Conditional probability and tree and Venn diagramsEdexcel A-Level Maths: Subtopic test

1. 1
   Of 50 sixth-form students, 28 study Mathematics, 20 study Physics and 8 study both subjects. One of the 50 students is chosen at random.

   (a)

   Find the probability that the student studies Physics, given that they study Mathematics.

   [1 mark]

   • A$\frac{2}{5}$
   • B$\frac{2}{7}$
   • C$\frac{4}{25}$
   • D$\frac{14}{25}$

   (b)

   Find the probability that the student studies at least one of the two subjects.

   [1 mark]

   • A$\frac{4}{5}$
   • B$\frac{24}{25}$
   • C$\frac{1}{5}$
   • D$\frac{4}{25}$

   (c)

   Given that the student studies exactly one of the two subjects, find the probability that it is Mathematics.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   On any day, the probability that it rains is $0.3$. When it rains, the probability that a certain bus is late is $0.4$. When it does not rain, the probability that the bus is late is $0.1$.

   (a)

   Find the probability that the bus is late on a given day.

   [1 mark]

   • A$0.12$
   • B$0.5$
   • C$0.19$
   • D$0.07$

   (b)

   Given that the bus is late, find the probability that it rained.

   [1 mark]

   • A$0.4$
   • B$0.12$
   • C$0.19$
   • D$\frac{12}{19}$

   (c)

   Given that the bus is not late, find the probability that it did not rain.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Events $A$ and $B$ are such that $P(A)=0.6$, $P(B)=0.5$ and $P(A\mid B)=0.4$.

   (a)

   Find $P(A\cup B)$.

   [3 marks]

   (b)

   Find $P(B\mid A')$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A disease affects $2\%$ of a population. A test gives a positive result for $95\%$ of people who have the disease and for $4\%$ of people who do not have the disease.

   (a)

   (i) Show that the probability that a randomly chosen person tests positive is $0.0582$. (ii) Find the probability that a person who tests positive has the disease. (iii) Explain why most people who test positive do not have the disease, even though the test is quite accurate.

   [6 marks]

   (b)

   Everyone who tests positive is tested a second time. For any one person the two results are independent. Find the probability that a person has the disease, given that both tests are positive.

   [6 marks]

   Total for question 4: 12 marks

End of questions