Aman has n keys on a key ring, one of which opens the door to his apartment. having celebrated a bit too much one evening, he returns home only to find himself unable to distinguish one key from another. resourceful, he works out a fiendishly clever plan: he will choose a key at random and try it. if it fails to open the door, he will discard it and choose at random one of the remaining n−1 keys, and so on. clearly, the probability that he gains entrance with the first key he selects is 1/n. show that the probability the door opens with the third key he tries is also 1/n.

Fanning company is considering the addition of a new product to its cosmetics line. the company has three distinctly different options: a skin cream, a bath oil, or a hair coloring gel. relevant information and budgeted annual income statements for each of the products follow. skin cream bath oil color gel budgeted sales in units (a) 110,000 190,000 70,000 expected sales price (b) $8 $4 $11 variable costs per unit (c) $2 $2 $7 income statements sales revenue (a × b) $880,000 $760,000 $770,000 variable costs (a × c) (220,000) (380,000) (490,000) contribution margin 660,000 380,000 280,000 fixed costs (432,000) (240,000) (76,000) net income $228,000 $140,000 $204,000 required: (a) determine the margin of safety as a percentage for each product. (b) prepare revised income statements for each product, assuming a 20 percent increase in the budgeted sales volume. (c) for each product, determine the percentage change in net income that results from the 20 percent increase in sales. (d) assuming that management is pessimistic and risk averse, which product should the company add to its cosmetics line? (e) assuming that management is optimistic and risk aggressive, which product should the company add to its cosmetics line?

(a) what do you think will happen if the price of non-gm crops continues to rise? why? (b) what will happen if the price of non-gm food drops? why?

Sailcloth & more currently produces boat sails and is considering expanding its operations to include awnings for homes and travel trailers. the company owns land beside its current manufacturing facility that could be used for the expansion. the company bought this land 5 years ago at a cost of $319,000. at the time of purchase, the company paid $24,000 to level out the land so it would be suitable for future use. today, the land is valued at $295,000. the company has some unused equipment that it currently owns valued at $38,000. this equipment could be used for producing awnings if $12,000 is spent for equipment modifications. other equipment costing $490,000 will also be required. what is the amount of the initial cash flow for this expansion project?