1. 1
   A bag contains 12 balls: 5 red, 4 blue and 3 green. A ball is selected at random from the bag.

   (a)

   Find the probability that the ball is not red.

   [1 mark]

   • A$\dfrac{7}{12}$
   • B$\dfrac{5}{12}$
   • C$\dfrac13$
   • D$\dfrac14$

   (b)

   A ball is drawn, its colour noted, and then replaced; this is repeated 60 times. Find the expected number of red balls drawn.

   [1 mark]

   • A$20$
   • B$25$
   • C$15$
   • D$5$

   (c)

   Two balls are selected at random from the bag, without replacement. Find the probability that both are red.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In a group of 50 students, 28 study French, 22 study Spanish, and 10 study neither language.

   (a)

   Find the probability that a randomly chosen student studies both French and Spanish.

   [1 mark]

   • A$0.56$
   • B$0.44$
   • C$0.2$
   • D$0.8$

   (b)

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

   [1 mark]

   • A$\dfrac{5}{14}$
   • B$0.56$
   • C$0.2$
   • D$\dfrac{5}{11}$

   (c)

   Determine, showing your reasoning, whether the events 'studies French' and 'studies Spanish' are independent.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The times, in minutes, taken by 9 students to complete a puzzle are: 12, 15, 9, 21, 14, 18, 11, 16, 13.

   (a)

   Find the mean and the median of these 9 times.

   [3 marks]

   (b)

   A tenth student's time of 45 minutes is added to the data set. State, with a reason, whether the mean or the median is a more appropriate measure of central tendency for this new set of 10 times, and find the value of the more appropriate measure.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A calculator may be used in this question. A factory produces light bulbs, of which 4\% are defective, independently of each other. A random sample of 25 bulbs is taken.

   (a)

   State a suitable distribution for the random variable $X$, the number of defective bulbs in the sample, including its parameters. Find $P(X=2)$ and $P(X\le1)$, and state the expected number of defective bulbs in the sample.

   [6 marks]

   (b)

   The mass of a bulb is normally distributed with mean 55 g and standard deviation 3 g. Find the probability that a randomly selected bulb has a mass between 50 g and 60 g. Find also the mass, in grams, that is exceeded by 10\% of bulbs, correct to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The discrete random variable $X$ has probability distribution given by $P(X=x) = \dfrac{x+1}{10}$, for $x\in\{0,1,2,3\}$.

   (a)

   Find $P(X=2)$.

   [1 mark]

   • A$0.3$
   • B$0.4$
   • C$0.2$
   • D$0.1$

   (b)

   Find $E(X)$.

   [1 mark]

   • A$1.5$
   • B$2.0$
   • C$2.5$
   • D$1.0$

   (c)

   Find $P(X\ge2)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The heights of a species of plant are normally distributed with mean $\mu$ cm and standard deviation $\sigma$ cm. It is known that 20\% of plants are taller than 85 cm, and 10\% of plants are shorter than 60 cm.

   (a)

   Find the $z$-value corresponding to the top 20\% of the distribution (i.e. the value $z_1$ such that $P(Z>z_1)=0.2$), correct to 3 significant figures.

   [1 mark]

   • A$-0.842$
   • B$1.282$
   • C$0.842$
   • D$0.253$

   (b)

   Find the $z$-value corresponding to the bottom 10\% of the distribution (i.e. the value $z_2$ such that $P(Z<z_2)=0.1$), correct to 3 significant figures.

   [1 mark]

   • A$1.28$
   • B$-0.842$
   • C$-1.645$
   • D$-1.28$

   (c)

   Using your answers to (a) and (b), form two equations in $\mu$ and $\sigma$, and hence find the value of $\sigma$, correct to 3 significant figures.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A researcher records, for 8 households, the number of occupants $x$ and the weekly grocery spend $y$ (in AED hundreds): $(2, 4.5)$, $(3, 5.6)$, $(4, 7.0)$, $(2, 4.0)$, $(5, 8.2)$, $(3, 5.9)$, $(6, 9.5)$, $(4, 6.8)$.

   (a)

   A calculator may be used. Find the equation of the regression line of $y$ on $x$, giving the coefficients correct to 3 significant figures, and interpret the value of the gradient in context.

   [3 marks]

   (b)

   Find the equation of the regression line of $x$ on $y$, correct to 3 significant figures, and hence estimate the number of occupants in a household with a weekly grocery spend of 750 AED, giving your answer to the nearest whole number.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A school has 900 students: 500 in the Lower School and 400 in the Upper School. Of the Lower School students, 300 study Art; of the Upper School students, 120 study Art. A student is selected at random from the whole school.

   (a)

   Let $L$ be the event that the student is in the Lower School, and $A$ the event that the student studies Art. Find $P(A)$, $P(A|L)$ and $P(L|A)$, each correct to 3 significant figures. Hence determine, showing your reasoning, whether the events $L$ and $A$ are independent.

   [6 marks]

   (b)

   The school wants a stratified sample of 45 students, proportional to year group (Lower School and Upper School). Find the number of students to be sampled from each year group. The school also wants a stratified sample of 45 students, this time proportional to whether or not a student studies Art. Find the number of Art students and the number of non-Art students to be sampled.

   [6 marks]

   Total for question 8: 12 marks

End of questions