1. 1
   In a school, a student is chosen at random. Let $A$ be the event that the student plays a musical instrument and $B$ the event that the student is in a sports team. It is known that $P(A) = 0.5$, $P(B) = 0.3$ and $P(A \cup B) = 0.65$.

   (a)

   Find $P(A \cap B)$.

   [1 mark]

   • A$0.8$
   • B$0.35$
   • C$0.15$
   • D$0.195$

   (b)

   Find $P(A \mid B)$.

   [1 mark]

   • A$0.5$
   • B$0.3$
   • C$0.15$
   • D$0.075$

   (c)

   Determine whether $A$ and $B$ are (i) mutually exclusive, (ii) independent. Justify your answers.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A drawer contains 7 white socks and 5 black socks. Two socks are taken from the drawer at random, one after the other, without replacement.

   (a)

   Find the probability that both socks are black.

   [1 mark]

   • A$\frac{25}{144}$
   • B$\frac{5}{36}$
   • C$\frac{10}{33}$
   • D$\frac{5}{33}$

   (b)

   Find the probability that the two socks are different colours.

   [1 mark]

   • A$\frac{35}{132}$
   • B$\frac{35}{66}$
   • C$\frac{35}{72}$
   • D$\frac{28}{33}$

   (c)

   Given that the two socks are the same colour, find the probability that they are both black.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A gym has 120 members. Of these, 70 use the swimming pool, 55 use the weights room and 20 use neither. A member is chosen at random. Let $S$ be the event that the member uses the swimming pool and $W$ the event that the member uses the weights room.

   (a)

   Find the number of members who use both the swimming pool and the weights room.

   [3 marks]

   (b)

   (i) Find $P(W \mid S)$.
   (ii) Determine whether the events $S$ and $W$ are independent. Justify your answer.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Each school day, Aisha either cycles to school or takes the bus. The probability that she cycles is 0.6. If she cycles, the probability that she is late is 0.1. If she takes the bus, the probability that she is late is 0.25. Let $L$ be the event that Aisha is late on a given day.

   (a)

   (i) Find $P(L)$.
   (ii) Given that Aisha is late, find the probability that she took the bus.
   (iii) Show that the events 'Aisha is late' and 'Aisha takes the bus' are not independent.

   [6 marks]

   (b)

   Whether Aisha is late on one day is independent of whether she is late on any other day. Consider Monday and Tuesday of one week.
   (i) Find the probability that she is late on exactly one of the two days.
   (ii) Find the probability that she is late on at least one of the two days.
   (iii) Given that she is late on at least one of the two days, find the probability that she is late on both days.

   [6 marks]

   Total for question 4: 12 marks

End of questions