Part II

1. A group of 6 friends are exchanging Christmas gifts. In how many ways can they exchange gifts so that no one receives their own gift? (15%)

2. Given that the lifetime of a particular device follows an exponential distribution with a mean of \( x \) hours, find the expected lifetime of the device, given that it has lasted at least \( Y \) hours. (15%)

3. A rare medical condition affects only 1 in 50,000 people in a specific population. A sophisticated diagnostic test for this condition has the following characteristics:●If a person has the disease, the test is \( 99.5\% \) accurate in detecting it (sensitivity).●If a person does not have the disease, the test has a \( 1.5\% \) false positive rate.●Two independent tests are performed on the same patient.What is the probability (%) that a person actually has the disease, given that they tested positive on the first test and also tested positive on the second test? (20%)(Note: You are asked to answer in percentages, counted to two decimal places.)

(1) Solve it by the backward induction approach. (8%)

(2) It turns out that most responders reject the selfish proposal. Are they being irrational? (8%)

(1) What is the relationship between average costs and marginal costs of a firm in the perfect competitive market? (3%)

(2) Why are the profits in a perfect competitive market usually zero? (3%)

(3) What are the individual firms' supply curves and the industry supply curve in a perfect competitive market? (3%)

(4) In the short run, a firm in the perfect competitive market should always leave the market as long as profits are negative. Do you agree the above statement? Why? (8%)

(a) If there is an increase in taxes on business firms in a large open economy, it causes the current account of the large open economy to rise.

(b) In a small open economy, if the government has a budget deficit due to a cut in current taxes, the nation's saving, investment and current account may not change.

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