Probability with Venn diagrams

Unit 14 Probability with Venn diagrams

Key unit competence: Determine the probability of an event from real life situations using Venn diagrams.

Key ideas

• The sample space $S$ is the rectangle; events are circles. With equally likely outcomes, $P(A)=\frac{n(A)}{n(S)}$.
• $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ (addition rule). For mutually exclusive events $A\cap B=\varnothing$, so $P(A\cup B)=P(A)+P(B)$.
• $P(A^{\prime })=1-P(A)$; $P(\text{only }A)=P(A)-P(A\cap B)$; $P(\text{neither})=1-P(A\cup B)$.
• Independent events: $P(A\cap B)=P(A)\times P(B)$.
• To solve: fill the Venn diagram with numbers (or probabilities) starting from the intersection, then read off the required region.

Formulas to know

Common misconceptions

• "$P(A\cup B)=P(A)+P(B)$ always." Only when $A$ and $B$ are mutually exclusive.
• "Mutually exclusive means independent." No: if $P(A),P(B)>0$, mutually exclusive events are dependent (one happening prevents the other).

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. In a class of 45, 25 play football, 20 play basketball and 8 play both. Draw a Venn diagram and find P(football only), P(basketball or football), P(neither).

   Show answer

   Football only 17, basketball only 12, both 8, neither 8. $P=\frac{17}{45}$, $\frac{37}{45}$, $\frac{8}{45}$.

   Common mistake: Putting 25 in the "football only" region.

2. A fair coin is tossed and a die is thrown. Use the multiplication rule to find P(head and an even number), and say why it applies.

   Show answer

   The two experiments are independent: $\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$.

   Common mistake: Adding the probabilities.

3. $P(A)=0.45$, $P(B)=0.35$, $P(A\cup B)=0.6$. Find $P(A\cap B)$ and $P(A^{\prime }\cap B^{\prime })$.

   Show answer

   $P(A\cap B)=0.2$; $P(A^{\prime }\cap B^{\prime })=0.4$.

   Common mistake: Taking $P(A^{\prime }\cap B^{\prime })=1-P(A\cap B)$.

4. Three events $A$, $B$, $C$ in a Venn diagram of 50 outcomes have regions: $A$ only 10, $B$ only 8, $C$ only 6, $A\cap B$ only 5, $A\cap C$ only 4, $B\cap C$ only 3, all three 2, outside 12. Find $P(A)$ and $P(\text{exactly two})$.

   Show answer

   $n(A)=10+5+4+2=21$, $P(A)=\frac{21}{50}$. Exactly two: $5+4+3=12$, $P=\frac{12}{50}=\frac{6}{25}$.

   Common mistake: Including the "all three" region in "exactly two".

5. Show that $A$: "even" and $B$: "greater than 4" on one fair die are independent.

   Show answer

   $P(A)=\frac{1}{2}$, $P(B)=\frac{1}{3}$, $A\cap B=\{6\}$: $P=\frac{1}{6}=\frac{1}{2}\times \frac{1}{3}$. Independent.

   Common mistake: Concluding dependence because they share the outcome 6.

6. In a group of 80, 50 speak English, 40 French and 15 neither. A person is chosen. Find P(speaks exactly one language).

   Show answer

   At least one: 65; both $=50+40-65=25$; exactly one $=65-25=40$; $P=\frac{40}{80}=\frac{1}{2}$.

   Common mistake: Forgetting to subtract those who speak both.

7. A survey of 100 households: 60 have electricity, 45 have piped water, 30 have both. Find the probability that a household has electricity but not water.

   Show answer

   $\frac{60-30}{100}=0.3$.

   Common mistake: Using $\frac{60}{100}$.

8. Are events with $P(A)=0.3$, $P(B)=0.4$, $P(A\cup B)=0.7$ mutually exclusive? Independent?

   Show answer

   $P(A\cap B)=0.3+0.4-0.7=0$: mutually exclusive. Not independent, since $0\ne 0.12$.

   Common mistake: Thinking mutually exclusive events must be independent.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. $A$ and $B$ are mutually exclusive, $P(A)=0.2$, $P(B)=0.45$. Find $P(A\cup B)$.	a. 0.5
   2. $P(A)=0.5$, $P(B)=0.4$ and $P(A\cup B)=0.7$. Find $P(\text{exactly one of }A,B)$.	b. 0.65
   3. $A$ and $B$ are independent with $P(A)=0.6$ and $P(B^{\prime })=0.7$. Find $P(A\cup B)$.	c. 0.72
   4. $P(A)=0.6$, $P(B)=0.5$, $P(A\cap B)=0.3$. Find $P(A\cup B)$.	d. 0.8

   Show answer

   1 → b (0.65) · 2 → a (0.5) · 3 → c (0.72) · 4 → d (0.8)

   • Add when mutually exclusive.
   • Step 1. $P(A\cap B)=0.5+0.4-0.7=0.2$.
     Step 2. Exactly one $=P(A\cup B)-P(A\cap B)=0.7-0.2=0.5$.
   • Step 1. $P(B)=0.3$ and $P(A\cap B)=0.6\times 0.3=0.18$.
     Step 2. $P(A\cup B)=0.6+0.3-0.18=0.72$.
   • $0.6+0.5-0.3$.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   independent  ·  complementary  ·  mutually exclusive  ·  $\frac{1}{6}$
   1. $P(A)=0.4$, $P(B)=0.3$, and $P(A\cap B)=0.12$. $A$ and $B$ are .
   2. Same die: $P(A\cap B)$ is .
   3. If $P(A\cup B)=P(A)+P(B)$, then $A$ and $B$ are .
   Show answer

   1: independent · 2: $\frac{1}{6}$ · 3: mutually exclusive

   • $0.4\times 0.3=0.12$.
   • Only 3.
   • The intersection has probability 0.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. $A$ and $B$ are independent, $P(A)=0.3$ and $P(A\cap B)=0.12$. Find $P(B)$.	a. 0.4
   2. Same cards: find $P(A\cup B)$.	b. $\frac{7}{15}$
   3. Rolling a die: $A=\{1,2,3\}$, $B=\{3,4\}$. Find $P(A\cup B)$.	c. $\frac{3}{5}$
   4. A number is chosen from 1 to 30. Find $P(\text{multiple of 3 or of 5})$.	d. $\frac{2}{3}$

   Show answer

   1 → a (0.4) · 2 → c ($\frac{3}{5}$) · 3 → d ($\frac{2}{3}$) · 4 → b ($\frac{7}{15}$)

   • Step 1. Independent: $P(A\cap B)=P(A)P(B)$.
     Step 2. $P(B)=\frac{0.12}{0.3}=0.4$.
   • $\frac{10+4-2}{20}=\frac{12}{20}$.
   • $\{1,2,3,4\}$: $\frac{4}{6}$.
   • Step 1. Multiples of 3: 10; of 5: 6; of both (15, 30): 2.
     Step 2. $\frac{10+6-2}{30}=\frac{14}{30}=\frac{7}{15}$.

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $\frac{1}{10}$  ·  0.5  ·  0.15  ·  0.9
   1. Independent events $A$, $B$ have $P(A)=0.3$, $P(B)=0.5$. Find $P(A\cap B)$. →
   2. $P(A)=0.7$, $P(A\cap B)=0.2$. Find $P(A\text{ only})$. →
   3. A card is chosen from 1 to 20. $A$: even; $B$: multiple of 5. Find $P(A\cap B)$. →
   Show answer

   1: 0.15 · 2: 0.5 · 3: $\frac{1}{10}$

   • $0.3\times 0.5$.
   • $0.7-0.2$.
   • 10 and 20: $\frac{2}{20}$.

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