Venn Diagrams with Conditional Probability (Cambridge International A-Level Mathematics)

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

  • 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

How do I solve conditional probability problems using Venn diagrams?

• 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 $\mathrm{P} ( A \vert B ) = \frac{ \mathrm{P} ( A \cap B )}{ \mathrm{P} ( B )}$ shade B first, then shade $A\cap B$

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

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

events $A$ and $C$ are mutually exclusive

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

Find

(i) $\mathrm{P} ( A \vert B )$

(ii) $\mathrm{P} ( A ' \vert C ' )$

(iii) $\mathrm{P} ( C \vert ( A \cup B ) ' )$

Answer:

Exam Hint

• 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