4.6 Combined events and conditional probabilityIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   In a class of 30 students, 18 study Spanish ($S$), 12 study French ($F$) and 5 study both languages. A student is chosen at random from the class.

   (a)

   Find $P(S\cap F)$.

   [1 mark]

   • A$\frac{5}{18}$
   • B$\frac16$
   • C$\frac{5}{12}$
   • D$\frac56$

   (b)

   Find the probability that the student studies Spanish or French or both.

   [1 mark]

   • A$1$
   • B$\frac35$
   • C$\frac25$
   • D$\frac56$

   (c)

   Find the probability that a student studies French, given that the student studies Spanish.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A box contains 4 red and 6 yellow pencils. Two pencils are taken at random from the box, one after the other, without replacement.

   (a)

   Find the probability that both pencils are red.

   [1 mark]

   • A$\frac{4}{25}$
   • B$\frac25$
   • C$\frac{2}{15}$
   • D$\frac{11}{15}$

   (b)

   Find the probability that the second pencil is yellow, given that the first pencil is red.

   [1 mark]

   • A$\frac23$
   • B$\frac35$
   • C$\frac13$
   • D$\frac59$

   (c)

   Find the probability that the two pencils are of different colours.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sports club has 120 members. Of these, 70 play tennis ($T$), 50 play badminton ($B$) and 20 play both sports. A member is chosen at random.

   (a)

   Find the number of members who play neither tennis nor badminton.

   [3 marks]

   (b)

   (i) Find $P(T\mid B)$.
   (ii) Determine whether playing tennis and playing badminton are independent events. Justify your answer.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Sara travels to school by bus. Let $L$ be the event that the bus is late and $R$ the event that it rains. On a school morning, $P(R)=0.2$, $P(L\mid R)=0.6$ and $P(L\mid R')=0.1$.

   (a)

   (i) Find the probability that it rains and the bus is late.
   (ii) Find the probability that the bus is late.
   (iii) Given that the bus is late, find the probability that it is raining.

   [6 marks]

   (b)

   (i) Given that $P(L)=0.2$, show that $L$ and $R$ are not independent.
   (ii) Assume that whether it rains on one morning is independent of whether it rains on the next. Find the probability that it rains on exactly one of two consecutive mornings.
   (iii) In a term of 40 school days, find the expected number of mornings on which it rains and the bus is late.

   [6 marks]

   Total for question 4: 12 marks

End of questions