1. Sometimes the street light outside my house works properly, but sometimes it is faulty. If it works properly one day, there is a 95% chance that it will work properly the next day as well. But if it is faulty one day, there is a 98% chance it will be faulty the next day as well.
On average, on how many days is it faulty each year? You should assume that a year is exactly 365 days long. Show your working and explain your answer.
2. Let A be the matrix
where α, β ∈ R.
a) Write down the characteristic polynomial, XA(t), of A. Show your working.
b) Write down the eigenvalues of A.
c) Fbr each of the eigenvalues you identified in (c), write down a corresponding eigen-vector. Explain your answer and show your working.
d) Write down an invertible matrix P and a diagonal matrix D such that .0 = P-1 AP is diagonal, and show by direct calculation that this equation is valid for your choice of P and D.
e) Suppose β = 1/a. For what values of a is A invertible? Explain your answer and show your working.
3. On average, I receive 25 emails each day, of which 60% are 'span'. What is the probability that I will receive exactly 15 'span' emails tomorrow? Explain your answer and show your working.
4. Until recently, all the residents of a small village have done their shopping at the town's only shop, Hobson's. However, a major discount supermarket chain, Cheapo, recently opened a new shop nearby. Mr Hobson is worried about the effect on his business, and has been collecting data. Analysis of this data suggests the following probabilities for customers switching where they prefer to go shopping each week:
|
|
From Hobson's
|
Cheapo
|
|
TO Hobson's
Cheapo
|
0.75
0.25
|
0.40
0.60
|
a) At the end of the first four weeks, what proportion of the town's population were still shopping at Hobson's? Explain your answer and show your working.
b) In the long-term, what proportion of the town's population will do their shopping at Cheapo? Explain your answer and show your working.
5. Approximately 64.1 million currently live in the UK, of whom 5.4 million are thought to suffer from the medical condition, asthma. A new test for asthma has recently been proposed, and clinical trials show that
• in patients with asthma, the test correctly returns positive 68% of the time;
• in patients who do not have asthma, the test correctly returns negative 82% of the time.
Suppose a patient goes for testing. What is the probability that the patient has asthma,
a) if the test returns positive?
b) if the test returns negative? Show your working and explain your answers.
6. a) Find the general solution to the equation 6x + 14y e 4 mod 5. Show your working and explain your answer.
b) Find the smallest positive integer x satisfying the equations
x ≡ 5 mod 7
x ≡ 7 mod 11
Show your working and explain your answer.
7. An insurance company has developed some new software to help it decide whether its customers' insurance claims are likely to be fraudulent. Having tested the software on 1000 recent claims, they find that
• there were 15 cases of fraud, and the software correctly predicted 10 of these;
• of the remaining 985 (honest) claims, the software incorrectly predicted that 15 were fraudulent.
a) What is the probability of the software correctly identifying whether a claim is or is not fraudulent? That is, what is the value of P(fraudulent I predicted-fraudulent) + P(honest I predicted-honest)? Show your working and explain your answer.
b) Suppose the company decides not to use the software, and simply assumes that all claims are honest. What would the new probability be that the company correctly identifies whether or not a claim is fraudulent? Show your working and explain your answer.
c) In your opinion, which approach should the company adopt (should they use the software, or simply assume that all claims are honest)? Explain your reasoning.
8. A friend of yours sits in a room rolling a fair 6-sided die Each he time he throws it, he looks to see what value he gets. If the value is greater than 4 he rolls the die again, and otherwise he stops. Let X be the number of times he throws the die before stopping. For example, if he rolls 5 -) 6 -> 1 -> stop, the corresponding value of X will be 3.
a) Write down the probability mass function px. Show your working and explain your answer.
b) Calculate the expectation value E(X). Show your working and explain your answer.
c) On average, is the value of (X2 + 1) odd or even? Show your working and explain your answer.
9. Suppose you go to Hillsborough Interchange to catch the 52 bus (which run every 10 minutes on average). What is the probability that you will wait for at least fifteen minutes before the bus arrives, and then three buses will arrive at once (i.e. the first and third buses will arrive in the same 5-minute period)?
Explain your answer, and show your working.
10. A gambler offers you the following deal. You have to keep tossing a fair coin until you get a heads, at which point you stop and collect your winnings: if it happens after ti throws, the gambler will give you 2" pence. However, in order to play the game you first have to agree to pay an entrance fee, but the amount you need to pay is not known to either of you. Instead, you each ask a stranger (before the game starts) to secretly write a random amount of money on a piece of paper and seal it in an envelope. Once the game has finished, and having paid you your winnings, the gambler will open both envelopes and you then have to pay him the average of the two amounts that the strangers have written (rounded up to the nearest penny).
From a purely financial point of view, is it worth playing the game? Show your working, and explain your answer.