1. 1
   A box contains 9 green pens and 6 orange pens. A pen is selected at random from the box.

   (a)

   Find $P(\text{green})$.

   [1 mark]

   • A$\dfrac35$
   • B$\dfrac25$
   • C$\dfrac96$
   • D$\dfrac69$

   (b)

   A pen is selected at random and then replaced; this is repeated 60 times. Find the expected number of times a green pen is selected.

   [1 mark]

   • A$24$
   • B$36$
   • C$45$
   • D$20$

   (c)

   Two pens are selected at random from the box, one after another, without replacement. Find the probability that both are green.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A biased coin is such that the probability of obtaining heads on any toss is $0.35$, independently of other tosses. The coin is tossed 10 times. Let $X$ be the number of heads obtained.

   (a)

   Find $E(X)$.

   [1 mark]

   • A$6.5$
   • B$0.35$
   • C$3.5$
   • D$3.15$

   (b)

   Find $\text{Var}(X)$.

   [1 mark]

   • A$1.508$
   • B$1.225$
   • C$3.5$
   • D$2.275$

   (c)

   Find $P(X=4)$, giving your answer correct to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The times, in minutes, taken by 9 volunteers to complete a jigsaw puzzle are: 14, 22, 17, 31, 19, 25, 17, 28, 21.

   (a)

   Using technology, find the mean and the standard deviation of these times, each correct to 3 significant figures.

   [3 marks]

   (b)

   Write down the median and the interquartile range of these times, and determine whether the value 31 is an outlier, using the definition that an outlier lies more than 1.5 times the IQR from the nearest quartile.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The heights of a species of plant are normally distributed with mean $\mu=42$ cm and standard deviation $\sigma=6$ cm.

   (a)

   (i) Find the probability that a randomly chosen plant has height greater than 50 cm. [3]
   (ii) Find the probability that a randomly chosen plant has height between 36 cm and 48 cm. [3]

   [6 marks]

   (b)

   A different species of plant has heights normally distributed with mean $\mu$ cm and standard deviation $8$ cm. It is known that $15\%$ of these plants have height greater than $60$ cm. Find the value of $\mu$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   In a class of 30 students, 18 study French, 14 study Spanish, and 6 study neither language.

   (a)

   Find $P(\text{studies both French and Spanish})$.

   [1 mark]

   • A$\dfrac15$
   • B$\dfrac{4}{15}$
   • C$\dfrac13$
   • D$\dfrac{1}{15}$

   (b)

   Find $P(\text{studies French only})$.

   [1 mark]

   • A$\dfrac35$
   • B$\dfrac{4}{15}$
   • C$\dfrac15$
   • D$\dfrac13$

   (c)

   A student is selected at random from the class. Given that the student studies French, find the probability that they also study Spanish.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The discrete random variable $X$ has probability distribution $P(X=x)=\dfrac{5-x}{10}$ for $x\in\{1,2,3,4\}$.

   (a)

   Find $P(X=2)$.

   [1 mark]

   • A$\dfrac{3}{10}$
   • B$\dfrac15$
   • C$\dfrac25$
   • D$\dfrac1{10}$

   (b)

   Find $E(X)$.

   [1 mark]

   • A$2.5$
   • B$1$
   • C$2$
   • D$3$

   (c)

   Hence find $\text{Var}(X)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A teacher records, for 6 students, the number of hours spent on a revision app in the week before a test, $x$, and their test score out of 50, $y$: $x$: 2, 3, 4, 5, 6, 7; $y$: 28, 31, 33, 38, 41, 44.

   (a)

   Using technology, find the equation of the regression line of $y$ on $x$, giving the coefficients correct to 3 significant figures.

   [3 marks]

   (b)

   Using technology, find the equation of the regression line of $x$ on $y$, and hence estimate, to the nearest hour, the number of hours spent on the app in a week where a student scored 36.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A factory has two machines, A and B, that produce components. Machine A produces 60\% of the components and Machine B produces the remaining 40\%. Of the components from Machine A, 3\% are defective; of those from Machine B, 5\% are defective. A component is selected at random from the factory's output.

   (a)

   (i) Find the probability that the component is defective. [3]
   (ii) Find the probability that the component was produced by Machine A and is defective. [1]
   (iii) Determine whether the events "produced by Machine A" and "defective" are independent, justifying your answer using probabilities. [2]

   [6 marks]

   (b)

   Given that a randomly selected component is defective, find the probability that it was produced by Machine B.

   [6 marks]

   Total for question 8: 12 marks

End of questions