Problem 4

A target is a square with length of a side 2. A child shoots by bow trying to strike in the center of a target. Deviation from the center along one side of the square has a normal distribution with standard deviation 1.5. Deviation from the center along another side of the square has a normal distribution with standard deviation 2.5. Find the probability that the child will strike in the target.

NAME
GROUP

Problem 5

At Las Vegas, roulette is played with a wheel that has 38 slots – 20 losing slots and 18 winning slots. Your chances of losing your dollar are therefore 20/38, and of winning a dollar are 18/38.

1. In the very long run, what is the average loss per play?

2. What are your chances of ending up a net loser, if you play a dollar

5 times 25 times 125 times.

Icef 2003/2004

STATISTICS

FIRST YEAR

EXAM

Section II

Part B

Problem 6

Spend about 25 minutes on this part of the exam

Percent of Section II grade 25

NAME
GROUP

It is well known that bad weather and the danger of avalanches is really a plague of a ski resorts touring business – many people decided not to go to the mountains if the probability of bad weather is high. The manager of a touring company selling 10-days tours to some winter ski resort asked you to help her in careful investigation of her idea to offer a all money back (for the whole tour) if during 10 days of vacation at least 6 days due to the bad weather the skiing is impossible because of the bad weather. Money back are offered also for those days (not for the tour as a whole) when all main pistes are closed due to the danger of avalanches), but only if money were not paid back as the result of the bad weather. For the ski resort under consideration the bad weather day is usually 4 days from ten, while the probability of the day when main pistes are closed due to the danger of avalanches is 0.04. The new system could be approved by the chief manager of the touring company only if the expected sum paid back to the client is no more than 20% of the contract.

a. You decide to do a simulation experiment setting up 40 ten-day trials. Explain how you set up the trials using a random number table below. Show on the random number table how you carry out the trial using some marks (underlying or circle around combinations of random numbers).

b. Present and explain the results of your simulation. What is the expected percentage paid back to the clients?

c. Evaluate the appropriate theoretical probabilities and expectations. Write a brief report. In this report, discuss your results and compare your results with the theoretical prediction for this situation. What is your final conclusion about possibility of introducing of suggested payment system?

Table. Random Digits (Blocked merely for convenience)

39	65	76	45	45		19	90	69	64	61		20	26	36	31	62		58	24	97	14	97
73	71	23	70	90		65	97	60	12	11		31	56	34	19	19		47	83	75	51	33
72	20	47	33	84		51	67	47	97	19		98	40	07	17	66		23	05	09	51	80
75	17	25	69	17		17	95	21	78	58		24	33	45	77	48		69	81	84	09	29
37	48	79	88	74		63	52	06	34	30		01	31	60	10	27		35	07	79	71	53
02	89	08	16	94		85	53	83	29	95		56	27	09	24	43		21	78	55	09	82
87	18	15	70	07		37	79	49	12	38		48	13	93	55	96		41	92	45	71	51
98	83	71	70	15		89	09	39	59	24		00	06	41	41	20		14	36	59	25	47
10	08	58	07	04		76	62	16	48	68		58	76	17	14	86		59	53	11	52	21
47	90	56	37	31		71	82	13	50	41		27	55	10	24	92		28	04	67	53	44