Answer all questions precisely.

The Chevalier de Mere, a seventeenth century gambler, thought that it paid to bet evens on the event: A : you will get one or more sixes when four unbiased dice are thrown In other words, the Chevalier thought that P(A) > 1/2. (i) (a) Show that the probability of this event occurring is, in fact, 0.5177. (b) The experiment of throwing four unbiased dice is performed 10 times and results in the event A occurring 8 times. (1) Write down an equation which must be satisfied by pz , a lower 95% confidence limit for p = P(A). (2) Verify that pz = 0.493 satisfies this equation. (3) Comment on this value of p relative to the true value of p as specified in part (i)(a). [6] (ii) The experiment of throwing four unbiased dice is performed 1,000 times and results in the event A occurring Y times. (a) Write down a large sample expression for PL , a lower 95% confidence limit for p = P(A). (b) Determine how large Y' would have to be for pz to be greater than 1/2. (c) Using the true value of p as specified in part (i), calculate the probability that py will, in fact, be greater than 12. [8] (iii) For the situation where the experiment is performed 10,000 times: (a) repeat part (ii)(b) (b) repeat part (ii)(c) and comment briefly on any difference between your two answers. [6] [Total 20]12 A company has, historically, paid a steadily increasing dividend from one year to the next. The company has had a difficult year and has generated very little profit and has had slight cash flow problems. The directors are considering borrowing cash in order to maintain the dividend for the current year in order to maintain the company's share price. Discuss the implications for the share price of borrowing to meet dividend expectations. [5] 13 In theory, companies exist to maximise shareholder wealth. In practice, the relationship between shareholders and the directors whom they appoint to manage their companies appears to suggest that the directors have other motives. Explain why it is difficult for shareholders to be assured that directors are consistently working to serve their interests. [5] 14 A firm of actuaries has previously operated as a partnership. The firm is considering raising a large loan in order to buy office premises instead of renting them. One of the partners has suggested incorporating the firm as a limited company in order to avoid personal risk for the partners at minimal cost. Explain whether incorporation as a limited company is likely to achieve the objectives suggested by the partner. [5]