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Glossary
Chapter 1: Number skills
- Overview
- 1.1 Place value and ordering numbers
- 1.2 The four operations
- 1.3 Negative numbers
- 1.4 Priority of operations and inverse operations
- 1.5 Rounding and estimation
- Review
Chapter 2: Factors, multiples and primes
- Overview
- 2.1 Factors, multiples and prime numbers
- 2.2 Prime factorisation, HCF and LCM
- 2.3 Systematic listing strategies
- 2.4 The product rule for counting
- Review
Chapter 3: Powers, roots and indices
- Overview
- 3.1 Powers and roots
- 3.2 Estimating powers and roots
- 3.3 Laws of indices
- 3.4 Negative and fractional indices
- Review
Chapter 4: Fractions, decimals and percentages
- Overview
- 4.1 Equivalent fractions and ordering
- 4.2 Adding and subtracting fractions
- 4.3 Multiplying and dividing fractions
- 4.4 Fractions and terminating decimals
- 4.5 Recurring decimals
- 4.6 Fractions and percentages as operators
- Review
Chapter 5: Ratio and proportion
- Overview
- 5.1 Ratio notation, simplifying and sharing
- 5.2 Ratio, fractions and linear functions
- 5.3 Direct proportion
- 5.4 Inverse proportion
- 5.5 Scale factors, scale diagrams and maps
- 5.6 Comparing lengths, areas and volumes
- Review
Chapter 6: Percentages and financial mathematics
- Overview
- 6.1 Percentage of an amount
- 6.2 Percentage change
- 6.3 Reverse percentages
- 6.4 Simple and compound interest
- 6.5 Growth, decay and repeated percentage change
- Review
Chapter 7: Measures, accuracy, standard form and surds
- Overview
- 7.1 Units of measure and conversions
- 7.2 Compound measures: speed, density and pressure
- 7.3 Rounding and error intervals
- 7.4 Upper and lower bounds
- 7.5 Standard form
- 7.6 Surds and exact calculations
- Review
Chapter 8: Algebraic expressions and manipulation
- Overview
- 8.1 Algebraic notation and vocabulary
- 8.2 Simplifying and collecting like terms
- 8.3 Expanding brackets
- 8.4 Factorising with common factors
- 8.5 Substitution into expressions and formulae
- 8.6 Rearranging formulae
- 8.7 Expanding products of three binomials
- Review
Chapter 9: Equations and inequalities
- Overview
- 9.1 Solving linear equations
- 9.2 Equations with unknowns on both sides
- 9.3 Forming and solving equations
- 9.4 Inequalities on a number line and set notation
- 9.5 Solving linear inequalities
- 9.6 Quadratic inequalities
- Review
Chapter 10: Sequences
- Overview
- 10.1 Term-to-term and position-to-term rules
- 10.2 Special sequences
- 10.3 The nth term of a linear sequence
- 10.4 The nth term of a quadratic sequence
- 10.5 Geometric sequences
- Review
Chapter 11: Straight-line graphs and rates of change
- Overview
- 11.1 Coordinates and plotting graphs
- 11.2 Gradient and intercept: y = mx + c
- 11.3 Finding the equation of a line
- 11.4 Parallel and perpendicular lines
- 11.5 Real-life and distance-time graphs
- 11.6 Gradients and areas under graphs
- Review
Chapter 12: Quadratic expressions, equations and graphs
- Overview
- 12.1 Expanding and factorising quadratics
- 12.2 Factorising harder quadratics
- 12.3 Quadratic graphs and their features
- 12.4 Solving quadratic equations by factorising
- 12.5 The quadratic formula
- 12.6 Completing the square
- Review
Chapter 13: Other functions and graphs
- Overview
- 13.1 Cubic, reciprocal and exponential graphs
- 13.2 Trigonometric graphs
- 13.3 Transformations of graphs
- 13.4 The equation of a circle and tangents
- Review
Chapter 14: Simultaneous equations and iteration
- Overview
- 14.1 Simultaneous equations by elimination
- 14.2 Simultaneous equations by substitution
- 14.3 Solving simultaneous equations graphically
- 14.4 Linear and quadratic simultaneous equations
- 14.5 Iteration and numerical methods
- Review
Chapter 15: Functions, algebraic fractions and proof
- Overview
- 15.1 Functions: inverse and composite
- 15.2 Simplifying algebraic fractions
- 15.3 Operations with algebraic fractions
- 15.4 Identities and equivalent expressions
- 15.5 Algebraic proof
- Review
Chapter 16: Angles, polygons and geometrical reasoning
- Overview
- 16.1 Angle facts at a point and on a line
- 16.2 Angles in parallel lines
- 16.3 Angles in triangles and quadrilaterals
- 16.4 Angles in polygons
- 16.5 Geometrical reasoning and proof
- Review
Chapter 17: Properties, constructions, loci and bearings
- Overview
- 17.1 Properties of 2D and 3D shapes
- 17.2 Plans and elevations
- 17.3 Constructions with ruler and compasses
- 17.4 Loci
- 17.5 Bearings and scale drawings
- Review
Chapter 18: Transformations, congruence and similarity
- Overview
- 18.1 Reflection, rotation and translation
- 18.2 Enlargement
- 18.3 Enlargement with fractional and negative scale factors
- 18.4 Congruence and congruence criteria
- 18.5 Similarity and similar lengths
- 18.6 Similarity: areas and volumes
- Review
Chapter 19: Perimeter, area, surface area and volume
- Overview
- 19.1 Perimeter and area of 2D shapes
- 19.2 Circumference and area of circles
- 19.3 Arc length and sector area
- 19.4 Surface area and volume of prisms and cylinders
- 19.5 Surface area and volume of spheres, cones and pyramids
- 19.6 Composite shapes and solids
- Review
Chapter 20: Pythagoras' theorem and trigonometry
- Overview
- 20.1 Pythagoras' theorem
- 20.2 Trigonometric ratios (SOHCAHTOA)
- 20.3 Angles of elevation and depression
- 20.4 Exact trigonometric values
- 20.5 3D Pythagoras and trigonometry
- 20.6 The sine rule
- 20.7 The cosine rule
- 20.8 The area of any triangle
- Review
Chapter 21: Circle geometry and circle theorems
- Overview
- 21.1 Parts of a circle
- 21.2 Circle theorems
- 21.3 Tangents and chords
- 21.4 Proof of circle theorems
- Review
Chapter 22: Vectors
- Overview
- 22.1 Vector notation and column vectors
- 22.2 Adding and subtracting vectors
- 22.3 Vectors in geometric problems and proofs
- Review
Chapter 23: Probability
- Overview
- 23.1 The probability scale and equally likely outcomes
- 23.2 Relative frequency and expected outcomes
- 23.3 Frequency trees, tables and possibility spaces
- 23.4 Combined events and tree diagrams
- 23.5 Venn diagrams and set notation
- 23.6 Conditional probability
- Review
Chapter 24: Statistics: representing and analysing data
- Overview
- 24.1 Collecting data, sampling and questionnaires
- 24.2 Tables, bar charts, pie charts and pictograms
- 24.3 Averages and range
- 24.4 Averages from frequency tables
- 24.5 Time series and line graphs
- Review
Chapter 25: Statistics: spread, grouped data and correlation
- Overview
- 25.1 Quartiles, interquartile range and box plots
- 25.2 Cumulative frequency
- 25.3 Histograms
- 25.4 Scatter graphs and correlation
- 25.5 Comparing distributions and outliers
- Review
About this book

23.4 Combined events and tree diagrams — Practise with help

Work each question first. Open the hint if you need it, then reveal the working to check.

Warm-up: remember

Try these from memory before opening the answers.

What is $\frac12\times\frac12$ ?
List the outcomes of two coin tosses.

Check

$\frac14$ . 2. HH, HT, TH, TT.

Practise with help

1. A fair coin is tossed twice. Find the probability of exactly two tails.

Hint

List the four equally likely outcomes. Which one has two tails?

Full working

Only TT has two tails. Its probability is $\frac12\times\frac12=\frac14$ .

2. A fair die is rolled twice. Find the probability of two numbers less than 3.

Hint

The results less than 3 are 1 and 2. The two rolls are independent.

Full working

For each roll the probability is $\frac26=\frac13$ . Multiply: $\frac13\times\frac13=\frac19$ .

3. A fair coin is tossed twice. Find the probability of at least one head.

Hint

Find the probability of no heads, then subtract it from 1.

Full working

Use the complement: $1-P(\text{no heads})=1-\frac14=\frac34$ .

4. A bag contains 4 red and 3 blue counters. One counter is drawn and not replaced. Find the probability of red then blue.

Hint

After a red counter is removed, count the blue counters and the total counters left.

Full working

The probabilities are $\frac47$ for red first and $\frac36$ for blue next. Multiply: $\frac47\times\frac36=\frac{12}{42}=\frac27$ .