Conditional Probability

What is conditional probability?

Conditional probability refers to situations where the probability of an event changes or is dependent on other events having already happened
For example, drawing names from a hat, without replacement
- If there are 10 (different) names in a hat to start with
- the first name drawn has the probability of $\frac{1}{10}$ of being a particular name
- the second name drawn has probability $\frac{1}{9}$ of being a particular name
  - or, if this particular name was the first one to be drawn, it would have probability 0 of being drawn second
- The probability has changed depending on what has happened already
Conditional probabilities often occur in the context of Venn diagrams, tree diagrams or two-way tables
- however questions may also be given in words only
- in such cases it may sometimes be easier to understand what is happening by drawing one of these diagrams
- unless a question tells you to though, drawing a diagram is not essential
- for many questions it is quicker simply to consider the possible options without drawing a diagram
Conditional probability questions are often in the form of "given that" questions
- e.g. Find the probability it will rain today given that it rained yesterday
  - It makes sense that whether or not it rained yesterday would affect the probability of whether or not it rains today
  - The phrase "given that" is not always used in conditional probability questions
  - Like AND/OR, you will need to interpret the phrases used in questions
Conditional probabilities are sometimes written using the 'straight bar' notation $P (A | B)$
- That is read as 'the probability of A given B'
- For example $P (passes | no revision)$ would indicate the probability that a student passes his exams, given that he has done no revision
- That probability is likely to be quite different from $P (passes | lots of revision)$ !

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 then set aside and not put back into the box.
A second counter is then taken at random, and its colour noted.

Write down the probability that

the second counter is red, given that the first counter was red

ii)

the second counter is blue, given that the first counter was red

iii)

the second counter is red, given that the first counter was blue

iv)

the second counter is blue, given that the first counter was blue.

If the first counter was red, then only 7 red counters remain in the box.
There are still 3 blue counters, and 10 counters in total.

$\begin{array}{rcl} \frac{7}{10} \end{array}$

ii)

If the first counter was red, then only 7 red counters remain in the box.
There are still 3 blue counters, and 10 counters in total.

$\begin{array}{rcl} \frac{3}{10} \end{array}$

iii)

If the first counter was blue, then only 2 blue counters remain in the box.
There are still 8 red counters, and 10 counters in total.

$\frac{8}{10} (= \frac{4}{5})$

iv)

If the first counter was blue, then only 2 blue counters remain in the box.
There are still 8 red counters, and 10 counters in total.

$\frac{2}{10} (= \frac{1}{5})$

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