Venn diagrams
Probability and statistics
- Crossover
- Non-calculator
- 8 questions
- 29 marks
Questions marked Higher are aimed at the Higher tier.
These are original practice questions, not a past paper. This site is independent and not affiliated with AQA or Pearson.
-
There are $40$ students. $22$ study French and $18$ study Spanish, including $8$ who study both. Complete the four regions of the Venn diagram.
Venn diagram of French and Spanish. The outside number is the number studying neither. 2 marks
-
Find the probability that a student picked at random studies French.
2 marks
-
Find the probability that a student studies both languages.
3 marks
-
Find the probability that a student studies French or Spanish or both.
3 marks
-
Find the probability that a student studies French given that the student studies Spanish.
4 marks
-
Find the probability that a student studies neither language, and the probability that a student studies exactly one of the languages.
4 marks
-
Find the probability that a student does not study French. A student is picked from those who study French. Find the probability that this student also studies Spanish. Higher
5 marks
-
Two different students are picked at random. Find the probability that both study neither language. Explain what the overlap in the diagram represents. Higher
6 marks
Mark scheme
-
Answer. French only $14$, both $8$, Spanish only $10$, neither $8$
- French only $= 22 - 8 = 14$.
- Spanish only $= 18 - 8 = 10$, and neither $= 40 - 32 = 8$.
-
Answer. $\dfrac{11}{20}$
- $\dfrac{22}{40}$
- $\dfrac{11}{20}$
-
Answer. $\dfrac{1}{5}$
- $\dfrac{8}{40}$
- $\dfrac{1}{5}$
-
Answer. $\dfrac{4}{5}$
- The union contains $32$ students.
- $\dfrac{32}{40} = \dfrac{4}{5}$
-
Answer. $\dfrac{4}{9}$
- Restrict to the $18$ Spanish students.
- $\dfrac{8}{18} = \dfrac{4}{9}$
-
Answer. $\dfrac{1}{5}$ and $\dfrac{3}{5}$
- Neither $= \dfrac{8}{40} = \dfrac{1}{5}$.
- Exactly one $= \dfrac{24}{40} = \dfrac{3}{5}$.
-
Answer. $\dfrac{9}{20}$ and $\dfrac{4}{11}$
- Not French $= 10 + 8 = 18$, so $\dfrac{18}{40} = \dfrac{9}{20}$.
- Of the $22$ French students, $8$ also study Spanish.
- $\dfrac{8}{22} = \dfrac{4}{11}$
-
Answer. $\dfrac{7}{195}$. The overlap is the students who study both French and Spanish.
- $\dfrac{8}{40} \times \dfrac{7}{39} = \dfrac{7}{195}$
- The overlap contains students counted in both subjects.