1. 1
   A box contains 5 red counters and 3 blue counters. A counter is taken at random from the box, its colour noted, and then replaced before a second counter is taken at random.

   (a)

   What is the probability that both counters are red?

   [1 mark]

   • A$\frac{5}{8}$
   • B$\frac{25}{64}$
   • C$\frac{10}{16}$
   • D$\frac{15}{64}$

   (b)

   What is the probability that the first counter is red and the second counter is blue?

   [1 mark]

   • A$\frac{9}{64}$
   • B$\frac{25}{64}$
   • C$\frac{3}{8}$
   • D$\frac{15}{64}$

   (c)

   Which statement correctly explains why the two draws are independent events?

   [1 mark]

   • ABecause the total number of counters in the box stays the same
   • BBecause red and blue counters are equally likely to be drawn
   • CBecause the counter is replaced, the composition of the box for the second draw is unchanged by the outcome of the first draw
   • DBecause red and blue are mutually exclusive outcomes

   Total for question 1: 3 marks

2. 2
   A drawer contains 4 black socks and 6 white socks. Two socks are taken at random from the drawer, one after the other, without replacement.

   (a)

   Calculate the probability that both socks are white.

   [2 marks]

   (b)

   Explain why the probability that the second sock is white is not simply $\frac{6}{10}$, and state the two possible values this probability could take.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In a survey of 120 students, each student was asked whether they play a musical instrument and whether they play a sport. 70 students play a sport. Of these 70 students, 28 also play a musical instrument. Of the 50 students who do not play a sport, 15 play a musical instrument.

   (a)

   Find the probability that a student chosen at random plays a musical instrument, given that they play a sport.

   [3 marks]

   (b)

   A student is chosen at random from the survey. Use the addition rule to find the probability that the student plays a sport or plays a musical instrument (or both).

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A factory tests every component for two faults, A and B. From past data, the probability that a component has fault A is 0.1, the probability that it has fault B is 0.05, and the probability that it has both faults is 0.02.

   (a)

   Show that faults A and B are not independent events.

   [4 marks]

   (b)

   Find the conditional probability that a component has fault B, given that it has fault A. Show your method clearly.

   [4 marks]

   (c)

   The factory produces 4000 components per day. A component is rejected if it has fault A or fault B (or both). Using the addition rule, calculate how many components the factory should expect to reject each day.

   [5 marks]

   Total for question 4: 13 marks

End of questions