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Unformatted text preview: Kvarna A new restaurant will offer light food, coffee, wine, and live music. The managers of Kvarna, Alice and Harry, face challenges in understandiang its opportunities. Menu To control costs, Kvarna will offer a limited menu of 4 panini, 2 salads, 3 pastries, and 6 wines. If a party of two orders two meals when a meal consists of a panini or salad, a pastry, and wine with each choosing something different than the other, in how many different ways may a couple order? If each couple may order the same meal or different meals, in how many ways may a couple dine? availablechoices panini 4 or salad 2 6 and pastry 3 3 and wine 6 6 108 first choice 107 second choice product= 11556 number of combinations o 108^2 if both alike 11664 Panini or salad gives 4+2 = 6 choices. Pastry 3. Wine 6. The product of the choices is 108 for one meal. There are then 107 unlike choices for the second meal. The product of 108 * 107 = 11556 is the # of unlike pairs. If they can order the same meal, 108^2 = 11,664 is the number of possible ways for the couple to dine. The product of 108 * 107 = 11556 is the # of unlike pairs. If they can order the same meal, 108^2 = 11,664 is the number of possible ways for the couple to dine. The following result was reached by several students on the assumption that all th are than "something different" as the question reads. If the second unlike meal means that the entre, dessert, and wine all mu then there are 5 * 2 * 5 = 50 choices for the second meal 108*50= 5400 is the number of c er of combinations of unlike pairs ssumption that all three elements of the meal had to differ ert, and wine all must differ, econd meal is the number of completely unlike pairs. Revenues What is the average monthly rate of increase in revenues at Kvarna during th up period shown? Monthly Revenues March $18,268 April $19,169 May $20,039 June $20,408 July $20,468 August $21,004 September $21,038 October $21,922 November $22,125 December $22,779 January $23,104 February $23,198 March $23,361 April $24,416 May $24,531 June $24,811 1.0493 1.0454 1.0184 1.0029 1.0262 1.0016 1.0420 1.0093 1.0296 1.0143 1.0041 1.0070 1.0452 1.0047 1.0114 Compute the growth rate as a r of each month's revenue to the month's revenue. Compute the geomean of the g rates = 1.0206. 1.02 average monthly growth rate 1.0208 This the incorrect arithmetic mean of the g enues at Kvarna during the start ute the growth rate as a ratio h month's revenue to the prior 's revenue. ute the geomean of the growth = 1.0206. y growth rate t arithmetic mean of the growth rates. Profit The business plan for Kvarna identifies several levels of possible profit with assoc probabilities for two scenarios. First is a lunch & dinner only program. A second program includes a breakfast menu of coffee and pastries. What is the expected of profit and the standard deviation of each program? Which program is the mor attractive investment? Lunch & Dinner Only # of sales probability ($15,000) 0.15 $0 0.15 $10,000 0.20 $20,000 0.15 $30,000 0.15 $40,000 0.10 $50,000 0.10 1.00 Breakfast, Lunch, & Dinner # of sales probability ($15,000) 0.10 $0 0.20 $10,000 0.25 $20,000 0.20 $30,000 0.10 $40,000 0.10 $50,000 0.05 1.00 product dev ($2,250) 31250 $0 16250 $2,000 6250 $3,000 3750 $4,500 13750 $4,000 23750 $5,000 33750 E[x]= $16,250 product dev ($1,500) 29500 $0 14500 $2,500 4500 $4,000 5500 $3,000 15500 $4,000 25500 $2,500 35500 E[x]= $14,500 Multiply each outcome by its probabilty and sum to find the expected value. Subtract the expected value from each outcome, square these deviations, and sum to get the variance. The square root of the variance is the Multiply each outcome by its probabilty and sum to find the expected value. Subtract the expected value from each outcome, square these deviations, and sum to get the variance. The square root of the variance is the standard deviation. Program one has a higher expected value ($16,250 versus $14,500) but a larger standard deviation ($19,876 versus $17,095). The extra return comes with higher risk. ssible profit with associated y program. A second What is the expected value ch program is the more dev^2*prob 146484375 39609375 7812500 2109375 28359375 56406250 113906250 394687500 var $19,867 stdev dev^2*prob 87025000 42050000 5062500 6050000 24025000 65025000 63012500 292250000 var $17,095 stdev ind the expected uare these deviations, e variance is the ind the expected uare these deviations, e variance is the versus $14,500) but ). The extra return Panini
Kvarna views a menu item as profitable if customers buy 25 of the item per week. Suppose the menu consists of four panini chosen at random from many possibilities and of two salads chosen at random from many possibilities. If the probability that a panini will be successful is 0.60 and the probability that a salad will be successful is 0.30, find the probability that all six will be profitable. What is the probability that none (of four) panini and neither of the two salads will be successful? What is the probability that exactly two panini and one salad will be successful while the others are not? a. b. c. all six successful (.6^4)*(.3^2) no panini but two salads (.4^4)*(.7^2) two and one (.6^2)*(.4^2)*(.3^1)*(.7^1) Probability that all are successful is (.6^4)*(.3^2) = 0.01166. Probability that none are successful is (.4^4) * (.7^2) = 0.01254. Probability that two panini and one salad are successful is (.6^2)*(.4^2)*(.3^1)*(.7^1) = 0.01210. 5 of the item n at random m from many ul is 0.60 and e probability er of the two d will be 0.01166 0.01254 0.01210 1254. Door Prize
At the grand opening, Kvarna will draw five names for a door prize from those who attend. One of five boxes holds a "Kvarna bottomless cup" with a promise of free refills for a year worth $150. The other four have a coupon for a dozen pastries worth $10. Cyril Reynaud draws first, is offered the choice of box. Before the box is opened, two of the unopened boxes are revealed to have pastry coupons. Cyril is invited to either stay with his first choice or to switch to one of the two remaining boxes not yet chosen or revealed. If Cyril stays with his first choice, what is his expected value? If he switches to a different box, what is his expected value? At the outset, each urn has a 20 percent chance. After two are revealed as pastries, the 40 percent on the unselected goes to the re $150 $10 coffee pastry stick 0.20 0.80 $38 switch 0.40 0.60 $66 When two choices are revealed as pastry, their probability loads on the unselected choices. The information revealed by exposing twp of the original unselected boxes applies only to the other unselected cans. With sticking, the probability of winning is 0.20 and the expected value if $38. By switching, the probability of winning is 0.40 and the expected value if $66. With sticking, the probability of winning is 0.20 and the expected value if $38. By switching, the probability of winning is 0.40 and the expected value if $66. or prize from mless cup" er four have raws first, is the l is invited to remaining ? elected goes to the remaining two not selected. y loads exposing her xpected pected xpected pected ...
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 Spring '08
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