Probability · Ordinary Level
Venn Diagrams
Ordinary Level · Class 8 · Tap NEXT to begin
Section 1 of 3
Venn Diagrams — Two Sets
Reading a Venn diagram
The box is everything ($U$). The overlap is both. To find a probability, count the region you want over the total in the box.
$6$ people play hurling only, $4$ play both, $2$ play soccer only, $1$ plays neither.
Worked example — five probabilities
A person is chosen at random. Find (i) $P(\text{hurling})$; (ii) $P(\text{hurling only})$; (iii) $P(\text{both})$; (iv) $P(\text{neither})$; (v) $P(\text{only one sport})$.
Total $= 6+4+2+1 = 13$
(i) $P(H) = \tfrac{6+4}{13} = \tfrac{10}{13}$
(ii) $\tfrac{6}{13}$; (iii) $\tfrac{4}{13}$; (iv) $\tfrac{1}{13}$
(v) only one $= 6+2 = \tfrac{8}{13}$
$\tfrac{10}{13},\tfrac{6}{13},\tfrac{4}{13},\tfrac{1}{13},\tfrac{8}{13}$
Section 2 of 3
A Venn Diagram of Probabilities
On a Venn diagram: $A$ only $= 0.3$, both $= 0.4$, $B$ only $= 0.1$, and $x$ outside.
Worked example — find x and the probabilities
Find $x$, then $P(A)$, $P(B)$ and $P(A \cap B)$.
$0.3+0.4+0.1+x = 1 \Rightarrow 0.8 + x = 1 \Rightarrow x = 0.2$
$P(A) = 0.3+0.4 = 0.7$; $P(B) = 0.4+0.1 = 0.5$
$P(A \cap B) = $ the middle $= 0.4$
$x=0.2;\ P(A)=0.7,\ P(B)=0.5,\ P(A\cap B)=0.4$
Section 3 of 3
Three Sets
Maths, English and French. Regions: Maths only $5$, Maths&English $3$, English only $2$, all three $1$, English&French $4$, French only $2$.
Worked example — three probabilities
Find $P(\text{studies Maths})$, $P(\text{Maths and French})$ and $P(\text{only one subject})$.
Total $= 5+3+2+1+4+2 = 17$
$P(\text{Maths}) = \tfrac{5+3+1}{17} = \tfrac{9}{17}$
$P(\text{Maths and French}) = \tfrac{1}{17}$ (the centre)
only one $= 5+2+2 = \tfrac{9}{17}$
$\tfrac{9}{17},\ \tfrac{1}{17},\ \tfrac{9}{17}$
You try
On a three-set Venn ($A, B, C$): $A$ only $5$, $A\cap B$ $1$, $B$ only $4$, $A\cap C$ $2$, centre $3$, $B\cap C$ $9$, $C$ only $6$, outside $8$. Find (i) $P(A)$; (ii) $P(A \text{ only})$; (iii) $P(\text{all three})$.
Total = add all regions including the outside.
Total $= 5+1+4+2+3+9+6+8 = 38$
(i) $P(A) = \tfrac{5+1+2+3}{38} = \tfrac{11}{38}$
(ii) $\tfrac{5}{38}$; (iii) $\tfrac{3}{38}$
$\tfrac{11}{38},\ \tfrac{5}{38},\ \tfrac{3}{38}$
That’s Class 8.
Venn diagrams for two and three sets, and Venns of probabilities. Class 9: the ‘or’ rule and mutually exclusive events.