A souvenir booklet is to be produced by a publishing firm to be sold on the occasion of the final of a major sporting competition. The manager of the firm wants to know how many booklets to print for this time-limited market. If she prints too few, she will miss the opportunity for profit. If she prints too many, she will have a lot of books left over after the competition which will have to be sold off at a loss. Suppose every book sold on the day makes a profit of £5 whilst the publishing firm loses £2 for every one left over. The expected number of people visiting the stadium and associated fan zones is 200,000. From past experience of such publications, on average about 5% of people will buy the booklet, but this varies a lot. Using this information, and previous sales figures, the manager draws up the probability distribution for the match-day demand shown in the table below.

(i) Calculate (in units of 1000 booklets) the expected demand, and the standard deviation of this demand.

(ii) Calculate figures for a third column to the table, equalling the profit (in units of £1,000) for each amount sold if the print run is 3,000 books. What is the return (expected profit) in this case? [Note that (a) even if the demand is greater than 3,000, only 3,000 booklets will be sold; (b) when the demand is less than 3,000, unsold booklets will have a negative impact on profits.]

(iii) By repeating (ii) for different print run sizes, calculate the print run size that will maximise the expected profit. What is the risk (standard deviation) associated with this strategy? Support your answers by calculating expected profit figures for each possible stock level in a table.

(iv) Use your answers to parts (i) to (iii) to produce a detailed report (maximum 200 words) advising the manager of the publishing firm on her strategy.Your report should include all information you think it is appropriate to give to the manager (such as profit, risk, and maximum loss) - do not assume she has access to your answers (i) to (iii).