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

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

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

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)