1. 1
   Events $A$ and $B$ are such that $P(A)=0.4$, $P(B)=0.5$ and $P(A\cap B)=0.12$.

   (a)

   Find $P(A\mid B)$.

   [1 mark]

   • A$0.3$
   • B$0.24$
   • C$0.2$
   • D$0.78$

   (b)

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

   [1 mark]

   • A$0.28$
   • B$0.6$
   • C$0.56$
   • D$0.44$

   (c)

   Determine whether $A$ and $B$ are independent. Justify your answer.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In a large school, 60% of students study Spanish. 18% of all students study both Spanish and music, and 12% of all students study music but not Spanish. A student is chosen at random. Let $S$ be the event that the student studies Spanish and $M$ the event that the student studies music.

   (a)

   Find the probability that the student studies music, given that the student studies Spanish.

   [1 mark]

   • A$0.18$
   • B$0.6$
   • C$0.108$
   • D$0.3$

   (b)

   Find $P(M\mid S')$.

   [1 mark]

   • A$0.3$
   • B$0.12$
   • C$0.2$
   • D$0.7$

   (c)

   Show that the events $S$ and $M$ are independent.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   On any working day, the probability that it rains in the morning is 0.35. If it rains in the morning, the probability that Kofi is late for work is 0.6. If it does not rain in the morning, the probability that Kofi is late for work is 0.2. Let $R$ be the event that it rains in the morning and $L$ the event that Kofi is late.

   (a)

   Find $P(L)$.

   [3 marks]

   (b)

   On a particular day Kofi is late for work. Find the probability that it rained that morning, and hence explain whether the events $R$ and $L$ are independent.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In a survey of 200 students, 80 play tennis and 120 travel to school by bus. Let $T$ be the event that a randomly chosen student plays tennis and $B$ the event that the student travels to school by bus. Let $n$ be the number of students who both play tennis and travel to school by bus.

   (a)

   (i) Show that $P(T\cup B)=\frac{200-n}{200}$.
   (ii) Find the value of $n$ for which $T$ and $B$ would be independent, and the value of $P(T\cup B)$ in this case.
   (iii) Write down the value of $n$ for which $T$ and $B$ would be mutually exclusive, and explain why $T$ and $B$ could not then be independent.

   [6 marks]

   (b)

   In fact $n=40$.
   (i) Find $P(T\mid B)$ and $P(T\mid B')$.
   (ii) Hence state, with a reason, whether $T$ and $B$ are independent, and interpret your answer in context.
   (iii) Find the probability that a student who does not play tennis travels to school by bus.

   [6 marks]

   Total for question 4: 12 marks

End of questions