1. 1
   A screening test is used for a condition that affects 2% of a population. If a person has the condition, the test is positive with probability 0.95. If a person does not have the condition, the test is positive with probability 0.10. A person is chosen at random and tested.

   (a)

   Find the probability that the test is positive.

   [1 mark]

   • A$0.019$
   • B$0.117$
   • C$0.098$
   • D$0.525$

   (b)

   Given that the test is positive, find the probability that the person has the condition, correct to three significant figures.

   [1 mark]

   • A$0.162$
   • B$0.950$
   • C$0.019$
   • D$0.838$

   (c)

   Given that the test is negative, find the probability that the person does not have the condition. Give your answer correct to three significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A factory makes bolts on three machines. Machine A makes 50% of the bolts, machine B makes 30% and machine C makes 20%. The proportions of defective bolts made by A, B and C are 2%, 3% and 5% respectively. A bolt is chosen at random from the day's output.

   (a)

   Find the probability that the bolt is defective.

   [1 mark]

   • A$0.1$
   • B$0.01$
   • C$0.0333$
   • D$0.029$

   (b)

   Given that the bolt is defective, find the probability that it was made by machine C.

   [1 mark]

   • A$0.05$
   • B$\frac{10}{29}$
   • C$0.2$
   • D$0.01$

   (c)

   Given that the bolt is not defective, find the probability that it was made by machine B. Give your answer correct to three significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   On a school morning the probability that it rains is 0.3. If it rains, the probability that Sam arrives late for school is 0.4. If it does not rain, the probability that Sam arrives late is $p$. Over a long period, Sam arrives late on 19% of school mornings.

   (a)

   Find the value of $p$.

   [3 marks]

   (b)

   Sam's teacher notices that Sam arrived late one morning. Find the probability that it rained that morning, and compare it with the probability that it rained on a morning when Sam arrived on time.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An email service sorts incoming messages into three types: 60% are genuine (G), 30% are marketing (M) and 10% are scams (S). The word “free” appears in 5% of genuine emails, 40% of marketing emails and 70% of scam emails. An email containing “free” is flagged.

   (a)

   (i) Show that the probability that a randomly chosen email is flagged is 0.22.
   (ii) Given that an email is flagged, find the probability that it is a scam.
   (iii) Given that an email is flagged, find the probability that it is genuine.

   [6 marks]

   (b)

   To reduce errors, every flagged email is now given a second check. For each type of email, the result of the second check is independent of whether the email contains “free”. The second check gives a warning for 2% of genuine emails, 10% of marketing emails and 90% of scam emails. An email is flagged and then receives a warning. Find the probability that it is a scam, and comment on how effective the second check is.

   [6 marks]

   Total for question 4: 12 marks

End of questions