5.3.1

Fairness and Bias

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Fairness and Bias

When an event occurs, it can either be completely fair, or the outcome can be impacted by bias.

Definitions

Definitions

  • Fair event: Every result has an even probability of happening.
  • Biased: One or multiple results have a higher probability of happening than the rest, so they are not all the same.
Example 1

Example 1

  • A 6-sided dice is thrown, what should the chance of rolling a 2 be?
    • There is a 1x\dfrac{1}{x} chance of each outcome, and in this case there are 6 possible outcomes
    • Therefore the chance of a 2, or any number, being rolled is 16\dfrac{1}{6}
Example 2

Example 2

  • A spinner with 4 sections is spun; however, it is biased, so the chance of getting a 3 is twice as much as 1, 2, or 4. What is the probability of spinning a 3?
    • With the extra weighted chance of spinning a 3, the probabilities are calculated as x5\dfrac{x}{5}
    • The unbiased sides of 1,2 and 4 all have the probability of \dfrac{1}{5}$$, and the biased side of 3 will have the probability of $$\dfrac{2}{5}
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1

Number

1.1

Place Value & Ordering

1.1.1Ordering Integers & Decimals1.1.2Equal Symbols1.1.3Place Value
1.2

Numerical Operations

1.2.1Adding Integers1.2.2Subtracting Integers1.2.3Multiplying Integers1.2.4Dividing Integers1.2.5Adding & Subtracting Decimals1.2.6Multiplying Decimals1.2.7Dividing Decimals1.2.8Adding & Subtracting Negative Numbers1.2.9Multiplying & Dividing Negative Numbers1.2.10Multiplying by Powers of Ten1.2.11Dividing by Powers of Ten
1.3

Fractions

1.3.1Writing Fractions1.3.2Constructing Fractions1.3.3Simplifying Fractions1.3.4Ordering Fractions1.3.5Fractions & Recurring Decimals1.3.6Mixed Numbers & Improper Fractions
1.4

Numerical Operations on Fractions

1.4.1Adding & Subtracting Fractions1.4.2Adding & Subtracting Mixed Numbers1.4.3Multiplying Fractions1.4.4Multiplying Mixed Numbers1.4.5Reciprocals1.4.6Dividing Fractions1.4.7Dividing Mixed Numbers (sample)1.4.8Dividing Mixed Numbers
1.5

Roots & Powers

1.5.1Integer Powers1.5.2Roots1.5.3Estimating Powers & Roots1.5.4Fractional Indices (1/n)1.5.5Fractional Indices (m/n)1.5.6Laws of Indices (Multiplication)1.5.7Laws of Indices (Division)
1.6

Structure of Calculations

1.6.1Order of Operations (BIDMAS)
1.7

Numerical Skills

1.7.1Using a Calculator - Powers, Roots & Fractions1.7.2Using a Calculator - Trigonometric Functions
1.8

Factors & Multiples

1.8.1Finding Factors1.8.2Finding Multiples
1.9

Systematic Listing

1.9.1Systematic Listing
1.10

Surds

1.10.1Multiplying & Dividing Surds1.10.2Simplifying Surds1.10.3Expanding Expressions With Surds1.10.4Rationalising the Denominator
1.11

Standard Form

1.11.1Converting to Standard Form (Positive Powers)1.11.2Converting to Standard Form (Negative Powers)1.11.3Converting From Standard Form (Positive Powers)1.11.4Converting From Standard Form (Negative Powers)1.11.5Adding & Subtracting Standard Form1.11.6Multiplying & Dividing Standard Form1.11.7Standard Form on a Calculator
1.12

Fractions, Decimals & Percentages

1.12.1Converting Fractions, Decimals & Percentages1.12.2Converting Recurring Decimals1.12.3Numbers as Percentages1.12.4Fractions of Amounts
1.13

Percentage Change

1.13.1Percentages of Amounts (Calculator)1.13.2Percentages of Amounts (No Calculator)1.13.3Percentage Change1.13.4Original Values After Percentage Change1.13.5Growth & Decay Using Multipliers
1.14

Rounding & Estimation

1.14.1Rounding Integers1.14.2Estimating Calculations1.14.3Rounding Decimals1.14.4Rounding With Significant Figures
1.15

Bounds & Accuracy

1.15.1Finding Error Intervals1.15.2Upper & Lower Bounds1.15.3Truncating Decimals1.15.4Error Intervals From Truncation
1.16

Using Numbers

1.16.1Highest Common Factor1.16.2Lowest Common Multiple1.16.3Prime Numbers
2

Algebra

2.1

Algebraic Notation

2.1.1Using Algebraic Notation
2.2

Substitution & Simplification

2.2.1Substituting into Expressions2.2.2Substituting into Real-Life Formulae2.2.3Simplifying Expressions by Collecting Like Terms
2.3

Brackets

2.3.1Expanding Single Brackets2.3.2Expanding & Simplifying with Single Brackets2.3.3Expanding Double Brackets2.3.4Expanding Multiple Brackets2.3.5Factorising into One Bracket2.3.6Factorising Quadratics of the Form x^2 + bx + c2.3.7Factorising Quadratics - AC Method2.3.8Factorising Quadratics - Middle Term Method2.3.9Factorising the Difference of Two Squares2.3.10Completing the Square
2.4

Algebraic Fractions

2.4.1Simplifying Algebraic Fractions by Cancelling2.4.2Simplifying Algebraic Fractions by Factorising2.4.3Operations with Algebraic Fractions
2.5

Rearranging

2.5.1Changing the Subject2.5.2Changing the Subject (Repeated Subject)
2.6

Functions

2.6.1Finding the Input of a Function2.6.2Finding Composite Functions2.6.3Inverse Functions
2.7

Solving Linear Equations

2.7.1Solving Linear Equations (One Step)2.7.2Solving Linear Equations (Multi-Step)2.7.3Linear Equations (Unknown Both Sides)2.7.4Solving Equations: (Unknown Denominator)2.7.5Forming & Solving Linear Equations
2.8

Inequalities

2.8.1Representing Inequalities on a Number Line2.8.2Solving Linear Inequalities2.8.3Linear Inequalities with Unknowns on Both Sides2.8.4Graphs of Linear Inequalities2.8.5Solving Quadratic Inequalities
2.9

Solving Quadratic Equations

2.9.1Solving Quadratics by Factorising2.9.2Solving Quadratics Using the Formula
2.10

Simultaneous Equations

2.10.1Solving Simultaneous Equations (Elimination)2.10.2Solving Simultaneous Linear & Quadratic Equations2.10.3Solving Simultaneous Equations Graphically
2.11

Iteration

2.11.1Iterative Formulae2.11.2Solutions Using Iteration
2.12

Sequences

2.12.1Term-to-Term Rules2.12.2Position-to-Term Rules2.12.3Finding the Nth Term (Linear Sequences)2.12.4Finding the Nth Term (Quadratic Sequences)2.12.5Quadratic Sequence Formulae2.12.6Simple Geometric Progressions
2.13

Coordinates

2.13.1Coordinate Reading & Plotting2.13.2Midpoints of Line Segments
2.14

Linear Graphs

2.14.1Plotting Straight Line Graphs2.14.2Equations of Straight Lines2.14.3Finding Line Equation from Point & Gradient2.14.4Finding Line Equation from Two Points2.14.5Parallel Line Equations2.14.6Perpendicular Line Equations
2.15

Quadratic Graphs

2.15.1Plotting Quadratics2.15.2Interpreting & Sketching Quadratics2.15.3Finding Turning Points by Completing the Square
2.16

More Non-Linear Graphs

2.16.1Cubic Graphs2.16.2Trigonometric & Exponential Functions2.16.3Equations of Circles2.16.4Equations of Tangents
2.17

Graph Transformations

2.17.1Translating Graphs2.17.2Reflecting Graphs2.17.3Transforming Graphs
2.18

Real-Life Graphs

2.18.1Distance-Time Graphs2.18.2Acceleration from Velocity-Time Graphs2.18.3Distance from Velocity-Time Graphs
3

Ratio

3.1

Unit Conversions

3.1.1Converting Units of Time3.1.2Metric Conversions3.1.3Converting Units of Area & Volume
3.2

Compound Units

3.2.1Calculating Speed3.2.2Best Buy & Unit Pricing3.2.3Density, Mass & Volume3.2.4Pressure, Force & Area
3.3

Ratio Fundamentals

3.3.1Ratio Notation & Simplification3.3.2Ratios as Fractions & Linear Equations3.3.3Sharing in a Ratio3.3.4Proportion Problems
3.4

Advanced Ratio & Scale

3.4.1Map Scales & Real Distances
3.5

Proportion

3.5.1Currency Conversions3.5.2Direct & Inverse Proportion3.5.3Graphs of Direct & Inverse Proportion
3.6

Algebraic Proportion

3.6.1Direct Proportion Equations3.6.2Inverse Proportion Equations3.6.3Direct Proportion with Powers & Roots3.6.4Inverse Proportion with Powers & Roots
3.7

Percentages & Growth

3.7.1Compound Interest & Depreciation
3.8

Manipulating Ratios

3.8.1Differences in Ratios3.8.2Fractions in Ratios
3.9

Percentage & Interest

3.9.1Simple Interest
4

Geometry

4.1

Area & Perimeter

4.1.1Perimeter of Simple Shapes4.1.2Perimeter of Compound Shapes4.1.3Area of Triangles4.1.4Area of Rectangles & Parallelograms4.1.5Area of Trapezia4.1.6Area of Compound Shapes4.1.7Area of Shapes Involving Triangles
4.2

Properties of Shapes

4.2.1Properties of Lines4.2.2Symmetry4.2.3Rotational Symmetry4.2.4Shortest Distance from a Point to a Line4.2.5Properties of 3D Shapes4.2.6Cross Section
4.3

Angles

4.3.1Classification of Angles4.3.2Angles on a Line & Around a Point4.3.3Combining Angle Facts in Triangles4.3.4Angles on Parallel Lines4.3.5Interior & Exterior Angles in Polygons
4.4

Trigonometry

4.4.1Sides in Right-Angled Triangles4.4.2Angles in Right-Angled Triangles4.4.3Trigonometric Graphs4.4.4Exact Values of Trigonometric Ratios4.4.5Trigonometry in 3D Shapes4.4.6Using the Sine Rule4.4.7Using the Cosine Rule4.4.8Area of a Triangle Using Sine
4.5

Similarity & Congruence

4.5.1Congruent Shapes4.5.2Length Scale Factors in Similar Shapes4.5.3Area & Volume Scale Factors
4.6

Circle Geometry

4.6.1Identifying Parts of a Circle4.6.2Calculating Circumference4.6.3Calculating Area of a Circle4.6.4Calculating Arc Lengths4.6.5Calculating Area of Sectors
4.7

Circle Theorems

4.7.1Standard Circle Theorems4.7.2Cyclic Quadrilaterals & Segment Theorems4.7.3Chord & Tangent Properties
4.8

Pythagoras

4.8.1Pythagoras' Theorem in 2D4.8.2Pythagoras’ Theorem in 3D
4.9

3D Representations

4.9.1Nets of 3D Shapes
4.10

Surface Area

4.10.1Surface Area of Cubes & Cuboids4.10.2Surface Area of Prisms4.10.3Surface Area of Cylinders4.10.4Surface Area of Spheres & Cones4.10.5Surface Area of Composite Solids
4.11

Volume

4.11.1Volume of Prisms4.11.2Volume of Cylinders4.11.3Volume of Frustums & Composite Solids
4.12

Vectors

4.12.1Vectors & Column Notation4.12.2Operations with Column Vectors
4.13

Transformations

4.13.1Translation Using Vectors4.13.2Rotations4.13.3Reflections4.13.4Enlargements & Positive Scale Factors4.13.5Enlargements & Negative Scale Factors4.13.6Combining Transformations
4.14

Scale Drawings

4.14.1Scale Diagrams
4.15

Bearings

4.15.1Bearings
4.16

Geometry Proof

4.16.1Geometric Proofs Using Vectors
4.17

Introduction to Geometry

4.17.1Geometry Basics4.17.2Geometry Basics 24.17.3Bisectors & Perpendiculars4.17.4Locus
4.18

Triangles & Quadrilaterals

4.18.1Quadrilaterals
4.19

Transformations

4.19.1Invariant Points
4.20

Measurements & Units

4.20.1Time
4.21

3D Shapes

4.21.1Types of 3D Shape4.21.2Plans & Elevations4.21.3Volume of a Cuboid4.21.4Volume of a Sphere4.21.5Volume of a Hemisphere4.21.6Volume of a Cone4.21.7Volume of a Pyramid
5

Probability

5.1

Single Events

5.1.1Writing Probabilities as Fractions5.1.2Mutually Exclusive Events5.1.3Expected Results
5.2

Combined Events

5.2.1Venn Diagrams5.2.2Using Set Notation5.2.3Using Set Notation 25.2.4Independent & Mutually Exclusive Events5.2.5Frequency Trees5.2.6Tree Diagrams (Independent)5.2.7Tree Diagrams (Dependent)
5.3

Experimental Probability

5.3.1Fairness and Bias5.3.2Relative Frequency & Convergence
5.4

Counting Outcomes

5.4.1Product Rule for Counting
5.5

Conditional Probability

5.5.1Conditional Probability: Venn Diagrams5.5.2Conditional Probability: Tree Diagrams5.5.3Conditional Probability from Tables
5.6

Representing Probabilities

5.6.1Probability Scale5.6.2Sample Space Diagrams
5.7

Probability Calculations

5.7.1Complex Tree Diagrams
6

Statistics

6.1

Averages & Range

6.1.1Range6.1.2Median6.1.3Mode6.1.4Mean6.1.5Averages from Frequency Tables6.1.6Interpreting Two-Way Tables & Frequency Tables
6.2

Bar Charts

6.2.1Bar Charts
6.3

Scatter Graphs

6.3.1Scatter Graphs6.3.2Lines of Best Fit
6.4

Histograms

6.4.1Interpreting Histograms6.4.2Drawing Histograms
6.5

Frequency Polygons

6.5.1Frequency Polygons
6.6

Cumulative Frequency

6.6.1Drawing Cumulative Frequency6.6.2Interpreting Cumulative Frequency
6.7

Box Plots

6.7.1Interpreting Box Plots & Comparing Distributions
6.8

Collecting & Presenting Data

6.8.1Types of Data6.8.2Designing Questionnaires6.8.3Capture-Recapture
6.9

Charts & Diagrams

6.9.1Pie Charts6.9.2Stem & Leaf Diagrams6.9.3Back to Back Stem & Leaf Diagrams6.9.4Pictograms6.9.5Time Series Graphs6.9.6Correlation & Causation

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Tree Diagrams (Dependent)

Relative Frequency & Convergence