Venn Diagrams with Conditional Probability (Edexcel IAL Maths)

Further Venn Diagrams

Spec point — 3.4

Venn diagrams with conditional probability

How do I find conditional probabilities from a Venn diagram?

• Interpreting questions in terms of OR ($\cup$), AND ($\cap$),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 $\mathrm{P} ( A \vert B ) = \frac{ \mathrm{P} ( A \cap B )}{ \mathrm{P} ( B )}$ shade B first, then shade $A\cap B$

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

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

events $A$ and $C$ are mutually exclusive

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

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

(b) Find

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

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

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

Answer:

Exam Hint

• 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