Probability

6.4 Probability Review The probability of an event is a measure of the chance of it happening. Probability is measured on a scale from 0 to 1. An event which is impossible has a probability of 0. An event which is certain has a probability of 1.

For simple events, like throwing a dice or selecting a ball from a bag, symmetry can be used to work out the expected probability of the event occurring.

E d b b.1. the number of ways the event can happen

xpecte pro a 11ty = the number of possible outcomes

When an experiment (like rolling a dice or tossing a coin) is repeated several times, we can calculate the number of times we expect an event to occur. Call the event in which we are interested a 'success'.

Expected number of successes = (probability of a success) x (number of trials)

(a) Seven discs numbered 3, 4, 5, 7, 9, 11, 12 are placed in a bag. One disc is selected at random.

(b) A fair dice is rolled 540 times. How many times would you expect to roll a '2'.

In this example there are 7 possible outcomes of a trial. (i) p (selecting a '5') = + (ii) p (selecting an odd number) = t (iii) p(selecting a '10') = 0

Some events can either 'happen' or 'not happen'.

p (rolling a 2) = i Expected number of 2's = i x 540

= 90

163

(a) The spinner shown has equal sectors.

(b) The probability of a drawing pin landing 'point up' is 061.

(i) p (spinning a 3) = t (ii) p (not spinning a 3) = 1 - t = f

Therefore, the probability of the drawing pin landing 'point down' is 1 - 061 = 039.

162

6. A gardening expert has succeeded in growing a new greenhouse plant which reproduces very quickly. In 1995 he already has 20 000 plants and he estimates that the number will treble every year for the next five years. Copy and complete the table.

Year 1995 1996 1997 1998 Plants 20000

Plot the points on a graph and join them up with a smooth curve. Use a vertical scale of 2 em to represent 1 million plants and a horizontal scale of 2 em to represent 1 year.

1999 2000

If one large greenhouse can hold 40 000 plants, how many greenhouses will he require in the year 2001?

7. The men's athletics world records in 1994 were as shown. For each distance work out the average

. . [ distance] runnmg speed m m/s speed = . . hme

For the longer distances you will first have to convert the times into seconds.

Draw a pair of axes as shown. The zig-zag section indicates that a part of the vertical axis has been cut out so that we can use a larger scale. Plot the speed for each distance, as accurately as possible, and draw a smooth curve through the points.

lOOm 985 s 200m 1973s 400m 4329 s 800m 1m 4173 s

1500m 3m 2882s 5000m 12m 5696

lOOOOm 26m 5223

average speed (rnls) 10

6

distance (m)

The world record for 3000 m has been left out. Use your graph to estimate the likely running speed needed to set the world record. Work out the time which this speed would give for 3000 m. Compare your answer with the actual record, which your teacher will know (from the answer book).

L. Burrell M. Marsh

B. Reynolds S. Coe

N. Morceli H. Gebresilasie

W. Sigei

10000

Part 6

USA USA USA G.B. Alg Eth Ken

Plotting Graphs

3. Most countries nowadays measure temperature in oc (Celsius), but some still prefer op (Fahrenheit). This country used op as the standard until about 20 years ago. Your grandparents will know that 75F is 'hot' but may have no idea what 25C means. (a) Draw axes, as shown, with a scale of 1 em to so.

Two equivalent temperatures are 32F = ooc and 86F = 30C.

(b) Draw a line through the points above and use your graph to convert: (i) 20C into op (ii) -10C into op (iii) 50F into oc

40

-20

(c) The normal body temperature of a healthy person is 98F. Susie's temperature is 39C. Should she stay at home today, or go to school as usual?

4. In the U.K. , petrol consumption for cars is usually quoted in 'miles per gallon' . In other countries the metric equivalent is 'km per litre'. (a) Convert 20 m.p.g. into km per litre. (b) Convert 5 km per litre into m.p.g. (c) At a steady speed of 50 m.p.h., a

Jaguar Sovereign travels 9 km on one litre of petrol. Convert this consumption into miles per gallon and hence work out how many gallons of petrol the car will use, if it is driven at 50 m.p.h. for 2~ hours.

5. Selmin and Katie make different charges for people wanting pages typed professionally.

Selmin 20 fixed charge plus 1 per page

161

oc

X (86, 30)

op

32 110

10 20 30 miles per gallon

(a) How much would Selmin charge to type 30 pages? (b) How much would Katie charge to type 10 pages? (c) Draw axes for the number of pages

Total cost,

typed and the total cost, using the scales given.

(d) On the same diagram, draw a graph for each typist to show their charges for up to 60 pages.

(e) Use your graphs to decide for what number of pages Selmin is the cheaper typist to choose.

100

0 ('l

. . ~ ~II ~ E

(.)

('l

(Scale: 2 em = 10 pages)

number of pages typed 60

Area of a circle 6 Around and around 172 Averages 132

Bearings 37 Brackets 89 Break the codes 140

Calculator 27 Centre of enlargement 113 Centre of rotation 34 Changing fractions to decimals Changing decimals to fractions Circumference 1 Cross numbers 118 Cylinders 105

Differences 68

Enlargement Equations Estimating Expressions

112 89 123 77

Finding a formula Forestry problem Formulas 72 Frequency table

Graphs 155

72 26

138

Hidden words 58

Investigating tables 142

'L' puzzles 99 Line of best fit 130 Liquids 104 Listing outcomes 169

9 9

INDEX

Mental arithmetic Mixed problems Multiple choice test

Negative numbers

Operator squares Order of operations

108 50

82

18

24 27

Percentages 9, 11 , 151 Perimeters and common edges 60 Prisms 103 Probability 161 Problem solving, equations 194 Pythagoras' theorem 62

Ratio 45 Recurring decimals 10 Reflection 13 Review exercises 82, 178 Rotation 30 Round the class (game) 120

Scatter graphs 127 Sequences 68 Speed 145 Substituting into a formula 75

Three-dimensional objects 79

Using letters for numbers 89

Vending machine problem 141 Views 79 Volume 101

Revision Exercises

15. (a) A post bag contains three letters A, B and C. Letter A is addressed to house number 1. Letter B is addressed to house number 2. Letter Cis addressed to house number 3. The 'postman' is a monkey who has been trained to put one letter into each house. The monkey has not yet learned how to read. Consequently the 3 letters could be posted into the 3 houses in any order. (i) List all the ways in which it is possible to

post one letter into each house. (ii) What is the probability that all the letters

are delivered to the correct houses? (iii) What is the probability that none of the

letters is delivered to the correct house? (b) Another post bag contains four letters A, B, C, D. Letter A

is addressed to house 1, letter B is addressed to house 2, letter C is addressed to house 3 and letter D is addressed to house 4. The monkey delivers the letters at random. (i) List all the ways in which it is possible to post one letter

into each house. (ii) What is the probability that all the letters are delivered

to the correct houses? (iii) What is the probability that just one out of the four

letters is delivered to the correct house?

185

184

12. Describe fully each of the following transformations. (a) 61 -t 62

y

(b) 61 -t 63 (c) 61 -t 64 (d) 64 -t 65

13. Draw x and y axes with values from -8 to +8. Plot and label 61 at ( -4, -3), ( -4, -6), ( -2, -3). (a) Draw the triangles 62, 63, 64, 65 as follows:

(i) 61 -t 62 rotation, 180, centre (0, - 3) (ii) 62 -t 63 rotation, 180, centre (3, 1) (iii) 61 -t 64 enlargement scale factor 3, centre ( -5, -7) (iv) 61 -t 65 reflection in y = 1

(b) Describe fully each of the following transformations: (i) 62 -t 63 (ii) 63 -t 64.

14. A school teacher thinks there is a connection between her pupils' test results and the average number of hours of television they watch per week. She thinks that those who watch the most television will do least well in the tests. Here are scatter graphs for her classes in year 8 and year 9.

Marks Year 8 Marks

X X X X

X X X

X X X X X X

hours of TV watched

Was the teacher' s theory correct for (a) Year 8 (b) Year 9?

X

X X

X X

Year9 X X

X X X

X X X

X

hours of TV watched

In both cases state briefly what the graphs show.

Part 7

Revision Exercises

5. The trapezium, the square and the circle below all have the same area.

4cm

CJ-T 4cm J 7cm

Calculate the values of x and r.

6. Solve the equations 4

(a)--= 3 x+l

4 (b)-+ 2 = 5

X

7. Use estimation to choose the odd one out. 315 X 97; 5874 7 1983; 1527 X 201; 30908 7 102

8. Copy the triangle on squared paper. Find the area of the triangle, giving your answer in square units.

9. Calculate the volume of each of the prisms shown below. All lengths are in em. (a) 4 (b)

12

10. Oil from the large drum is used to fill many of the small cans. How many cans may be filled from the drum?

11. The diagram shows an equilateral triangle of side 8 em with a line of symmetry drawn through A. (a) Calculate the vertical height of the triangle. (b) Calculate the area of the triangle.

(c) semi circle

183

I 70cm

!

182

Revision exercise 3

1. The graph shows the journeys of Mark and Meera, who drove from their home to an hotel 120km away. (a) What was Meera's

speed? (b) What was Mark's

speed after his short stop?

(c) How far apart were they at 1 030?

(d) What was Mark's average speed for the whole journey?

0900 . \ ............. ; .. . ! .......... +. 1 1 000 I ........... J 1 ............ + ............ + . ............ ! ............ + 2. Abi is thinking of a number. Three times the number plus 5

gives the same answer as when the number is added to 4 and then the result is doubled. Find the number she is thinking of.

3. Different shapes can be drawn on a grid vertices of each shape are drawn at any dot. Here are two examples: a rectangle and a trapezium.

of nine dots. The

CJ

Draw four grids and label them A, B, C, D. [You can use dotty paper but it is not necessary] . (a) On grid A draw any parallelogram. (b) On grid B draw any isosceles triangle. (c) On grid C draw another isosceles triangle, different to the

one you drew on grid B. (d) On grid D draw a trapezium, different to the one in the

example above.

4. (a) Write each of the following as decimals (i) 22% (ii) i (iii)7% (b) Simon got 52 out of 80 in a science test.

What was his mark as a percentage? (c) Write these numbers in order of size, smallest first:

011 , t, 10%, 001.

Part 7

1 100 I 1 200 i . ............ ~ ... .......... + ......... ; -~- ............. t

D

Revision Exercises

8. In these diagrams a letter 'V' is drawn across each rectangle.

(a) Count the black squares, b, and the white squares, w, in each diagram and write the results in a table. * _13

(b) Draw the next diagram in the sequence. Count the black squares and the white squares and add the results to your table.

(c) Predict the number of white squares in the next diagram (that is the fifth diagram) in the sequence.

(d) In a later diagram, there are 136 white squares. How many black squares are there in that diagram?

9. The test results of eight pupils are recorded in the table below.

Pupil A B c D E Maths 25 10 35 45 20 Geography 10 15 15 25 30 History 10 15 20 22 27

Draw scatter graphs for (a) the Maths and Geography marks. (b) the History and Geography marks.

F G

10 40 35 -45 30 36

(c) What correlation, if any, is there in the results?

H I

15 ? 50 38 40 ?

(d) Pupil I got 38 marks in Geography. Estimate, if possible, her marks in Maths and History.

10. A shopkeeper buys a computer for 620 and sells it for 999. Calculate the profit made by the shopkeeper as a percentage of the cost price (620).

11. A spinner is spun and a dice is rolled at the same time. (a) List all the possible outcomes

on a grid. (b) Find the probability of

obtaining a total score of (i) 4 (ii) 10

(c) What is the probability of obtaining the same number on the spinner and the dice?

12. Describe the single transformation equivalent to reflection in the line y = x followed by reflection in the line x = 4.

181

50Laths

50

0 Geography

SOt History

~ 0 Geography