Combined Probability

What do we mean by combined probabilities?

In general this means there is more than one event to bear in mind when considering probabilities
- these events may be independent or mutually exclusive
- they may involve an event that follows on from a previous event
  - e.g. Rolling a dice, followed by flipping a coin

How do I work with and calculate combined probabilities?

In your head, try to rephrase each question as an AND and/or OR probability statement
- e.g. The probability of rolling a 6 followed by flipping heads would be "the probability of rolling a 6 AND the probability of flipping heads"
- In general,
  - AND means multiply ( $\times$ ) and is used for independent events
  - OR mean add ( $+$ ) and is used for mutually exclusive events
The fact that all probabilities sum to 1 is often used in combined probability questions
- In particular when we are interested in an event "happening" or "not happening"
  - e.g. $P (rolling a 6) = \frac{1}{6}$ so $P (NOT rolling a 6) = 1 - \frac{1}{6} = \frac{5}{6}$
Tree diagrams can be useful for calculating combined probabilities
- especially when there is more than one event but you are only concerned with two outcomes from each
  - e.g. The probability of being stopped at one set of traffic lights and also being stopped at a second set of lights
- however unless a question specifically tells you to, you don't have to draw a diagram
- for many questions it is quicker simply to consider the possible options and apply the AND and OR rules without drawing a diagram

Worked example

A box contains 3 blue counters and 8 red counters.
A counter is taken at random and its colour noted.
The counter is put back into the box.
A second counter is then taken at random, and its colour noted.

Work out the probability that

both counters are red,

ii)

the two counters are different colours.

This is an "AND" question: 1st counter red AND 2nd counter red.

$\begin{array}{rcl} P (both red) & = & P (R) \times P (R) \\ = & \frac{8}{11} \times \frac{8}{11} \\ = & \frac{64}{121} \end{array}$

$\begin{array}{rcl} P (both red) & = & \frac{64}{121} \end{array}$

ii)

This is an "AND" and "OR" question: [ 1st red AND 2nd green ] OR [ 1st green AND 2nd red ].

$\begin{array}{rcl} P (one of each) & = & [P (R) \times P (G)] + [P (G) + P (R)] \\ = & \frac{8}{11} \times \frac{3}{11} + \frac{3}{11} \times \frac{8}{11} \\ = & \frac{24}{121} + \frac{24}{121} \\ = & \frac{48}{121} \end{array}$

$\begin{array}{rcl} P (one of each) & = & \frac{48}{121} \end{array}$

In the second line of working in part (ii) we are multiplying the same two fractions together twice, just 'the other way round'.

It would be possible to write that instead as $2 \times (\frac{8}{11} \times \frac{3}{11}) = 2 \times \frac{24}{121} = \frac{48}{121} .$

That sort of 'shortcut' is often possible in questions like this.

Worked example

The probability of winning a fairground game is known to be 26%.

If the game is played 4 times find the probability that there is at least one win.
Write down an assumption you have made.

At least one win is the opposite to no losses so use the fact that the sum of all probabilities is 1.

$P (at least 1 win) = 1 - P (0 wins)$

Use the same fact to work out the probability of a loss.

$\begin{array}{rcl} P (lose) & = & 1 - P (win) \\ = & 1 - 0.26 \\ = & 0.74 \end{array}$

The probability of four losses is an "AND" statement; lose AND lose AND lose AND lose.
Assuming the probability of losing doesn't change, this is $0.74 \times 0.74 \times 0.74 \times 0.74 = {(0.74)}^{4}$ .

$\begin{array}{rcl} P (at least 1 win) & = & 1 - P (0 wins) \\ = & 1 - P (4 loses) \\ = & 1 - {(0.74)}^{4} \\ = & 0.700 134 . . . \end{array}$

P(at least 1 win) = 0.7001 (4 d.p.)

The assumption that we made was that the probability of winning/losing doesn't change between games.
Mathematically this is described as each game being independent.
I.e., the outcome of one game does not affect the outcome of the next (or any other) game.

It has been assumed that the outcome of each game is independent.

1. Number

Number Toolkit

Mathematical Operations
Negative Numbers
Money Calculations
Number Operations
Related Calculations
Counting Principles

Prime Factors, HCF & LCM

Types of Number
Prime Factor Decomposition
HCF & LCM

Powers, Roots & Standard Form

Powers, Roots & Indices
Standard Form

Fractions

Basic Fractions
Operations with Fractions

Percentages

Basic Percentages
Working with Percentages

Simple & Compound Interest, Growth & Decay

Interest & Depreciation
Exponential Growth & Decay

Fractions, Decimals & Percentages

Converting between FDP
Converting between FDP

Rounding, Estimation & Bounds

Rounding & Estimation
Bounds

Surds

Simplifying Surds
Rationalising Denominators

Using a Calculator

Using a Calculator

2. Algebra

Algebra Toolkit

Algebraic Notation & Vocabulary
Algebra Basics

Algebraic Roots & Indices

Algebraic Roots & Indices

Expanding Brackets

Expanding Single Brackets
Expanding Multiple Brackets

Factorising

Factorising
Factorising Quadratics
Quadratics Factorising Methods

Completing the Square

Completing the Square

Rearranging Formulae

Rearranging Formulae

Algebraic Proof

Algebraic Proof

Linear Equations

Solving Linear Equations

Solving Quadratic Equations

Solving Quadratic Equations
Quadratic Equation Methods

Simultaneous Equationsr

Simultaneous Equations

Iteration

Iteration

Forming & Solving Equations

Forming Equations
Equations & Problem Solving

Functions

Functions Toolkit
Composite & Inverse Functions

Coordinate Geometrys

Coordinates
Coordinate Geometry

Linear Graphs y = mx + c

Straight Line Graphs (y = mx + c)
Parallel & Perpendicular Lines

Graphs of Functions

Types of Graphs
Graphical Solutions
Trig Graphs

Equation of a Circle

Equation of a Circle
Equation of a Tangents

Estimating Gradients & Areas under Graphs

Finding Gradients of Tangents
Finding Areas under Graphs

Real-Life Graphs

Distance-Time & Speed-Time Graphs
Conversion Graphs
Rates of Change of Graphs

Solving Inequalities

Solving Linear Inequalities
Conversion Graphs
Solving Quadratic Inequalities

Graphing Inequalities

Graphing Inequalities

Transformations of Graphs

Reflections of Graphs

Sequences

Introduction to Sequences
Types of Sequences
Linear Sequences
Quadratic Sequences

3. Ratio, Proportion & Rates of Change

Ratio Toolkit

Simple Ratio
Working with Proportion

Ratio Problem Solving

Ratios & FDP
Multiple Ratios

Direct & Inverse Proportions

Direct & Inverse Proportion

Standard & Compound Units

Time
Unit Conversions
Compound Measures

Exchange Rates & Best Buys

Exchange Rates & Best Buys

4. Geometry & Measures

Geometry Toolkit

Symmetry
2D & 3D Shapes
Plans & Elevations

Angles in Polygons & Parallel Lines

Basic Angle Properties
Angles in Polygons
Angles in Parallel Lines

Bearings, Scale Drawing, Constructions & Loci

Bearings
Scale & Maps
Constructing Triangles
Constructions & Loci

Circle Theorems

Angles at Centre & Semicircles
Chords & Tangents
Cyclic Quadrilaterals
Segment Theorems
Circle Theorem Proofs

Area & Perimeter

Area & Perimeter
Problem Solving with Areas

Circles, Arcs & Sectors

Area & Circumference of Circles
Arcs & Sectors

Volume & Surface Area

Volume
Surface Area

Congruence, Similarity & Geometrical Proof

Congruence
Similarity
Geometrical Proof

Area & Volume of Similar Shapes

Similar Area & Volumes

Right-Angled Triangles – Pythagoras & Trigonometry

Pythagoras Theorem
Right-Angled Trigonometry
Exact Trig Values

Sine, Cosine Rule & Area of Triangles

Sine & Cosine Rules
Area of a Triangle
Applications of Trigonomet

3D Pythagoras & Trigonometry

3D Pythagoras & Trigonometry

Vectors

Introduction to Vectors
Working with Vectors

Transformations

Translations
Reflections
Rotations
Enlargements
Combination of Transformations

5. Probability

Probability Toolkit

Basic Probability
Relative & Expected Frequency

Simple Probability Diagrams

Two Way Tables
Frequency Trees
Set Notation & Venn Diagrams

Tree Diagrams

Tree Diagrams

Combined & Conditional Probability

Combined Probability
Conditional Probability
Combined Conditional Probabilities

6. Statistics

Statistics Toolkit

Mean, Median & Mode
Averages from Tables
Range & Quartiles
Comparing Distributions
Population & Sampling

Statistical Diagrams

Bar Charts & Pictograms
Pie Charts
Time Series Graphs
Working with Statistical Diagrams

Histograms

Histograms

Cumulative Frequency & Box Plots

Cumulative Frequency Diagrams
Box Plots

Scatter Graphs & Correlation

Scatter Graphs

Top Rated by Parents/Students Nationwide