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MR ALI GCSE REVISION NOTES (HIGHER) - 2016 It’s about… Arithmetic with whole numbers Decimals Approximation Negative numbers Multiples and Factors Fractions Percentages Can you…? Do long multiplication and division e.g. 26 × 5634 or 26 5364 Do above but with decimals e.g. 2.6 × 56.34 or 1.44 ÷ 1.2 Round using decimal places e.g. 3.14159 = 3.142 (2 d.p.) Round using significant figures e.g. 3.14159 = 3.1 (2 s.f.) Use rounding to do approximate calculations e.g. 37.01 ÷ 7.99 ≈ 40 ÷ 8 Choose a suitable degree of accuracy Calculate with negative numbers e.g. 7 × -2 = -14 or -4 – -7 = +3 Understand the words: multiple, factor, prime, square, cube, root Write a number in prime factor form e.g. 36 = 22 × 32 , 54 = 2 × 33 Find the hcf (highest common factor) of 2 numbers e.g. 36, 54, hcf = 18 Find the lcm (lowest common multiple) of same e.g. 36, 54, lcm = 108 Work out a fraction of a number e.g. ¾ of 72 = 54 + and – fractions and mixed numbers e.g. ¾ + ⅔ = 15/12 2½ - 1¼ = 1¼ × and ÷ the above e.g. ¾ × ⅔ = ½ 2½ ÷ 1¼ = 5/2 ÷ 5/4 = 5/2 × 4/5 = 2 Convert between fracs, % and decs e.g. 0.015 = 1.5% = 15 3 Work out a % of a number e.g. 18% of £42 = 0.18 × £42 = £7.56 and calculate % change e.g. decrease £25 by 10%, £25-2.50=£22.50 work out one number as a % of another e.g. buy for £12, sell for £14, profit is £2 so percentage profit is 2÷12 × 100 = 16.7% work out the original value after a % change e.g. After a 25% cut, price is £405 75% of original price = £405 1% of original price = £405 ÷ 75=£5.40 Original price = 100 × £5.40 = £540 Do compound interest e.g. £500 for 3 years at 4.5% After year 1 £500 + 4.5% = £522.50 After year 2 £522.50 + 4.5% = £546.01 After year 3 £546.01 + 4.5% = £570.58 Compound Interest the quick way. £500 for 3 years at 4.5%. Ratio Proportion 1000 200 Amount of interest is £570.58 - £500 = £70.58 Scale according to a ratio e.g. An alloy is mixed in the ratio 5:7 If I have 12kg of metal A How much metal B do I need? 12 ÷ 5 × 7 = 16.8kg Share in a given ratio e.g. share £400 in the ratio 3:2 £400 ÷ 5 = £80, 3 × £80 = £240 2 × £80 = 160 Apply ratio to ‘best value’ e.g. 300ml @ 50p or 400ml @ 65p 300 ÷ 50 = 6ml per penny BEST VALUE 400 ÷ 65 = 6.15ml per penny Apply to speed and density problems (TRICKIER) . Know how to rearrange into and What is the weight of a piece of rock which has a volume of and a density of (which means 2.25 grams per cubic cm). 1 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 Powers with numbers Powers with algebra Fractional Powers Standard Form Surds Work with fractional indices e.g. 81 / 3 2 Add and subtract with standard form e.g. Substitution Expand brackets Factorise into one Linear Equations Equations from problems Do equations with letters on both sides e.g. Solve equations with brackets e.g. Use trial and Solve simultaneous 3/ 2 27 8 Substitute numbers into formulae e.g. x = 5 and y = -3 3x2 – y = 3 × 52 - -3 = 75 + 3 = 78 Expand and simplify brackets e.g. 3(2x + 1) - 2(x – 4) = 6x + 3 - 2x + 8 = 4x + 11 Factorise into a single bracket e.g. t2 – 5t = t(t – 5) Solve simple linear equations e.g 3x + 7 = 25 3x = 25 – 7 3x = 18 x = 6 Solve equations with negative coefficients e.g. 10 – 2x = -4 improvement 9 4 Note: turn into regular numbers with adding and subtracting Multiply and divide with standard form e.g. 6.3 105 2.1 103 3 102 Note: calculate in standard form when multiplying and dividing Use a calculator for standard form i.e. use the EXP button Work with surds e.g 3√2 +5√8= 3√2 + 10√2 = 13√2 bracket 3 / 2 Can you…? 4 9 543000 5.4 105 2.3 104 540000 23000 563000 5.63 105 It’s about… 54 53 1 5 2 9 5 25 4 2 3 12a b c Tidy up powers with algebra e.g. 3a 2 c 4a 2 b 2 c 2 Tidy up powers of numbers e.g. 10 10+4 14 14÷ - 2 -7 5x + 4 2x + 4 2x x = -4+2x = -2x = -2x = x = x = 3x +12 = 12 = 8 =4 3(4 – x) = 2(x + 1) 12 – 3x = 2x + 2 12 = 5x + 2 10 = 5x 2 =x Form an equation from a problem e.g. perimeter is 26cm, find x equations (x-1) Rearrange Formulae (2x+2) x-1+x-1+2x+2+2x+2 = 26 6x +2 = 26 6x = 24 x =4 solve x3+2x=20 x x3 2x x3+2x (trial and 2 8 4 12 low improvement) 3 27 6 33 high keep going until answer is accurate enough 2 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 4x + 3y = 37 2x + y = 17 ×2 4x + 2y = 34 - y =3 Substitute y = 3 into 4x + 9 = 37 4x = 28 x =7 Rearrange a simple formula e.g. y = 3x – 7 make x the subject y + 7 = 3x Do simultaneous equations e.g. y7 x x and that’s the answer! Rearrange formulae with squares e.g e = ½mv2 2e = mv2 2e v2 m 2e v m Quadratics and that’s the answer! Expand 2 brackets e.g. (5x + 4)(3x – 2) = 15x2 + 2x – 8 Factorise into 2 brackets e.g. x2 + 5x – 24 = (x + 8)(x – 3) Solve quadratic equations by factorising e.g. x2 + 5x – 24 = 0 so (x + 8)(x – 3)=0 so x = -8 or x = 3 Solve quadratic equations using the formula e.g. 2x2 + 3x – 5 = 0 so a = 2, b = 3 and c = -5 a 2 and use b b 4ac 2 4 4 2 (5) 2 44 4.63 or -8.63 Linear graphs Further graphs Algebraic Fractions Sequences Harder rearranging 2a 4 4 Draw any straight line graph e.g. draw y = 3x – 2 from x=-2 to x=4 find the gradient of any straight line i.e. count units and do rise run Find the equation of a graph (Note: remember y = mx+ c) e.g. if gradient = 2 and y-intercept = 3, y = 2x + 3 Solve simultaneous equations using graphs i.e. draw both graphs and see where they cross Draw graphs of quadratic equations e.g. draw x2 + x – 6 = 0 between x = -3 and x = +4 (fill in a table) Use quadratic graphs to solve equations e.g. solve x2 + x – 6 =0.5 using the above graph (see where y=0.5) 3 1 3x ( x 4) 2x 4 Work with algebraic fractions e.g. ( x 4) Find the nth term of a sequence e.g. Finding rules from diagrams eg. Make a table of values x x( x 4) x( x 4) 20, 17, 14, 11…. Going down in 3s so try -3n 1st term comes out at -3×1=-3 We need 20, so have to add 23 Formula is tn = -3n + 23 Tables 1 2 3 4 5 Seats 4 7 10 13 16 Seats going up by 3 so try 3t. 1st term is 3 x 1 = 3. 2nd term is 3 x 2 = 6. Need to add + 1 so Rule is: s = 3t + 1 To find how many seats for 50 tables s = 3(50) + 3 = 153 seats. To find how many tables are needed for 100 guests 100 = 3t + 3 3 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 Quadratic sequences e.g. 5, 11, 21, 35, 53 6 10 14 18 (1st diff) 4 4 4 (2nd diff) nd Halve the 2 diff (4) so the rule has ‘2x2’ in it. 1st term: 2x 12=2 (need 5 so plus 3). Eg. 2x2 + 3. 2nd term: 2 x 22 = 8 (add 3 is 11 so we are right) Rule is tn = 2x2 + 3 Factorise to rearrange formulae e.g a – b = ax (A-grade stuff) a = ax + b a – ax = b a(1-x) = b b and that’s the answer! a 1 x Rearrange with fractions and √s e.g a – b = ax Check that the dimensions of a formula are consistent e.g. do these expressions represent length, area or volume?: πr2 + ab a + 2b a + bc 1) Replace each letter with the letter m (π and numbers don’t count – cross off). 2) Simplify as much as possible. 3) Decide if the formula is a length, area or volume. Dimensions a + 2b = m + m = 2m πr2 + ab = m2 + m2 = 2m2 = =m3 LENGTH AREA VOLUME a + bc = m + m2 (can’t be simplified) NONE OF THESE Convert a α b to a = kb and a α 1/b to a = k/b and a α 1/b2 to a α k/b2 Solve problems of direct proportion e.g. A is directly proportional to t A is 45 when t is 5 Find A when t is 8 Aαt A kt 45 5k k 9 A 9t Direct and inverse proportion If t = 8 A = 9×8 = 72 Solve problems of inverse prop e.g. C is inversely proportional to f2 C is 20 when f is 3 Find C when f is 5 1 f2 k C 2 f k k 20 2 9 3 180 k C C Inequalities So when f = 5 Solve simple inequalities e.g Solve double inequalities e.g. 180 f2 C = 180/25 = 7.2 3x+4<5 3x < 1 x < ⅓or –8 < 5x+2 < 22 -10<5x or 5x < 20 -2 < x or x < 4 4 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 So -2 < x < 4 -2 < x < 4 x is an integer so x= -1, 0, 1, 2, 3 Do above for integer values e.g. Illustrate above on a number line remember solid dots for ≤ or ≥, hollow dots for < or > moves the graph by +a in the y direction (i.e. moves the graph up by a). moves the graph by -a in the x direction (i.e. moves the graph by a to the left). moves the graph by –a in the y direction (i.e. moves the graph down by a). moves the graph by +a in the x direction (i.e. moves the graph by a to the right). Transforming graphs 1. Area 2. Volume of Prisms 3. Volume of Pyramids 4. Volume and surface area of spheres stretches the graph by a factor of a in the y direction. stretches the graph by a factor of 1/a in the x direction (squashes by a factor of a). is the graph of reflected in the y axis. is the graph of reflected in the x axis. Calculate area of a rectangle: b × h (LEARN) Calculate area of a triangle: ½ × b × h (LEARN) Calculate area of a trapezium: ½ ×( l1×l2) × h (LEARN) Calculate circumference and area of a circles: C = πd = 2πr (LEARN) A = πr2 (LEARN) Calculate area and perimeter of sectors of circles e.g. 4cm P = 4 + 4 + (60/360 x 2 x π x 4) = 12.2cm 60° A = 60/360 x π x 42 = 8.37cm2 Use the above to calculate the areas of compound shapes 1. Chop into shapes you know 2. Work out areas separately 3. Add the separate areas Use cross-sectional area × length to calculate volume of prism 5 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 Area = A Use ⅓ × base area × height to calculate volume of pyramids (LEARN) Calculate volume and surface area of cone: V = 1/3 πr2h (ON Curved surface area = πr3l SHEET) Calculate volume and surface area of a sphere: A = 4 πr 2 (ON V = 4/3 πr3 SHEET) Calculate volumes of compound shapes (chop into shapes you know) Use Pythagoras to find missing sides a2 + b2 = c2 (c is the long side) 62 + 82 = x2 52 + y2 = 132 2 6cm x 36+64= x 5cm 13cm 25+ y2 = 169 2 100 = x y2 = 169-25 8cm x = 10cm y y2 = 144 y = 12cm Use Pythagoras to solve problems e.g. a 4m ladder is leaning against a wall, the base is 1m from the wall. How far up the wall does the ladder reach? DRAW A DIAGRAM a 2 + b 2 = c2 h 4m 12 + h2 = 42 1 + h2 = 16 h2 = 16 – 1 = 15 1m h = √15 = 3.87m Use Trig to find missing sides and angles: (always labels sides first) (o) ? 12cm (h) 42° opposite sin x hypotenuse opposite sin 42 12 Opposite = 12sin42° = 8.01cm adjacent hypotenuse opposite tan x adjacent cos x Angles (LEARN) Calculate surface areas of prisms/pyramids (draw nets to count faces) e.g. note it has 5 faces – work out area of each one and add Volume = A x l 1. Pythagoras 2. Trigonometry (right angled triangles SOHCAHTOA: opposite sin x hypotenuse l (a) 2m x° tan x tan x h (o) 3m opposite adjacent 3 1.5 2 tan-1(1.5) = 56.3° Use Trig to solve problems: (draw a diagram and labels sides first) e.g. a 4m ladder is leaning against a wall, the base is 1m from the wall. What angle does the ladder make with the ground? DRAW A DIAGRAM cosx = adjacent AND LABEL SIDES 4m hypotentuse cos x = ¼ = 0.25 x° x = cos-1(0.25) a 1m = 75.5° Use angle sum of a triangle = angles on a straight line = 180° (1×180°) Use angle sum of a quadrilateral = angle sum around a point = 360° Use angle sum of a pentagon = 540 (3×180°) Use angle sum of an n-sided polygon add up to (n-2) × 180° Remember that where two lines cross, opposite angles are equal In parallel lines, alternate (Z) and corresponding (F) angles are equal 6 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 Recall that in parallel lines, interior or allied (C) add to 180o. Alternate ( Z ) angles are equal therefore a = 74o Angles on a straight line add to 180o therefore b = 180o – a b = 180o – 74o b = 106o Allied (C) angles add to 180o therefore c = 180o – a c = 180o – 74o c = 106o (Also note that b and c are corresponding (F) angles which means they are equal) Corresponding angles are equal therefore d = 65o Opposite angles are equal therefore e = 65o Circle Theorems (all 7) draw and label the radius, diameter and circumference of a circle draw and label a chord, sector, segment or tangent to a circle Radius : the distance from centre to circumference (Diameter = 2 x radius) Tangent: a straight line which touches the circumference of the circle (once only) Chord: a straight line which connects any two points on the circumference Segment: a chord divides a circle into two segments (major and minor) Circumference: distance around the edge of a circle Use angles subtended by the same chord are equal a=b=c Use angle at centre = 2 × angle at circumference 7 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 Use opposite angles in a cyclic quadrilateral add up to 180° Use angle between a tangent and a radius is 90° Use tangents to a circle from an external point to the points of contact are equal in length. Use angle subtended (standing on) a diameter is 90° Use the alternate segment theorem Reflect a shape in a given line Reflect shape A in the line x = -1 x=-1 A NOTE: you need to be able to draw x=…-3, -2, -1, 0 1, 2, 3, 4 etc y= …-3, -2, -1, 0, 1, 2, 3, erc y=x and y=-x Rotate a shape around a given Rotate shape A 90° clockwise About the point (0,0) point A NOTE: use tracing paper to do this Transformations Translate a shape by a given vector Translate shape A by the vector A 1 3 Enlarge a shape with a given centre and scale factor Enlarge by scale factor 2 centre (2, 4) x A NOTE: find the distance of each corner from the centre of enlargement and multiply by the scale factor Carry out two of the above transformations in succession e.g. reflect the triangle A in the line x = -1, call the new shape B THEN rotate triangle B 180 about the 8 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 point (-1,0), call this shape C Fully describe a transformation e.g. describe the SINGLE TRANSFORMATION that maps A to C answer: reflection in the x-axis B A C Constructions and Loci Construct any triangle Given 3 sides e.g. To construct a triangle of sides 8 cm, 7cm and 6 cm 1. Draw a line of 8cm long with a ruler 2. Set compass to 7 cm, place at either end of the line and draw an arc 3. Set compass to 6 cm, place at the other end of the line and draw an arc to intersect the first one 4. Draw straight lines from the point of intersection to both ends of the line Given 1 side and 2 angles e.g. To construct a triangle of side, 9 cm with angles of 35 o and 65o 1. Draw a straight line 9cm long 2. Use a protractor to draw angles of and 650 on either end of line 3. Draw straight lines from both until they intersect to form the triangle 350 ends Given 2 sides and with an included angle e.g. To construct a triangle of sides, 9 cm and 7cm with an o angle of 40 1. Draw a straight line 9cm long 2. Use a protractor to draw an angle of 400 on either end of line 3. Mark off a length of 7cm along that line 4. Join end points to form the triangle 5. Construct angle bisectors of a line 1. Place compass at A, and draw an arc 2. Place compass at X and draw an arc 3. Do the same at Y (with the compass set in the same distance) 4. Draw the angle bisector from A through the point of intersection, B Construct perpendicular bisectors of a line 1. Place compass at one end of the line, set it over halfway and draw 2 arcs above and below the line 9 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 2. 3. Without changing the compass, do exactly the same at the other end of the line Draw a straight line through the points of intersection. This is the perpendicular bisector Solving locus problems The locus of a point that remains a constant distance form a fixed point is a circle The locus of a point on a straight line is a pair of parallel lines The locus of a point on a finite line is a pair of parallel lines joined by 2 semicircles Real Life Graphs Interpret distance-time graphs 10 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 Graph shows Jamie’s journey form home to his friend’s house. Left home at 10:10 Stopped for 30 mins Arrived home at 11:50 BE CAREFUL WITH UNITS! So, speed on the way there = Speed on way home = = = 40 = 0.5 Total distance travelled = 40km Conversion graphs Odd graphs (e.g. containers filling with water) Find missing lengths m Don’t forget to note when the missing value is only part of the unknown length: Similarity m Advanced Trigonometry Use scale factors to find areas and volumes: E.g. If length scale factor = 2 then Area scale factor = 2x2 = 22 = 4 and Volume scale factor = 2x2x2 = 23 = 8 Solve 3D problems Calculate the length of the diagonal. Length of diagonal across the base = = 10.77cm Length of diagonal = = 12.33cm use the sine and cosine rules for triangles without right angles a b c sin A sin B sin C or sin A sin B sin C a b c 11 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 x 30 sin 50 sin 60 x 30cm 60o x 50o 30 sin 50 sin 60 x 26.54cm 10cm θ 40o sin sin 40 14 10 14 sin 40 sin 10 sin 0.8999 14cm 64.15 a 2 b 2 c 2 2bcCosA 5m or b2 c2 a 2 CosA 2bc x 2 4.52 52 2 4.5 5 Cos115 115o 4.5m x 2 64.2678 x 8.02m x 8cm 9cm Cos Cos 0.7454 θ 12cm area 1 ab sin C 2 41.81 area 1 13 15 sin 30 2 area 48.75cm 2 13cm 30o 9 2 12 2 82 2 9 12 15cm Recognise the graphs of sin cos and tan and solve simple trig equations (A*) y = sinx y = cosx 12 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 y = tanx Recognise transformed trigonometric graphs Solve simple trig equations (A*) Degree of accuracy Rule of thumb For problems involving x & ÷ round to lowest number of significant figures of the numbers given in the question For problems involving + & - round to lowest number of decimal places of the numbers given in the question E.g. One side of a rectangle is measured as 12.53cm, the other side is 17.3cm Perimeter: 12.53 + 17.3 + 12.53 + 17.3 = 59.66. Round to one decimal place, 59.7 cm Area: 12.53 x 17.3 = 216.769. Round to 3 significant figures, 217 cm 2 Accuracy Upper and lower bounds. Given a measurement e.g. 17.6m, The lower bound is the lowest possible measurement that would be rounded up to 17.6 i.e. 17.55. The upper bound is the highest possible measurement that would be rounded down to 17.6 i.e. . Solve problems involving upper and lower bounds eg. A square playground has a side of length 25m to the nearest metre. (a) What is the upper bound for the perimeter? (b) What is the lower bound for the area? Vectors Use vectors written a, b, c Add, subtract and multiply vectors in component form Recognise parallel vectors (one must be a scalar multiple of the other) and use to determine whether points are on the same straight line (vectors defined using these points must be parallel and have a point in common). Translate between the above and AB etc B (7,3) A (4,1) 13 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 Use vectors to solve geometrical problems. a+b b + 3a a + b – 4a + b = 2b – 3a Questionnaires A good question will not be: Too personal. E.g. how old are you? Full of technical jargon. Leading. E.g. don’t you think that ________? Open. E.g. what do you think of _______? A good question will: Have a yes/no, number, or multiple choice answer. Multiple choices will cover all possible answers and will not have overlapping intervals. Questionnaires Sampling Averages Frequency polygons & Histograms Cumulative frequency Bad e.g. How much time do you spend watching TV on average per day? (THESE ARE NO GOOD –THEY OVERLAP) o 0 – 30 minutes o 30 – 60 minutes o 1-2 hours o 2 hours or more HERE’S HOW YOU FIX THE PROBLEM How much time do you spend watching TV on average per day? 1. Less than 30 minutes 2. More than 30 minutes, less than one hour 3. At least one hour but less than two hours 4. More than two hours hours Sampling Describe how to take a sample using the following methods Random Systematic Cluster Stratified random Describe the pros and cons of the above sampling methods Calculate the required group sizes for a stratified random sample of 100 students Year group Boys Girls 7 152 160 8 145 155 9 130 133 Sample 14 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 Year group 7 8 9 Boys 17 17 15 Girls 18 18 15 Size of sample group = __group size__ x desired sample size Population size Given a list of numbers eg. Test scores: 9, 10, 12, 15, 15, 15, 17, 18, 18, calculate the… Mean (14.22) Median (15) Mode (15) Upper quartile (17.5) Lower quartile (11) Inter-quartile range (6.5) Range (9) Draw a box-plot Given a frequency table eg. Number of pets Frequency 0 8 1 12 2 16 3 4 Calculate the… mean (1.4) median (1.5) mode (2) Draw a bar chart Given a grouped frequency table eg. Height Frequency 140 ≤ h < 150 4 150 ≤ h < 160 9 160 ≤ h < 170 11 170 ≤ h < 180 6 Calculate an estimate of the mean? (161.33) Which class contains the median? 160 ≤ h < 170 Which is the modal class? 160 ≤ h < 170 Draw a histogram and frequency polygon 15 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 15 10 5 140 150 160 170 180 Draw a cumulative frequency graph Estimate the median and quartiles from the cumulative frequency graph and draw a box plot 30 25 20 15 10 5 140 150 160 170 180 16 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 Given a grouped frequency table with uneven class intervals Height Frequency 100 ≤ h < 150 4 150 ≤ h < 160 9 160 ≤ h < 170 11 170 ≤ h < 180 6 Calculate an estimate of the mean? (161.33) Draw a histogram Complete the table and the graph Time (t) in minutes 0 < t ≤ 10 Frequency 20 Frequency density 10 < t ≤ 15 15 < t ≤ 30 30 < t ≤ 50 62 50 < t ≤ 60 23 0 20 30 40 Time (minutes) 50 60 70 Bar charts Data Presentation 10 Gaps between the bars The horizontal axis represents the type of data, it is NOT a continuous number scale The vertical axis represents frequency Histograms No gaps between the bars The horizontal axis has a continuous scale The vertical axis represents frequency density, which is (frequency of the class interval) (width of class interval) Area of each bar represents the class frequency of the bar 17 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 Pie charts Working out Total frequency: 22+36+8+24 = 90 Midlands: (22/90) x 3600 = 880 London:(36/90)x 3600 = 1440 Southern England (8/90) x 3600 = 320 Northern England: (24/90) x 3600=960 Scatter Graphs Plot points Draw line of best fit Comment on correlation Relative frequency Rob throws a biased dice 100 times. The table shows his results. Probability Score Frequency 1 8 2 10 3 15 4 12 5 21 6 34 Find an estimate for the probability that he will get a 6. 18 MR ALI GCSE REVISION NOTES (HIGHER) - 2016 Probability trees Loren has two bags. The first bag contains 3 red counters and 2 blue counters. The second contains 2 red counters and 5 blue counters. Loren takes one counter at random from each bag. Complete the probability tree diagram. Counter from first bag Counter from second bag Red 2 7 Red 3 5 ...... Blue Red ...... ...... Blue ...... Blue What is the probability that she … takes 2 red counters? (6/35) takes 2 different coloured counters? (19/35) 19