Sum and product lawsEdexcel International A Level Maths: Subtopic test

1. 1
   Events $A$ and $B$ are such that $P(A)=0.4$, $P(B)=0.5$ and $P(A\cap B)=0.15$.

   (a)

   Find $P(A\cup B)$.

   [1 mark]

   • A$0.90$
   • B$0.75$
   • C$0.20$
   • D$0.65$

   (b)

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

   [1 mark]

   • A$A$ and $B$ are independent, because $P(A\cap B)\neq0$
   • B$A$ and $B$ are mutually exclusive, because $P(A\cup B)<1$
   • C$A$ and $B$ are not independent, because $P(A)\times P(B)=0.2\neq P(A\cap B)$
   • D$A$ and $B$ are independent, because $P(A\cap B)<P(A)+P(B)$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A bag contains 5 red discs and 3 blue discs. Two discs are taken from the bag at random, one after the other, without replacement.

   (a)

   Find the probability that both discs are red.

   [1 mark]

   • A$\frac{25}{64}$
   • B$\frac{5}{16}$
   • C$\frac{20}{49}$
   • D$\frac{5}{14}$

   (b)

   Find the probability that the two discs are different colours.

   [1 mark]

   • A$\frac{15}{28}$
   • B$\frac{15}{56}$
   • C$\frac{15}{32}$
   • D$\frac{13}{28}$

   (c)

   Find the probability that at least one of the two discs is red.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A factory has two machines. Machine $X$ makes 60% of the components and machine $Y$ makes the other 40%. Each component from $X$ is defective with probability 0.05, and each component from $Y$ is defective with probability 0.02, independently of all others.

   (a)

   Find the probability that a component chosen at random from the factory's output is defective.

   [3 marks]

   (b)

   Three components from machine $X$ are tested one after another. Find the probability that at least one of them is defective.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A survey of 60 sixth-form students found that 35 study mathematics ($M$), 28 study physics ($P$) and 15 study both. The remaining students study neither subject. A student is chosen at random.

   (a)

   (i) Find the probability that the student studies at least one of the two subjects.
   (ii) Show that $M$ and $P$ are not independent.
   (iii) Two students are chosen at random without replacement. Find the probability that neither studies mathematics or physics.

   [6 marks]

   (b)

   (i) Two students are chosen at random without replacement. Find the probability that exactly one of them studies mathematics.
   (ii) Three students are chosen at random without replacement. Find the probability that at least one of them studies neither subject.

   [6 marks]

   Total for question 4: 12 marks

End of questions