1. 1
   Thirty raffle tickets are numbered 1 to 30. One ticket is chosen at random.

   (a)

   Find the probability that the number on the ticket is a multiple of 4.

   [1 mark]

   • A$\frac{1}{4}$
   • B$\frac{7}{30}$
   • C$\frac{4}{15}$
   • D$\frac{7}{29}$

   (b)

   Find the probability that the number on the ticket is not a prime number.

   [1 mark]

   • A$\frac{1}{3}$
   • B$\frac{19}{30}$
   • C$\frac{7}{10}$
   • D$\frac{2}{3}$

   (c)

   Find the probability that the number on the ticket contains the digit 2.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A factory tests a random sample of 400 light bulbs from its production line and finds that 14 of them are faulty.

   (a)

   Find the relative frequency of a faulty bulb in the sample.

   [1 mark]

   • A$0.965$
   • B$0.14$
   • C$0.035$
   • D$0.0363$

   (b)

   The factory produces a batch of 6000 bulbs. Use the relative frequency to find the expected number of faulty bulbs in the batch.

   [1 mark]

   • A$210$
   • B$2100$
   • C$14$
   • D$5790$

   (c)

   A second random sample of 600 bulbs from the same production line contains 27 faulty bulbs. Find the best estimate of the probability that a bulb from this line is faulty, and give a reason for your choice.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two fair six-sided dice, one red and one blue, each numbered 1 to 6, are rolled. The random outcome $D$ is the positive difference between the two scores, so that, for example, a red 2 and a blue 5 give $D = 3$, and two equal scores give $D = 0$.

   (a)

   Show that $P(D = 1) = \frac{5}{18}$.

   [3 marks]

   (b)

   Find the probability that $D$ is at least 3. Hence find the expected number of times that $D$ is less than 3 when the two dice are rolled 90 times.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A bag contains red, blue and yellow counters only. A counter is taken from the bag at random. The probability that it is red is 0.28, and the probability that it is blue is twice the probability that it is yellow. There are 36 blue counters in the bag.

   (a)

   (i) Find the probability that the counter is yellow.
   (ii) Write down the probability that the counter is not red.
   (iii) Find the total number of counters in the bag, and the number of red counters.

   [6 marks]

   (b)

   Sam takes a counter at random, records its colour and replaces it. He does this 150 times.
   (i) Find the expected number of times Sam takes a yellow counter.
   (ii) Sam takes a red counter 49 times. He says this proves that there are more than 21 red counters in the bag. Comment on his claim.
   (iii) Instead, $k$ extra red counters are added to the original bag, so that the probability of taking a red counter becomes 0.4. Find $k$.

   [6 marks]

   Total for question 4: 12 marks

End of questions