Conditional probability and independenceEdexcel International A Level Maths: Subtopic test

1. 1
   A bag contains 5 red counters and 3 blue counters. Two counters are drawn at random, one after the other, without replacement.

   (a)

   Find the probability that both counters are red.

   [1 mark]

   • A$\frac{25}{64}$
   • B$\frac{25}{56}$
   • C$\frac{5}{14}$
   • D$\frac{3}{28}$

   (b)

   Given that the first counter is blue, find the probability that the second counter is red.

   [1 mark]

   • A$\frac57$
   • B$\frac58$
   • C$\frac47$
   • D$\frac27$

   (c)

   Find the probability that the two counters are of different colours.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Events $A$ and $B$ are such that $P(A)=0.5$, $P(B)=0.4$ and $P(A\cap B)=0.2$.

   (a)

   Find $P(B\mid A)$.

   [1 mark]

   • A$0.5$
   • B$0.4$
   • C$0.2$
   • D$0.1$

   (b)

   Which statement about $A$ and $B$ is correct?

   [1 mark]

   • A$A$ and $B$ are mutually exclusive, since $P(A\cap B)=0.2$.
   • B$A$ and $B$ are not independent, since $P(A)+P(B)\neq1$.
   • C$A$ and $B$ are not independent, since $P(A\cup B)\neq P(A)+P(B)$.
   • D$A$ and $B$ are independent, since $P(A)\times P(B)=0.2=P(A\cap B)$.

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Events $A$ and $B$ are independent, with $P(A)=0.3$ and $P(A\cup B)=0.58$.

   (a)

   Find $P(B)$.

   [3 marks]

   (b)

   Find the probability that (i) neither event occurs, (ii) exactly one of the events occurs.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A screening test is used for a disease that affects 2% of a population. The test gives a positive result for 95% of people who have the disease and for 6% of people who do not have the disease. A person is chosen at random from the population. Let $D$ be the event that the person has the disease and $T$ the event that the test result is positive.

   (a)

   (i) Find $P(D\cap T)$.
   (ii) Find $P(T)$.
   (iii) Given that a person's test result is positive, find the probability that they have the disease.

   [6 marks]

   (b)

   (i) Find the probability that a person whose test result is negative has the disease.
   (ii) A newspaper claims that a positive result means that a person almost certainly has the disease. Use your answers to (a)(iii) and (b)(i) to evaluate this claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions