Chapter 7Introduction to Probability 7.1
Chapter 7Introduction to Probability
Warm-up Exercise1.In each of the following figures, O is the centre of the circle, what fraction of the circle is shaded?
(a)(b)(c)
2.Find the area of each of the following sectors. (Correct your answers to 3 significant figures.)
(a)(b)(c)
3.In each of the following figures, find the area of the shaded region. (Express your answers in terms of if necessary.)
(a)(b)(c)
Build-up Exercise[This part provides two extra sets of questions for each exercise in the textbook, namely Elementary Setand Advanced Set. You may choose to complete any ONE set according to your need.]
Exercise 7A
Elementary Set
Level 1
1.There are 38 students in S3A and Herbert is one of the students of the class. Mr. Chan chooses a student at random to be the class monitor, what is the probability of choosing Herbert?
2.There is an autumn lucky draw held by a department store with the prize of a private car. Among the 12300lucky draw tickets received by the department store, 15 of them belong to Mr. Wong. If a lucky drawticket is chosen at random, find the probability that it belongs to Mr. Wong.
3.If a letter is chosen at random from the word ‘SUPPLEMENT’, find the probability of getting
(a)a letter ‘M’.(b)a letter ‘E’.
4.The following frequency distribution table shows the months of birth of32students in S3A. If a student is chosen at random, find the probability that the student was born in February, April, June, August, October or December.
Month / FrequencyJanuary / 3
February / 2
March / 3
April / 2
May / 1
June / 2
July / 5
August / 2
September / 4
October / 1
November / 4
December / 3
5.There are only orange and black goldfish in fishponds A and B. In fishpondA, there are 8orange goldfish and 13black goldfish. In fishpondB, there are18orange goldfish and 26black goldfish. If a goldfish is drawn from each fishpond at random, which fishpond has a higher probability of getting a black goldfish?
6.Over the past hour,328 tourists have entered OceanPark. If one of them is chosen at random, the probability of choosing a tourist from Beijing is.How many tourists from Beijing have entered OceanPark over the past hour?
7.In S3A, 24 students wear glasses. If a student is selected at random from the class, the probability of selecting a student with glasses is, find the number of students in S3A.
Level 2
8.A Christmas party is held by a company with 24staff members joining it. 4 of the staff members are from personnel department, 3 of them are from information technology department, 12 of them are from sales department and the rest are from warehouse department. If a staff member is chosen at random for a prize, what are the probabilities of getting a staff member from the following department?
(a)Either from information technology department or personnel department.
(b)Neither from personnel department nor warehouse department.
9.There are 27customers subscribing newspapers from a news-stand. 15 of them subscribe one Chinese newspaper only, 9 of them subscribe one English newspaper only and the rest subscribe both Chinese and English newspapers.If one of these customers calls the news-stand,
(a)what is the probability that the customer has subscribed both Chinese and English newspapers?
(b)what is the probability that the customer has subscribed one newspaper only?
10.The following cumulative frequencypolygon shows the results of 100studentsin a Mathematics examination. It is known that students who score 80 or above get a grade A each. If one of the students is chosen at random, find the probability that the chosen student gets a grade A. /11.In a box of toy trains, x of them are made by machineA and the rest are made by machineB. If one toy train is chosen at random, the probability of getting a toy train made by machineA is. Express the number of toy trains made by machineB in terms of x.
12.Among 100lucky draw tickets available from soft drinks, prizes of 10 of them are a doll each. To make the probability of selecting a ticket at random with a prize of doll as, how many tickets with prizes of a dolleach should be added?
13.There are two types of drinks in carton, lemon tea and apple juice, on a table. The number of cartons of lemon tea is 5 more than that of apple juice. If a drink in carton is selected at random, the probability of getting a carton of lemon tea is. How many cartons of lemon tea are there on the table originally?
14.There are some red ball pens and blue ball pens in a box, where the number of blue ball pens is 3 more than that of red ball pens. If a ball pen is selected at random, the probability of selecting a red ball pen is 0.4. Find the total number of ball pens in the box originally.
Advanced Set
Level 1
1.There are 24members in the choir in which 3 of them are under 16. If the choir leader chooses a member at random, find the probability that the chosen one is under 16.
2.There are 600 free concert tickets available for the public, where each person can apply for one ticket only. If there are excess of applicants, the organizer will distribute the tickets to them at random. Given that there are 1764 applicants and Rachel is one of them, find the probability for her to obtain a ticket.
3.Robert has the following cards. If he selects a card randomly, find the probability of getting a ‘J’.
4.In a group of 40students, 2 of them are under 14, 37 of them are under 16, and the rest are 16 or above. If a student is chosen at random, find the probability that the student is 16 or above.
5.There are 280 flats in a building, where 36 of them are occupied by families of two, 80 of them are occupied by families of three, 64 of them are occupied by families of four, 40 of them are occupied by families of five, 28 of them are occupied by families of six or above, and the rest are occupied by peopleliving alone. If a flat is chosen at random, find the probability that the flat is occupied by less than four family members.
6.42 members of a Mathematics club are boys and 12 are girls. 30 members of a Physics club are boys and 8 are girls. If a member is selected from each of the two clubs at random, which club has a higher probability of selecting a girl?
7.There are 204 passengers in an aeroplane. If one of the passengers is selected at random, the probability of selecting an American is. How many American passengers are there in the aeroplane?
8.There are three types of candies, lemon flavour, grape flavour and orange flavour, in a box, where 9packs of them are orange flavour. If a pack of candies is chosen at random, the probability ofchoosing a pack of orange flavour candiesis, how many packs of candiesare therein the box?
Level 2
9.There are 56 students in a music centre. 24 students are learning to play pianos only, 22 are learning to play violins only. The rest of them are learning to play both pianos and violins. If a student is chosen at random,
(a)find the probability that the student is learning to play both pianos and violins.
(b)find the probability that the student is learning to play only one kind of musical instruments.
10.There are 36 students in S3B. Each of them can join only one club. It is given that 6 students have joined the Sport club, 8 students have joined the Computer club, and 12 students have joined the Mathematics club. If a student is chosen at random from S3B, find the probabilities of the following events.
(a)The student is a member of Computer club or Mathematics club.
(b)The student is neither a member of Computer club nor a member of Sport club.
11.The following cumulative frequency curve shows the results of a group of students in an English Language examination. It is known that students fail the examination if their scores are below 50. If one of the students is chosen at random, find the probability of choosing a student who passes the examination.
12.In a factory, there are three light bulb production lines, A, B and C. If a bulb is chosen at random, the probabilities of choosing a bulb produced by production lines A and B areandrespectively.
(a)If production line Aproduced x bulbs, express the number of bulbs produced by production line B in terms of x.
(b)If production line C produced 440 bulbs, find the number of bulbs produced by production line A.
13.For a batch of lottery tickets, the probability of getting one with a prize is. If 240 tickets without prizes are added, the probability of getting one with a prize becomes. Find the original number of lottery tickets.
14.There are some VCDs, DVDs and CDs on a shelf. The number of VCDs is 10 more than that of DVDs, and 4 less than that of CDs. If a disc is selected at random, the probability of selecting a VCD is. Find the number of VCDs on the shelf.
15.In a refrigerator, there are 4 pieces of chocolate cakes, 5 pieces of cheesecakes and 1 piece of mango cake. Suki, Billy and then Jessica each gets a piece of cake at random.
(a)Find the probability for Suki to get a piece of chocolate cake.
(b)If Suki gets a piece of chocolate cake, find the probability for Billy to get a piece of chocolate cake.
(c)If Suki gets a piece of chocolate cake and Billy gets a piece of mango cake, find the probability for Jessica to get a piece of mango cake.
Exercise 7B
Elementary Set
Level 1
1.Over the past 40 school days, Philip was late for school in 2 days. Find the relative frequency of the number of days for being late for school.
2.A chocolate manufacturer inspected 100packets of chocolates and obtained the following results.
Number of chocolates in a packet / 40 / 41 / 42 / 43 / 44 / 45 / 46Frequency / 14 / 10 / 12 / 15 / 16 / 14 / 19
(a)Find the experimental probability of getting a packet with 42chocolates.
(b)If any packet with less than 42chocolates is below standard, find the experimental probability of getting a standard packet.
3.The following frequency distribution table shows the number of daily bus trip of a student in April.
Number of daily bus trip / 0 / 1 / 2 / 3 / 4Frequency / 8 / 10 / 6 / 4 / 2
(a)For this student, find the relative frequencyof not travelling by bus in one day.
(b)For this student, find the relative frequencyof travelling by bus in one day.
4.The following table shows the distribution of the IQ of S3 students of a school in the past three years.
IQ / Number of students86 - 90 / 7
91 - 95 / 74
96 - 100 / 203
101 - 105 / 214
106 - 110 / 106
111 - 115 / 1
(a)Find the relative frequencyofS3 students with IQ liesbetween 96 and 105 inclusive.
(b)Find the relative frequencyofS3 studentsin the school with IQ higher than 105.
5.xeggs are chosen at random from 1600 eggs, of which 4 of them are rotten.
(a)If the relative frequency of rotten eggs out of the chosen ones is, find x.
(b)Among the 1600 eggs, how many rotten eggs are you expecting based on the above situation?
Level 2
6.The following table shows the results of tossing a coin 100 times, 1000 times and 10000 times. Do you think that the coin is fair? Explain briefly.
Number of tosses / Number of heads / Number of tails100 / 64 / 36
1000 / 487 / 513
10000 / 5014 / 4986
7.The following frequency distribution table shows the weights of 100moon cakes measured by a moon cake manufacturer in a survey.
Weight of a moon cake (g) / 201 - 210 / 211 - 220 / 221 - 230 / 231 - 240 / 241 - 250 / 251 - 260Frequency / 11 / 13 / 21 / 16 / 18 / 21
(a)Find the experimental probabilities of each of the following events.
(i)The weight of a moon cake lies between 221g and 230g inclusive.
(ii)The weight of a moon cake lies between 241g and 250g inclusive.
(iii)The weight of a moon cake is more than 230.5g.
(b)Ifthere are 20000mooncakes, estimate the number of moon cakes which weigh more than 230.5g.
8.The following frequency distribution table shows the result of telephone interviews with 1000 people about the paid channel they watchmost frequently in the evening.
Channel / Drama channel / Entertainment channel / News channel / Movie channel / OthersFrequency / 220 / 250 / 200 / 230 / 100
(a)If a person is chosen at random, find the experimentalprobability of each of the following events.
(i)The person watches the news channel most frequently.
(ii)The person watches the dramachannel or entertainment channel most frequently.
(b)If you want to advertise on one of the paid channels in the evening, which channel will you choose? Why?
(c)Given that 300000 people watch paid channels in one evening, estimate the number of people who watch the news channel most frequently.
Advanced Set
Level 1
1.The following table shows the number of times of a group of Japanese tourists visiting Hong Kong.
Number of times / 1 / 2 / 3 / 4 / 5 / 6 or aboveFrequency / 644 / 576 / 325 / 195 / 107 / 153
A Japanese tourist arrives Hong Kong,
(a)find the relative frequency that it is his/her first time visiting Hong Kong.
(b)find the relative frequency that it is his/her fourth time visiting Hong Kong.
2.The following cumulative frequency table shows the number of sleeping hours of Derek each night in November.
Sleeping hours less than / 5 / 6 / 7 / 8 / 9 / 10 / 11Cumulative frequency / 1 / 4 / 10 / 22 / 29 / 29 / 30
(a)Find the relative frequency for Derek to sleep less than 8 hours at night.
(b)Find the relative frequency for Derek to sleep more than or equal to 7 hours but less than 9 hours at night.
3.The following table shows the distribution of the heights of S3 students in a school.
Height (cm) / Number ofstudentsless than 130 / 2
130-139 / 8
140-149 / 32
150-159 / 84
160-169 / 76
170-179 / 30
180 or above / 8
(a)Find the experimental probability that the height of a S3 student lies between 140cm and 169cm inclusive.
(b)Find the experimental probability that the height of a S3 student is less than 140cm.
4.From 2400 electronic components, x of them are chosen at random, of which 3 of them are defective.
(a)If the relative frequency of defective electronic components out of the chosen ones is, find x.
(b)Among the 2400 pieces of electronic components, estimate the number of defective ones based on the above situation.
5.From 1200 copies of a new book, 50 of them are chosen at random, of whichx of them are misprinted,
(a)If the relative frequency of misprinted copies out of the chosen ones is, find x.
(b)Among the 1200 copies of the new book, estimate the number of misprinted copies based on the above situation.
Level 2
6.There are some red balls and black balls in a bag, and they are identical in size and weight. A ball is drawn out at random and then put back into the bag. The following table shows the respective number of red balls and black balls obtained in 100 draws, 1000 draws and 10000 draws.
Number of draws / Number of red balls / Number of black balls100 / 28 / 72
1000 / 358 / 642
10000 / 3052 / 6948
Do you think that there is the same number of red balls and black balls in the bag? Explain briefly.
7.To investigate the number of crows in a district, scientists caught 100 crows, put a ring around a foot of each crow and then released them. After a period of time, 100 crows in the district were caught again of which 8 of them had the rings.
(a)What is the relative frequency of the crows in the district with the rings?
(b)Estimate the number of crows in the district.
8.The following table shows the resultof a survey about the most frequently read newspaper with people.
NewspaperAge of reader / Southern Daily / Moon Daily / Orange Daily / Ping Daily / ChiuDaily
30 or below / 225 / 186 / 174 / 167 / 260
Above 30 / 288 / 94 / 190 / 294 / 122
According to the survey, answer each of the following questions.
(a)Find the probability that a person reads Orange Daily most frequently.
(b)Find the probability that a person above 30 years old reads Southern Daily most frequently.
(c)Among those who read Moon Dailymost frequently, find the probability that the reader is 30 years old or below.
(d)A company wants to advertise in one of the above five newspapers. The cost of the advertisement in Southern Daily, Moon Daily, Orange Daily, Ping Daily and Chiu Daily are $2400, $1600, $2000, $2200 and $1800 respectively. If the company wants to target on youngsters, which newspaper should be chosen for posting the advertisement? Explain briefly.
Exercise 7C
Elementary Set
Level 1
1.List out the sample space for each of the following.
(a)A coin is tossed once.
(b)A coin is tossed twice.
2.Ella is a S3 student. She decides to choose a subject from Chinese History and History, and a subject from Computer Studies, Geography and Economics when she promotes to S4. How many possible combinations are there?
3.There are1 yellow ball and 2white balls in a bag. A ball is drawn at random and then put back into the bag. Then a ball is drawn at random again.
(a)Use a tree diagram to represent the possible outcomes about the two draws.
(b)Find the probability that both balls drawn are white.
4.It is given that the probabilities of giving birth to a baby boy and a baby girl are the same. Vivian has 3children.
(a)Use a tree diagram to represent the possible outcomes about the sex of the 3children.
(b)How many possible outcomes are obtained in (a)?
(c)Find the probability that Vivian has 2sons and 1daughter.
5.Two fair dice are tossed together, find the probabilitythat
(a)the difference is 3.
(b)the product is less than 10.
6.A letter is chosen at random from each of the words ‘APPLE’ and ‘ORANGE’.
(a)List out the sample space in the following table.
O / R / A / N / G / EA
P
P
L
E
(b)Find the probability of each of the following events.
(i)The two letters are the same.
(ii)The two letters are vowels.
Level 2
7.In a game, a player needs to draw a banknote from box A and then a number card from box B. The amount of the cash prize obtained by the player is equal to the product of the face value of the banknote and the number on the card drawn. It is known that there aresix banknotes, $10, $20, $50, $100, $500 and $1000 inbox A, and five number cards, 0, 0.5, 1, 5 and 10 inbox B.
(a)Find the probability of obtaining acash prize of $0.
(b)Find the probability of obtaining a cash prize of $100.
(c)Find the probability of obtaining a cash prize over $500.
8.There are 4 cards labelled with ‘F’, ‘O’, ‘U’ and ‘R’. Two cards are chosen at random one by one without replacement.
(a)List out the sample space in the following table.
2nd card1st card / F / O / U / R
F
O
U
R
(b)Find the probability of each of the following events.
(i)The two cards chosen are the same.
(ii)The two cards chosen can form the word ‘OR’.
9.May has three $20banknotes, two$50banknotes and one $100banknote inside her wallet. Two banknotes are chosen together at random. Find the probability of getting