21998904 Probability Statistics

ASSIGNMENT #2 COUNTING TECHNIQUES AND PROBABILITY

Counting techniques Q1. There are 4 buses running between two cities. In how many ways can a man go from one city to other and comes back by a different bus. Q2.How many lunches are possible consisting of soup, sandwich and drink if one can select from 4 soups, 3 kinds of sandwiches and 2 soft drinks. Q3.In how many ways can a cricket eleven be chosen out of 14 players? How many of them will (i) include a particular player (ii) exclude a particular player Q4. A man has 4 suits, 3 ties and 2 pairs of shoes. In how many different ways can he dress up. Q5.A person travels from Karachi to Lahore and comes back to Karachi, 5 trains are available. In how many ways he can commute such that: (i) he does not like to return by the same train

(ii) he does not mind to return by the same train.

Q6. How many fractions can be made using the number 2, 3, 5,7,11 and 13? Q7. There are 3 men and 2 women. In how many ways they can sit alternately. Q8.Seven players of Pakistan’s hockey team can play in any of the five forward line position, how many ways can these position be filled. Q9.Four people enter in a bus in which 6 seats are vacant. In how many ways can they be seated? Q10.

In how many ways can 8 mangoes of different sizes be distributed among 8 boys of

different ages so that the largest one is always given to the youngest boy? Q11.

In how many ways 4 mathematics, 5 statistics and 7 English books can be arranged

on a shelf in a row if one can not distinguish between same types of books. Q12.

A cricket eleven is to be chosen from 13 players of whom only 4 can bowl, in how

many ways the team can be selected so as to include at least 2 bowlers. Q13.

A candidate is required to answer 6 out of 12 questions of the midterm paper which

are divided into two groups each containing 6 questions and he is permitted to attempt not more than 4 from ant group. In how many different ways can he make up his choice?

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Q14. A pizza place offers 3 choices of salad, 20 kinds of pizza and 4 different desserts. How many different 3-course meals can one order? Q15. A university student must take a modern language, a natural science, a social science, and English. If there are four different modern languages, five natural sciences, three social sciences , but each student must take the same English course, how many different ways can the student select his course of study? Q16. The executive of the Manitoba Association of Mathematics Teachers consists of 3 women and 2 men. In how many ways can a president and secretary be chosen if: (a) The president must be female and the secretary male? (b) The president must be male and the secretary female? (c) The president and secretary are of opposite sex? Q17.

Simplify: (a)

n! ( n +1)!

(b)

n +1! n!

(e)

n! ( n −1)!

(c)

n ( n +1)! ( n +1) n!

(d)

n 2 ( n −1)! n!

Q18. How many distinguishable permutations are there of the letters in the following words? (a) AARDVARK (b) BOOKKEEPERS Q19. How many distinct ways can 3 red flags, 2 blue flags, 2 green flags and 4 yellow flags be arranged in a row? Q20. How many different arrangements can be made using the letters in the word TEETH, if the word must start with H? Q21. Using the letters of the word FACTOR (without repetitions), how many four-letter code words can be formed: (a) starting with R? (b) with vowels in the two middle positions? (c) With only consonants? (d) With vowels and consonants alternating? Q22. Manitoba license plate numbers consist of 3 letters followed by 3 digits. How many different plates could be issued? Q23. Consider the digits 1, 3, 5, 7, 9. If repetitions are allowed, find: (a) How many 3 digit numbers can be formed. (b) How many 3 digit numbers can be formed if the number must be less than 600 and divisible by 5.

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Q24. How many ways can 5 people sit in a row if two of them insist on sitting together? Q25. How many ways can 5 people sit in a row if two of them insist on not sitting together? Q26. How many ways can four married couples sit on a park bench if (a) every husband and wife must sit together? (b) The men and women must alternate? Q27. Three brothers and 3 sisters are lining up for a picture. How many arrangements are there: (a) Altogether? (b) With brothers and sisters in alternating positions? Q28. Five students walk into a French classroom with 10 desks. How many different seating arrangements are possible? Q29. Winnipeg stadium has 4 entrances and 9 exits. In how many ways can 2 people enter together but leave by different exits? Q30.How many ways can a chairperson, a secretary and a treasurer be selected from a committee of 8 people? Q31.How many 5 digit numbers can be found from 1, 2, 3, 4, 5 if: ( no repetition) (a) The odd digits occupy the odd places? (b) The odd digits occupy the odd places in ascending order? Q32. Using the letters in THURSDAY: (a) how many 4 letter words are possible? (b) how many 4 letter words end in Y? (c) How many do not start with an R and end with a Y? Q33. (a) How many ways can 8 people be seated around a circular table? (b) How many way can they be seated if 2 of them insist on sitting next to each other? Q34. Five men and 5 women sit around a circular table, men and women alternating. How many ways can this be done? Q35. How many 4 digit numbers larger than 5600 can be made using the digits 0, 1, 2, 5, 6, 8, 9 (no repetitions); Q36. Using the numbers 1, 2, 3, 5, 6, 8, 0 (no repetitions): (a) how many 4 digit numbers are possible? (b) How many are divisible by 5? (c) How many are even?

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Q37. A book collector has 5 different books by Dickens, 3 different plays by Shakespeare, and 3 different novels by D. Steele. She also has an original story by Margaret Laurence. How many ways can they be arranged on a shelf if the books by each author are to be kept together? Q38. (a)How many ways can 8 people be seated around a circular table if Nicky and Monica insist on sitting together? (b) if, in addition, George and Brent refuse to be seated together, how many ways can this be done? Q39. How many ways can 4 girls and 2 boys sit at a movie theatre row (which contains 6 seats), if: (a) One girl must be seated on each end? (b) All the girls insist on sitting together? Q40. (a) How many 5 letter words are possible using the letters in WINTER? (b) How many contain I as the second letter? (c) How many do not start with an E? Q41. The letters in the word BARRIER are arranged all at a time. (c) Find the number of permutations (d) Find the number of arrangements beginning with the letter R (e) Find the number of arrangements beginning with exactly one R. Q42. An investment club has a membership of 4 women and 6 men. A research committee of 3 is to be formed. In how many ways can this be done if: (a) There are to be 2 women and 1 man on the committee? (b) There is to be at least 1 woman on the committee? (c) All three have to be the same sex?

PROBABILITY Q1.

A card is drawn at random from an ordinary pack of 52 playing

cards. Find the probability that the card (i) is a jack (ii) is not a jack Q2.

A bag contains 3 white, 4 black and 5 red balls. If 3 balls are drawn

at random determine the probability that (i) all 3 are red (ii) all 3 are black (iii) 2 red and one is black (iv) one of each color is drawn. -4-

Q3.

A room has3 lamp. From a collection of 10 light bulbs of which 6

are not good, a person selects 3 at random and puts them in the sockets. What is the probability that he will have light from all the three lamps? Q4.

A class contains 16 boys and 10 girls of which half of the boys and

half of the girls are fat. Find the probability that a student chosen at random is a boy or a fat student Q5.

A bag contains 25 balls marked 1 to 25. One ball is drawn at

random. What is the probability that it marked with a number of multiple of 5 or 6. Q6.

The probability that a contractor gets an electric contract is 2/7 and

that of his getting a plumbing contract is 3/5. If the probability of his getting at least one of the contract is 5/7. what is the probability of his getting both the contracts. Q7.

let

the population of

adults of a small village be categorized

according to sex and employment status as:

Employed Unemployed Total Male

460

40

500

Female

140

260

400

Total

600

300

900

If a person is chosen at random find the probability that it is (i)Male (ii)Employed Person (iii)Male and Employed Person (iv)Male or Employed Person Q8.

From a well shuffled pack of 52 playing cards, one card is drawn.

Find the probability that it is (i) a King (ii) a diamond (iii) the queen of hearts (iv) either the

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queen of heart or the jack of spades (v) either a two or three (vi) not the ace of spades (vii) not a club (viii) not a face card (ix) either a king, queen or a jack. Q9.

If two dice are thrown or rolled find the probability that: a. Two faces shows the same number b. The sum of the two faces are greater than 10 c. A total more than 7 d. at least one ace e. a total of at most 4 f. The number on the first die is greater than the number of the second die. g. A sum of 5 or a sum of 9 h. A total of 8. i. At most a total of 5.

Q10.

A committee of four members is to be selected at random from a

group consisting of six men and four women. What is the probability that on his committee men and women will get equal representation? Q11.

Of 150 people, 60 men and 90 are women, 30 of the men and 60 of

the women read newspaper daily. A person is to be selected at random from the lot. What is the chance that selected would be a man or read newspaper. Q12.

from each of the three married couples one of the partner is selected

at random. What is the probability of their being all of one sex? Q13.

In a certain firm 6 persons (4female and 2 male) are eligible for

promotion to 2 higher positions. How many different combinations of these employees are eligible for promotions? List them a. Assume that persons are selected at random for promotion then find the probability that: i. At least one of the persons promoted would be a female ii. Exactly one female would be promoted iii. No more than one female promoted iv. No female would be promoted.

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Q14. If each coded item in a catalog begins with 3 distinct letters followed by 4 distinct non-zero digits, Find the probability of randomly selecting one of these coded items with the first letter a vowel and the last digit even. Q15. Two cards are drawn in succession from a deck without replacement. What is the probability that both cards are greater than 2 and less than 8? Q16. In a poker hand consisting of 5 cards, find the probability of holding a. 3 aces; b. 4 hearts and 1 clubs Q17. Dom’s Pizza company uses taste testing and statistical analysis of the data prior to marketing any new product. Consider a study involving three types of crusts ( Thin , Thin with garlic and oregano, and Thin with bits of cheese). Dom’s is also studying three sauces, ( standard, a new sauce with more garlic, and a new sauce with fresh basil) a. How many combinations of crust and sauce are involved? b. What is the probability that a judge will get a plain thin crust with a standard sauce for his first taste test? Q18. According to consumer digest (July/August 1996), the probable location of personal computers (PC) in the home is as follows: Adult bedroom: 0.03 Child bedroom : 0.15 Other bedroom: 0.14 Office or den: 0.40 Other rooms: 0.28

(a) What is the probability that a PC is in a bedroom? (b) What is the probability that it is not in a bedroom? (c) Suppose a house hold is selected at random from households with a PC; in what room would you expect to find a PC? Q19. Interest centers around the life of an electric component . Suppose it is known that the Probability that the component survives for more than 6000 hours is 0.42. Suppose also that the probability that the component survives no longer than 4000 hours is 0.04. (a) what is the probability that the life of the component is less than or equal to 6000 hours? (b) What is the probability that the life is greater than 4000 hours?

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Q20. Consider the situation in Q no.19. Let A be the event that the component fails a particular test and B be the event that the component that the component displays strain but does not actually fail. P(A)= 0.20 and P(B)=0.35. (a) What is the Probability that the component does not fails the test? (b) What is the probability that the component works perfectly well ( i.e neither displays strain nor fails the test)? (c) What is the probability that the component either fails or shows strain in the test? Q21. Factory workers are constantly encouraged to practice zero tolerance when it comes to accidents in factories. Accidents can occur because the working environment or conditions themselves are unsafe. During last year, 300 accidents have occurred. The percentages of the accidents for the condition combinations are as follows: Shift Day Evening Graveyard

Un safe conditions 5% 6% 2%

Human error 32% 25% 30%

(a) what is the probability that the accident occurred on the grave yard shift? (b) What is the probability that the accident occurred due to human error? (c) What is the probability that the accident occurred due to unsafe condition? (d) What is the probability that the accident occurred on either the evening or grave yard shift? Q22. probability that a mechanic serviced 3,4,5,6,7,8 cars on a given day is as follows: Cars serviced 3 4 5 6 7 8

Percentage chance 12% 19% 28% 24% 10% 7%

(a) What is the probability that no m more than 4 cars will be serviced by the mechanic?

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(b) What is the probability that he will service fewer than 8 cars? (c) What is the probability that he will service either 3 or 4 cars? Q23. Prove that: P(A’∩B’)= 1 + P(A∩B) – P(A) – P(B)

SOLUTION

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