CD
(1387 June 2004)
95
CHAPTER 5
Probability
12 A bag contains 3 black beads, 5 red beads and 2 green beads.
Gianna takes a bead at random from the bag, records its colour and replaces it.
She does this two more times.
Work out the probability that of the three beads Gianna takes, exactly two are the same colour.
(1387 June 2003)
13 Amy is going to play one game of snooker and one game of billiards.
The probability that she will win the game of snooker is 13
The probability that she will win the game of billiards is 34
The probability tree diagram shows this information.
Snooker
Billiards
3
4
1
3
2
3
Amy wins
Amy wins
1
4
3
4
Amy does not win
1
4
Amy does not win
Amy wins
Amy does not win
Amy played one game of snooker and one game of billiards on a number of Fridays.
She won at both snooker and billiards on 21 Fridays.
Work out an estimate for the number of Fridays on which Amy did not win either game.
(1388 June 2005)
14 A bag contains 10 coloured discs.
4 of the discs are red and 6 of the discs are black.
Asif is going to take two discs at random from the bag,
without replacement.
a Copy and complete the tree diagram.
b Work out the probability that Asif will take two black discs.
c Work out the probability that Asif takes two discs of the same colour.
15 Ali has twenty socks in a sock drawer.
10 of them are grey, 6 of them are black and 4 of them are red.
Ali takes two socks at random without replacement from the drawer.
Calculate the probability that he takes two socks that have the same colour.
16 There are n beads in a bag.
6 of the beads are black and all the rest are white.
Heather picks one bead at random from the bag and does not replace it.
She picks a second bead at random from the bag.
The probability that she will pick 2 white beads is 12
Show that n2 25n 84 0
96
Red
Red
Black
Red
Black
Black
(1385 May 2002)
(1385 November 2001)
(1385 June 2002)

Contents MODULE 2 1 Scatter graphs 1

Transcription

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