Mutually exclusive and independent eventsAQA A-Level Maths: Subtopic test

1. 1
   Events $A$ and $B$ are mutually exclusive, with $P(A)=0.4$ and $P(B)=0.35$.

   (a)

   Find $P(A\cup B)$.

   [1 mark]

   • A$0.14$
   • B$0.05$
   • C$0.75$
   • D$0.25$

   (b)

   Find the probability that neither $A$ nor $B$ occurs.

   [1 mark]

   • A$0.75$
   • B$0.25$
   • C$0.65$
   • D$0.60$

   (c)

   Determine whether $A$ and $B$ are independent.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Events $A$ and $B$ are independent, with $P(A)=0.6$ and $P(B)=0.5$.

   (a)

   Find $P(A\cap B)$.

   [1 mark]

   • A$0.1$
   • B$0.5$
   • C$1.1$
   • D$0.3$

   (b)

   Find $P(A\cup B)$.

   [1 mark]

   • A$0.8$
   • B$1.1$
   • C$0.3$
   • D$0.2$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A factory has two machines, $X$ and $Y$. An item from $X$ is defective with probability $0.04$ and an item from $Y$ is defective with probability $0.06$, independently of each other. A pair consists of one item from each machine.

   (a)

   Find the probability that exactly one item in a pair is defective.

   [3 marks]

   (b)

   Three pairs are chosen independently. Find the probability that none of the six items is defective.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Events $C$ and $D$ are mutually exclusive, with $P(C)=0.3$ and $P(D)=0.45$. Event $E$ is independent of $C$ and independent of $D$, with $P(E)=0.4$.

   (a)

   (i) Find $P(C\cup D)$.
   (ii) Show that $P(E\cap(C\cup D))=0.3$.
   (iii) Hence find $P(C\cup D\cup E)$.

   [6 marks]

   (b)

   (i) Show that $C$ and $D$ are not independent.
   (ii) Find the probability that none of $C$, $D$ and $E$ occurs.
   (iii) Find the probability that $C$ occurs and $E$ does not.

   [6 marks]

   Total for question 4: 12 marks

End of questions