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SPECIAL PROBABILITY DISTRIBUTIONS CHAPTER 10 (DM045)
TUTORIAL QUESTIONS

 1. If X ~B 8,0.3 , determine the following probability using the binomial formula and

table. (b) PX  7
(d) P3 X 5
(a) PX 3
(c) PX  2 (f) the mean and the variance.

(e) P1X 4

 2. If W~B 20,0.25 , determine the following probability by using binomial formula and

table. (b) P (W = 5)
(a) P(W ≤ 3) (d) P (W < 6)
(c) P (W ≥ 4) (f) P (5 < W < 10)
(e) P (4 < W ≤ 9) (h) P (7 ≤ W <10)

(g) P (5 < W < 11)

3. If X ~ B (15, 0.75), find (b) P (X = 8)
(a) P (X ≤ 6) (d) P (X < 7)

(c) P (X > 3)
(e) P (X ≥ 5)

4. Four unbiased dice are thrown. Find the probability that there are:

(a) Exactly two sixes. (b) At least two sixes.

5. Six coins are tossed. Find the probability that there are:
(a) Not more than four heads.
(b) At least two heads.

6. Find the mean and variance of the binomial distribution B (6, 0.25).

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SPECIAL PROBABILITY DISTRIBUTIONS CHAPTER 10 (DM045)

7. It is claimed that 95 % of first class mail with the same city is delivered within two days
of the time of mailing. Six letters are randomly sent to different locations.
(a) What is the probability that all six will arrive within two days.
(b) What is the probability that exactly five will arrive within two days.
(c) Find the mean number of the letters that will arrive within two days.
(d) Find the variance number and standard deviation of the letters that will arrive
within two days.

8. Ten percent of the new automobiles will require warranty service within the first year.
John sells 12 automobiles in April.
(a) What is the probability that nine of these automobiles require warranty service.
(b) Find the probability that exactly one of them requires warranty service.
(c) Determine the probability that exactly two of them need warranty service.
(d) Compute the mean and standard deviation of this probability distribution.

9. At the local swimming club, the expected number of people that can swim a mile is 4.5
and the variance is 3.15. Find the probability that at least three people can swim a mile.

10. In a local youth group the expected number of people who can play a musical instrument
is 4 and the variance is 3.2. Find the probability that:
(a) Five people can play a musical instrument.
(b) Less than six people can play a musical instrument.

11. Given that Y ~ B (10, 0.8). Determine the mean value of Y.

12. The random variable Y  Po (1.0). Find :

(a) P (Y = 7) (b) P (Y = 4)

(c) P (Y ≤ 5) (d) P (Y  3)

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13. Find the values of a, b, c and d given that X  Po (2.5) :

(a) P ( X ≤ a) = 0.8912 (b) P (X  b) = 0.5438

(c) P ( X ≤ c) = 0.9997 (d) P (X  d) = 0.0420

14. The number of letters arriving at a particular address each day can be modeled by a
Poisson Distribution with a mean of seven letters per day, find the probability that
tomorrow:
(a) no letters will arrive.
(b) four letters will arrive.
(c) at least two letters will arrive.
(d) fewer than five letters will arrive.

15. (a) The number of swimming accidents per week at a particular stretch of coastline is
a Poisson variable with parameter 3.
(i) Find the probability that in any one week chosen at random exactly two
swimming accidents occur.
(ii) Find the probability that in any four-week period there are fewer than four
swimming accidents.

(b) In a particular document there are an average of eight typing errors a page.
Find the probability that:
(i) on a certain page there are fewer than four typing errors.
(ii) on a half page section there are six typing errors.

16. In the manufacture of commercial carpet, small faults occur at random in the carpet at an

average rate of 0.95 per 20 m2. Find the probability that in a randomly selected 20 m2

area of this carpet,

(a) there are no faults. (b) there are at most 2 faults.

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17. The mass, M grams, of a batch of commutative coins is such that M ~ N ( 50, 9 ). Each
coin is weighed individually before packaging and will be rejected if its mass is less than
47g. What percentage of coins would you expect to be rejected?

18. Using Z ~ N ( 0 , 1 ) , find k such that : (b) P z k  0.1

(a) P z k 0.85

19. The scores X, in a Mathematics Competition are such that X ~ N ( 80 , 20 ). Find
(a) The probability that Ali, a student chosen at random, scores between 75 and 90
on the test.
(b) The probability that Erica, a student chosen at random, scores less than 72.
(c) The probability that Intan, a student chosen at random, scores more than 85, given
that she scored less than 90.
(d) The score needed to obtain a high Distinction Certificate, if those are awarded to
the top 8 % of the students.

 20. If X ~ N (100, 2 ) and P X 106 0.8849.Find the variance 2 .

21. The masses of packets of sugar are normally distributed. In a large consignment of
packets of sugar, it is found that 99.76 % of them have a mass greater than 286g and
probability of each pack is less than 312g is 0.9918. Estimate the mean and the variance
of this distribution.

22. The numbers of days taken for a sick leave by factory workers is normally distributed

with a mean of 12 days and variance 2 in a year. If the probability of less than 10 days

 off is 0.1587, find the variance 2 . Finally, find a if P X 12 a 0.950.

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SPECIAL PROBABILITY DISTRIBUTIONS CHAPTER 10 (DM045)

23. The mathematics marks of the first semester examination are normally distributed with a
mean of 60 and a variance of 9. If a student is randomly selected, find the probability that
his/her Mathematics mark is
(a) less than 60
(b) at least 55
If 10 % of the students obtained grade A, find the minimum mark to achieve a grade A.

24. (a) If X ~ B (500, 0.002), find (ii) P (X = 1)
(i) P (X = 0)
(iii) P (X = 4)

(b) If X ~ B (200, 0.06), find (ii) P (X  5)
(i) P (X < 20)

25. The number of bacteria in one milliliter of a liquid is known to follow a Poisson
distribution with mean 3. Find the probability that one milliliter sample will contain no
bacteria. If 100 samples are taken, find the probability that at most ten will contain no
bacteria (Use a Poisson approximation and give your answer correct to 3 decimal places).

26. If X  B( 200,0.7), use the normal approximation to find

(a) P ( X ≥ 130) (b) P ( 136 ≤ X < 148 )

(c) P ( X < 142 ) (d) P ( X > 152 )

(e) P ( 141 < X < 146 )

27. 10% of the chocolates produced in a factory are mis-shapes. In a sample of 1000
chocolates find the probability that the number of mis-shapes is
(a) less than 80
(b) between 90 and 115 inclusive
(c) 120 or more.

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SPECIAL PROBABILITY DISTRIBUTIONS CHAPTER 10 (DM045)
TUTORIAL ANSWERS

1. (a) 0.2541 (b) 0.00129 (c) 0.2553 (d) 0.1828

(e) 0.7484 (f) mean = 2.4, variance = 1.68

2. (a) 0.2252 (b) 0.2024 (c) 0.7748 (d) 0.6172

(e) 0.5713 (f) 0.3689 (g) 0.3789 (h) 0.2003

3. (a) 0.0042 (b) 0.0393 (c) 1 (d) 0.0042

(e) 0.9999

4. (a) 0.1157 (b) 0.1319

5. (a) 0.8906 (b) 0.8906

6. mean = 1.5 variance = 1.125

7. (a) 0.7351 (b) 0.2321 (c) 5.7 (d) 0.285, 0.5339

8. (a) 1.6038 x 10-7 (b) 0.3766 (c) 0.2301 (d) mean = 1.2  = 1.0392

9. 0.8732

10. (a) 0.1746 (b) 0.8042

11. Y = 8

12. (a) 0.000073 (b) 0.0153 (c) 0.9994 (d) 0.0803

13. (a) a = 4 (b) b = 3 (c) c = 9 (d) d = 5

14. (a) 0.0009 (b) 0.0912 (c) 0.9927 (d) 0.173

15. (a) (i) 0.2241 (b) (i) 0.0424

(ii) 0.0023 (ii) 0.1042

16. (a) 0.3867 (b) 0.9286

17. 15.87 %

18. (a) k = 1.44 (b) k = 1.64

19. (a) 0.856 (b) 0.0367 (c) 0.1203 (d) a = 86.31

20. 2 = 25

21. µ = 300.05 2 = 24.80

22. 2 = 4 a = 3.92

23. (a) 0.5 (b) 0.9525 min. mark = 63.8448

24. (a) (i) 0.3679 (ii) 0.3678 (iii) 0.0153

(b) (i) 0.9787 (ii) 0.9924

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25. 0.050, 0.986 (c) 0.591 (d) 0.0268 (e) 0.2113
26. (a) 0.9474 (b) 0.6319 (c) 0.0197
27. (a) 0.0154 (b) 0.8149

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SPECIAL PROBABILITY DISTRIBUTIONS CHAPTER 10 (DM045)
PSPM QUESTIONS (PST)

1. The distribution of the number of car breakdowns on a highway in any one day is
Poisson with mean 3.5.

(a) Find the probability that

(i) Exactly 2 cars break down on a particular day. [2M]

(ii) At most 5 cars break down on a particular day. [2M]
(iii) *Out Of Syllabus – Between 100 to 111 cars break down on the

day for the month of April. [6M]

(b) An auto repair company places 3 of its trucks to provide assistance in car

breakdowns along the above highway everyday. Find the probability that on a

particular day the company could not provide any such assistance. [2M]

[PSPM 2004]

2. It is known that 10 % of the patients with high fever are confirmed to be suffering from
dengue fever.

(a) If 15 patients with high fever are randomly chosen, find the probability that

(i) Less than 6 are confirmed to be suffering from the dengue fever. [3M]

(ii) Exactly 10 patients with high fever are confirmed to be free of

dengue fever. [3M]

(b) If 100 patients with high fever are randomly chosen,

(i) Approximate the probability that 9 to 14 patients are confirmed

to be suffering from dengue fever. [5M]

(ii) Find the value of m such that the probability of more than m patients

that are confirmed to be suffering from dengue fever is 0.025. [4M]

[PSPM 2004]

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3. The number of short messages (SMS) received by a teenager in half an hour has a

Poisson distribution with mean  .

(a) If the probability of receiving no SMS within half an hour is 0.0025, [2M]

show that  6 (to the nearest integer).

(b) Using the value of  6, find the probability that

(i) He receives less than six SMS in half an hour.

(ii) He receives less than six SMS in one hour.

(iii) Two teenagers selected at random will receive at least six SMS in half an

hour. [8M]

[PSPM 2005]

4. In any of its shipments, a company found that the probability of bad oranges it supplies is
0.2. At the receiving terminal, a sample is taken at random and the number of bad
oranges is recorded.

(a) A shipment will be rejected if there are more than 10% bad oranges in the sample

taken. Calculate the probability that a particular shipment will be accepted if a

sample of size 20 is taken. [4M]

(b) Using the normal approximation, estimate the probability of obtaining 180 to 210

bad oranges if 1000 oranges are inspected at random. [7M]

In another shipment, the probability of obtaining bad oranges is 0.03. The probability of

rejecting this shipment is 0.022. Using the Poisson approximation, determine the

maximum allowable number of bad oranges in a sample of size 300 such that the

shipment is accepted. [4M]

[PSPM 2005]

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SPECIAL PROBABILITY DISTRIBUTIONS CHAPTER 10 (DM045)

5. In a delivery of microchips, it is known that the number of defective is 2 out of 10.

(a) If 15 microchips are delivered, calculate the probability that [3M]
(i) at least 5 microchips are defective. [2M]
(ii) exactly 11 microchips are good.

(b) If 500 microchips are delivered, find n such that the probability of obtaining

the number of defective microchips exceeding n is 0.147. [5M]

(c) *Out Of Syllabus – Suppose in another shipment of microchips, the probability

of defective is 0.01.

If a sample of 300 microchips is taken from the shipment, estimate the probability

of getting 1 to 3 defective microchips. [3M]

[PSPM 2006]

6. Assume that the number of e-mails received by a student daily has Poisson distribution
with a mean of 5.

(a) (i) Determine the probability that the student receives between 5 and 13

e-mailsdaily. [2M]

(ii) If the probability of a student receiving not more than m e-mails in a day is

0.616, determine the value of m. [4M]

(b) *Out Of Syllabus – If 15 days are randomly chosen, find the probability that the
student receives between 5 and 13 e-mails daily for a period of 9 days. [3M]

(c) *Out Of Syllabus – If 150 days are randomly chosen, use the normal

approximation to find the probability that the student receives between 5 and 13

e-mails daily for the less than 70 days. [6M]

[PSPM 2007]

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SPECIAL PROBABILITY DISTRIBUTIONS CHAPTER 10 (DM045)

7. *Out Of Syllabus – Compact discs produced by a factory are packed in boxes. Each box
contains 100 compact discs. It is known that 4% of the compact discs produced are
defective.

(a) Show that the probability that a box chosen at random will contain at most 3

defective compact discs is approximately 0.43. [3M]

(b) Find the probability that among 12 boxes chosen at random, there will be

4 boxes which contain at most 3 defective compact discs. [3M]

(c) Seventy boxes are chosen at random. Find the probability that between 20 boxes
and 40 boxes, inclusively, which contain at most 3 defective compact discs. [6M]
[PSPM 2008]

8. In any large shipment from a particular orchard, it is known that 2% are unripe. Upon
arrival of shipment at a receiving depot, random samplings with replacement are
conducted.

(a) Calculate the probability of getting at most one unripe watermelon in a sample of

size 20. [4M]

(b) *Out Of Syllabus – Approximate the probability of getting one to three unripe

watermelons in a sample of size 50. [5M]

(c) If the sample size is 1000, approximate the probability of getting not more than

eight unripe watermelons. [6M]

[PSPM 2009]

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9. The distribution of the weights of all sugar sachets produced by a particular factory is
assumed to be normal with mean 25 gm and standard deviation 2 gm.

Show that the probability of a randomly selected sachet weighs within 1 gm of the

mean is 0.383. [4M]

(a) If ten sachets are randomly selected, find the probability that between four and

seven sachet weigh within 1 gm of the mean. [3M]

(b) Determine the sample size, n such that the probability that none of the sachet

weigh within 1 gm of the mean is 0.021. [4M]

(c) If one hundred sachets are randomly selected, approximate the probability that

less than 40 sachets weigh within 1 gm of the mean. [4M]

[PSPM 2010]

10. On the average, a hospital receives 6 emergency calls in 15 minutes. It is assumed that
the number of emergency calls received follows the Poisson distribution.

(a) Find the probability that

(i) not more than 15 emergency calls are received in an hour. [3M]

(ii) the hospital will receive the first emergency call between 9.00 am

and 9.05 am. [3M]

(b) Find the number of emergency calls received, m, if it is known that the
probability at most m emergency calls received in half an hour is 0.155. [4M]
[PSPM 2011]

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11. The probability that a type of antibiotics can cure a certain disease is 0.95.

(a) If five patients are given the antibiotics, find the probability that

(i) exactly three patients are cured after finishing the course of antibiotics.

[3M]

(ii) at least one patient is cured after finishing the course of antibiotics.

[2M]

(b) If 500 patients are given the antibiotics, find the

(i) probability that more than 480 patients are cured. [4M]

(ii) largest possible value n such that the probability that at least n patients

recovered after finishing the course of antibiotics is 0.9. [4M]

[PSPM 2011]

12. The registration record of a private college indicates that 40% of its new intakes are
international students and the remaining are local students.

(a) If 20 new students are randomly selected and the number of local students are

noted, find the probability that there are

(i) equal number of local and international students. [2M]

(ii) not less than 9 local students. [4M]

(b) Exactly 100 new students are randomly selected. By using a suitable approximate

distribution,

(i) find the probability that between 38 and 46 are international students.

[5M]

(ii) determine the value m such that the probability that the number of

international students is at most m is 0.993. [4M]

[PSPM 2012]

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13. The number of motorcycles arriving at the main entrance of a university during peak
hours has a Poisson distribution with mean three per minute.

Find the probability that [3M]
(a) at most one motorcycle will arrive in one minute. [3M]
(b) exactly five motorcycles will arrive in two minutes. [PSPM 2012]

14. The lifetime of D sized batteries produced by a local factory is normally distributed with
mean 11.5 months and standard deviation 0.8 months.

(a) Suppose a battery is selected at random from the factory’s production line.
(i) Calculate the probability that the battery’s lifetime is between 9.5 and 11.5

months, correct to one decimal place. [3M]

(ii) If the probability that the battery’s lifetime is less than h months is 0.975,

determine the value of h. [4M]

(b) Suppose ten batteries are selected at random from the factory’s production line,

calculate the probability that at most three batteries have lifetime between 9.5 and

11.5 months. [3M]

(c) If 100 batteries are selected at random from the factory’s production line,

approximate the probability that from 48 to 51 batteries have lifetime between

9.5 and 11.5 months. [5M]

[PSPM 2013]

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SPECIAL PROBABILITY DISTRIBUTIONS CHAPTER 10 (DM045)
PSPM ANSWERS (PST)

1. (a) (i) 0.1849 (ii) 0.8576

(iii) 0.3754 (ii) 0.0105
(ii) m = 15
(b) 0.4634 (ii) 0.0203

2. (a) (i) 0.9978 (b) 0.7441

(b) (i) 0.6247 (ii) 0.1876
(c) 0.5974
3. (b) (i) 0.4457 (ii) m = 5
(c) 0.9798
(iii) 0.3072 (c)
(b) 0.6131
4. (a) 0.2061
(a) 0.2818
(c) 15 (c) 0.5987
(ii) 0.271
5. (a) (i) 0.1642
(ii) 1
(b) 109 (ii) n = 469
(b) 0.1606
6. (a) (i) 0.382 (ii) 0.9435
(ii) 52
(b) 0.0483 (ii) h=13.068
(c) 0.3094
7. (b) 9

8. (a) 0.94

(c) 0.005

9. 0.383

(b) n=8

10. (a) (i) 0.0344
(b) m=8

11. (a) (i) 0.0214
(b) (i) 0.1292

12. (a) 0.1991
13. (a) (i) 0.1171

(b) (i) 0.4903
14. (a) (i) 0.5

(b) 0.1719

END OF TUTORIAL

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