Section: A

1. Six dice are thrown 729 times. How many times do you expect at least three dice to show a five or a six?

2. The headless matches per box of 50 matches in a total of 100 boxes is given in the following table:

No. of headless Matches per box 0 1 2 3 4 5 6 7No. of boxes 12 27 29 19 8 4 1 0

Assuming the binomial distribution, calculate the expected frequencies

3. Fit the binomial distribution, to the following data :

X: 0 1 2 3 4 5Y: 2 14 20 34 22 8

4. Show that the Poisson distribution is a limiting case of Binomial distribution.5. Six coins are tossed 6400 times. Using the Poisson distribution find the probability of getting six

heads x times.6. Using Poisson distribution, find the probability that the ace of spades will be drawn from a pack

of well shuffled cards at least once in 104 trials.7. A skilled typist, on routine work, kept a record of mistakes made per day during 300 working

days.

Mistakes per day

0 1 2 3 4 5 6

Number of days

143 90 42 12 9 3 1

Compute the frequencies of the Poisson distribution which has the same mean and total frequencies as the above distribution.

8. The marks of 500 candidates appearing in a certain examination have a mean 41.6 and a S.D 11.2. Assuming a normal distribution, find how many students obtain first class (for which a minimum of 60 marks is necessary).

9. In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and S.D of the distribution.

10. A certain number of articles manufactured in one batch were classified into three categories, according to a particular characteristic being less than 50, between 50 and 60, or greater than 60.

University of Petroleum & Energy StudiesDepartment of Mathematics

Statistics-Assignment/Tutorial(B.Tech_II sem All Branches)

If this characteristic is known to be normally distributed, determine the mean and the S.D for this

batch if 60%, 35% and 5% were found in categories. Given that if P=1

√2π∫

0

z

e−( 1

2 )t2dt, the value

of Z corresponds to P=0.1 and 0.45 are 0.25 and 1.65 respectively.11. A simple binary communication channel carries messages by using only two signals, say 0 and 1.

We assume that, for a given binary channel, 40% of the time a 1 is transmitted; the probability that a transmitted 0 is correctly received is 0.90, and the probability that a transmitted 1 is correctly received is 0.95. Determine (a) the probability of a 1 being received, and (b) given a 1 is received, the probability that 1 was transmitted.

12. A homeowner has just installed 20 light bulbs in a new home. Suppose that each has a probability 0.2 of functioning more than three months. What is the probability that at least five of these function more than three months? What is the average number of bulbs the homeowner has to replace in three months?

13. In oil exploration, the probability of an oil strike in the North Sea is 1 in 500 drillings. What is the probability of having exactly 3 oil-producing wells in 1000 explorations?

14. Owing to many independent error sources, the length of a manufactured machine part is normally distributed with m=11 cm andσ=0.2cm. If specifications require that the length be between 10.6 cm and 11.2 cm, what proportion of the manufactured parts will be rejected on average?

15. A laboratory blood test is 95 percent effective in detecting a certain disease when it is, in fact, present. However, the test also yields a “false positive” results for 1 percent of healthy persons tested. ( That is, if a healthy person is tested, then, with probability 0.01, the test result will imply he or she has the disease.) If 0.5 percent of the population actually has the disease, what is the probability a person has the disease given that the test result is positive?

16. Suppose the number of typographical errors on a single page of your book has a Poisson

distribution with parameter λ=12

. Calculate the probability that there is at least one error on this

page.

Section: B

Choose the correct distribution for the following problems and justify your answer:

17. What type of probability distribution will the consulting firm most likely employ to analyze the insurance claims in the following problem? An insurance company has called a consulting firm to determine whether the company has an unusually high number of false insurance claims. It is known that the industry proportion for false claims is 6%. The consulting firm has decided to randomly and independently sample 50 of the company’s insurance claims. They believe the number of these 50 that are false will yield the information the company desires.(a) Binomial distribution(b) Poisson distribution(c) Normal distribution(d) None of the above

18. What type of probability distribution will most likely be used to analyze warranty repair needs on new cars in the following problem?

The service manager for a new automobile dealership reviewed dealership records of the past 20 sales of new cars to determine the number of warranty repairs he will be called on to perform in the next 30 days. Corporate reports indicate that the probability any one of their new cars needs a warranty repair in the first 30 days is 0.035. The manager assumes that calls for warranty repair are independent of one another and is interested in predicting the number of warranty repairs he will be called on to perform in the next 30 days for this batch of 20 new cars sold.(a) Binomial distribution(b) Poisson distribution(c) Normal distribution(d) None of the above

19. The quality control manager of Marilyn’s Cookies is inspecting a batch of chocolate chip cookies. When the production process is in control, the average number of chocolate chip parts per cookie is 9.0. The manager is interested in analyzing the probability that any particular cookie being inspected has fewer than 10.0 chip parts. What probability distribution should be used?(a) Binomial distribution(b) Poisson distribution(c) Normal distribution(d) None of the above

20. Based on past experience, your time spent on e-mails per day has a mean of 10 minutes and a standard deviation of 3 minutes. To compute the probability of spending at least 12 minutes on e-mails, you might use what probability distribution?(a) Binomial distribution(b) Poisson distribution(c) Normal distribution(d) None of the above

21. A computer lab at a university has 100 personal computers. The probability that any one of them will require repair on a given day is 0.05. To find the probability that exactly 20 of the computers will require repair on a given day, you will use what probability distribution?(a) Binomial distribution(b) Poisson distribution(c) Normal distribution(d) None of the above

22. On the average, 1.8 customers per minute arrive at any one of the checkout counters of a grocery store. What probability distribution can be used to find out the probability that there will be no customers arriving at a checkout counter in the next minute?(a) Binomial distribution(b) Poisson distribution(c) Normal distribution(d) None of the above

23. A multiple-choice test has 30 questions. There are 4 choices for each question. A student who has not studied for the test decides to answer all questions randomly. What probability distribution can be used to figure out his chance of getting at least 20 questions right?(a) Binomial distribution(b) Poisson distribution(c) Normal distribution

(d) None of the above