2.3.3 Further Venn Diagrams

Further Venn Diagrams

The Venn diagrams used here are no more complicated than those in the first Venn Diagrams revision note, however
- questions may use set notation as well, or alongside contextual questions
- there may be more detailed use of conditional probability
- mutually exclusive, three events and other unusual Venn diagram setups may be involved

2-3-3-cie-fig0-special-venn

Interpreting questions in terms of AND ( $\cup$ ), OR ( $\cap$ ), complement ( ‘ ) and
“given that” ( | )
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 $\frac{" double shading "}{" single shading "}$

plx9zFR~_3-2-2-fig1-set-notation-examples

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

events $A$ and $C$ are mutually exclusive

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

Find

(i)

P (A | B)

(ii)

P (A' | C')

(iii)

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

2-3-3-ciie-fig2-we-solution_part1

2-3-3-ciie-fig2-we-solution_part2

Although very versatile, Venn diagrams may not always be the best option
- use a tree diagram if one event follows another

e.g. two beads drawn from a bag without replacement

- use a two-way table for showing how many items/people are in associated categories
  e.g. A school year group split by gender selecting which sport they wish to play