Further Venn Diagrams

What do you mean by further Venn diagrams?

The Venn diagrams used here are no more complicated than those in the first Venn Diagrams revision note, however
- Venn diagrams, and the wording/terminology in questions, may now involve the use of set notation including the symbols $\cup$ (union),
  $\cap$ (intersection), ‘ (complement) and | (“given that”)
- Conditional probability problems can be solved using Venn diagrams

How do I solve conditional probability problems using Venn diagrams?

Interpreting questions in terms of AND ( $\cup$ ), OR ( $\cap$ ), complement ( ‘ ) and
“given that” ( | )
Conditional probability may now be involved too
Use mini-Venn diagrams to sketch and shade the regions you are dealing with – use different colours if available or different styles of shading if not
- Shading can help you see the answer

since $P (A | B) = \frac{P (A \cap B)}{P (B)}$ shade B first, then shade $A \cap B$

the answer will then be $\frac{" double shading "}{" all shading "}$

3-2-2-fig1-set-notation-examples

Worked example

Three events, $A, B$ and $C$ are such that

events $A$ and $C$ are mutually exclusive

$P (A \cap B) = 0.2 P (B \cap C) = 0.3 P ((A \cup B \cup C)') = 0.1 P (B) = 0.7 P (A') = 0.75 .$

(a)

Draw a complete Venn diagram to show the probabilities connecting the three events.

(b)

Find

(i)

P (A | B)

(ii)

P (A' | C')

(iii)

P (C | (A \cup B)')

(a)

Draw a complete Venn diagram to show the probabilities connecting the three events.

3-2-2-fig2-we-solution-part-1

(b)

Find

(i)

P (A | B)

(ii)

P (A' | C')

(iii)

P (C | (A \cup B)')

3-2-2-fig2-we-solution-part-2-luggage-removed

Examiner Tip

Always draw the box in a Venn diagram; it represents all possible outcomes of the experiment so is a crucial part of the diagram, the bubbles merely represent the events we are particularly interested in
You may be able to answer some questions by applying formulae or you may prefer to use shaded mini-Venn diagrams; complicated questions tend to be easier with mini-Venn diagrams

Further Venn Diagrams (AQA A Level Maths: Statistics)

Revision Note

What do you mean by further Venn diagrams?

How do I solve conditional probability problems using Venn diagrams?

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1. Statistical Sampling

1.1 Sampling & Data Collection

1.1.1 Sampling & Data Collection

2. Data Presentation & Interpretation

2.1 Statistical Measures

2.1.1 Basic Statistical Measures

2.1.2 Frequency Tables

2.1.3 Standard Deviation & Variance

2.2 Data Presentation

2.2.1 Data Presentation

2.2.2 Box Plots & Cumulative Frequency

2.2.3 Histograms

2.3 Working with Data

2.3.1 Outliers & Cleaning Data

2.3.2 Intrepreting Data

2.3.3 Skewness

2.4 Correlation & Regression

2.4.1 Correlation & Regression

2.5 Further Correlation & Regression (A Level only)

2.5.1 PMCC & Non-linear Regression

2.5.2 Hypothesis Testing for Correlation

3. Probability

3.1 Basic Probability

3.1.1 Calculating Probabilities & Events

3.1.2 Venn Diagrams

3.1.3 Tree Diagrams

3.2 Further Probability (A Level only)

3.2.1 Set Notation & Conditional Probability

3.2.2 Further Venn Diagrams

3.2.3 Further Tree Diagrams

3.2.4 Probability Formulae

4. Statistical Distributions

4.1 Probability Distributions

4.1.1 Discrete Probability Distributions

4.2 Binomial Distribution

4.2.1 The Binomial Distribution

4.2.2 Calculating Binomial Probabilities

4.3 Normal Distribution (A Level only)

4.3.1 The Normal Distribution

4.3.2 Normal Distribution - Calculations

4.3.3 Standard Normal Distribution

4.4 Choosing Distributions (A Level only)

4.4.1 Modelling with Distributions

4.4.2 Normal Approximation of Binomial

5. Hypothesis Testing

5.1 Hypothesis Testing

5.1.1 Hypothesis Testing

5.2 Hypothesis Testing (Binomial Distribution)

5.2.1 Binomial Hypothesis Testing

5.3 Hypothesis Testing (Normal Distribution) (A Level only)

5.3.1 Sample Mean Distribution

5.3.2 Normal Hypothesis Testing

6. Large Data Set

6.1 Large Data Set

6.1 Large Data Set

Author: Paul