A disease has a 5% prevalence in the population. You need to choose between two tests.

1) If the disease is present, the first test will give a positive result with probability 0.90. If the disease is not present, it will give a positive result with probability 0.05.

2) If the disease is present, the second test will give a positive result with probability 0.98. If the disease is not present, it will give a positive result with probability 0.10.

## A disease has a 5% prevalence in the population

**Question 1 (9 Marks)**

A disease has a 5% prevalence in the population. You need to choose between two tests.

1) If the disease is present, the first test will give a positive result with probability 0.90. If the disease is not present, it will give a positive result with probability 0.05.

2) If the disease is present, the second test will give a positive result with probability 0.98. If the disease is not present, it will give a positive result with probability 0.10.

a) For each of the tests, what is the probability that someone has the disease given a positive test result?

b) The disease is highly contagious, and so those with the disease need to be quarantined. Which test would you recommend? Explain your reasoning.

### Question 2 (9 Marks)

Poker is a game played with a standard deck of 52 cards.[1] The aim of a hand of poker is to finish with the strongest five card combination. In straight draw poker, each player is dealt five cards. Players are then given a single opportunity to discard as many cards as they wish and have them replaced with new cards off the top of the deck. In the questions that follow, only your hand is being dealt from the deck of cards (i.e. you do not need to worry about cards dealt to other players).

On the initial deal:

1) How many five-card hands can be dealt from a deck of 52 cards?

2) What is the probability that a five-card hand contains four aces?

What to keep:

3) You currently have two pairs. If you discard your fifth card, what is the probability that you will complete a full house (three of a kind plus a pair)?

4) From the initial deal you have the Four of Spades, Five of Spades, Six of Spades, Seven of Hearts and Jack of Spades. In deciding what to discard, you can either discard the Seven of Hearts and try for a flush (five cards from the same suit, in any order) or discard the Jack of Spades and try for a straight (five cards in order, in any combination of suits). Which hand are you more likely to complete?

Section 2 – Discrete Probability Distributions

#### Question 3 (7 Marks)

You wish to form an investment portfolio, and have three assets to choose from, a high-risk asset A, and medium-risk asset B, and a zero-risk asset C.

The returns per $1,000 invested for each of these assets in each of three possible economic conditions is give in the table below:

1) What is the expected return and variance of returns for each asset?

2) Derive the general formula for the variance of the sum of three random variables.

3) What is the expected return and variance of returns for a portfolio that invests $300 in A, $400 in B, and $300 in C?

**Question 4 (5 Marks)**

A cloth manufacturer measures defects per metre of cloth, and estimates that on average, there is one flaw per five metres of cloth. What is the probability that:

a) a 1-metre length of cloth has two or more flaws?

b) a 10-metre length of cloth has one or more flaws?

c) a 50-metre roll will have between five and 15 (inclusive) flaws?

Question 5 (5 Marks)

A government tax office auditor is to select and audit six tax returns from a batch of 100. If two or more of the selected returns contain errors, the whole batch will be audited.

a) What is the probability that the whole batch will be audited if the true number of errors in the batch is:

i. 25?

ii. 30?

iii. 5?

iv. 10?

b) What factors would you consider in choosing the size of the sample that you check (assuming you do not have to stick to checking 6)? How would it vary with the true error rate?

Section 3 – Continuous Probability Distributions

##### Question 6 (5 Marks)

A trucking company was worked out that on average its trucks drive 100,000 kilometres a year, with a standard deviation of 20,000 kilometres. The distances driven are normally distributed.

a) What proportion of the trucks will drive between 80,000 and 120,000 kilometres a year?

b) What percentage of trucks will drive less than 60,000 or more than 140,000 kilometres per year?

c) What minimum distance will be driven by at least 80% of the trucks?

Section 4 – Sampling Distributions

Question 7 (5 Marks)

In the first quarter of 2014, the rental cost of a three-bedroom house in a regional town was $300 with a standard deviation of $30. Assume that the rental costs are normally distributed. If you select a random sample of ten rental properties, what is the probability that the sample will have a mean rental cost of:

a) Less than $290?

b) Between $285 and $290?

c) Greater than $308?

[1] If you are not familiar with playing cards, see here: https://en.wikipedia.org/wiki/Standard_52-card_deck . If you are not familiar with poker, see here: https://en.wikipedia.org/wiki/Poker . You need to know the composition of a deck, but the questions should not require knowledge of the game.