第1章：有向數data.pakkau.edu.hk/~fankawing/maths_supplementaryex/s3... · web view(ii) the...
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第1章：有向數
1Supplement
7.14數學新里程 中三下 — 初中附加練習
第7章 概率的認識7.41
第7章 概率的認識
【本章各練習均中英對照，以供參考。】
熱身練習
1.下列各圖中，O 是圓心。問圖中陰影部分是圓的幾分之幾？
1.In each of the following figures, O is the centre of the circle, what fraction of the circle is shaded?
(a)
1
8
0
°
O
(b)
6
0
°
O
(c)
2
1
0
°
O
2.求下列各扇形的面積。(答案準確至 3 位有效數字。)
2.Find the area of each of the following sectors. (Correct your answers to 3 significant figures.)
(a)
6
0
°
7
c
m
(b)
4
c
m
1
2
0
°
(c)
2
1
0
°
6
c
m
3.求下列各圖中陰影部分的面積。(如有需要，答案以 ( 表示。)
3.In each of the following figures, find the area of the shaded region. (Express your answers in terms of ( if necessary.)
(a)
1
2
c
m
1
2
c
m
1
8
c
m
1
8
c
m
(b)
4
c
m
4
c
m
(c)
1
6
c
m
2
0
c
m
9
c
m
強化練習
【本部分為書中每個練習額外提供兩種不同的題目組合：「初級組合」和「高級組合」。同學可按其需要選擇完成其中一組 題目。】
練習7A
(((((((((((((((( 初級組合 ((((((((((((((((
程度一
1.中三甲班有 38 名學生，志強是該班的學生。陳老師隨意選出一名學生擔任班長，問選中志強的概率是多少？
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.某百貨公司舉行秋季大抽獎，獎品是一輛私家車。已知該公司收到 12 300 張抽獎券，其中 15 張是王先生的。若該公司從該批抽獎券中隨意抽出一張，求該獎券屬於王先生的概率。
2.There is an autumn lucky draw held by a department store with the prize of a private car. Among the 12 300 lucky draw tickets received by the department store, 15 of them belong to Mr. Wong. If a lucky draw ticket is chosen at random, find the probability that it belongs to Mr. Wong.
3.從「SUPPLEMENT」這個英文字中隨意抽出一個字母，求抽出下列字母的概率。
(a)字母「M」
(b)字母「E」
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.下表是中三甲班 32 名學生生日月份的頻數分佈表。若從該班中隨意抽出一名學生，求抽出學生的生日之月份是二月、四月、六月、八月、十月或十二月的概率。
4.The following frequency distribution table shows the months of birth of 32 students 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.
月份
Month
頻數
Frequency
一月
January
3
二月
February
2
三月
March
3
四月
April
2
五月
May
1
六月
June
2
七月
July
5
八月
August
2
九月
September
4
十月
October
1
十一月
November
4
十二月
December
3
5.魚池 A 和魚池 B 均只飼養了橙色和黑色兩種金魚。已知魚池 A 有 8 條橙色金魚和 13 條黑色金魚，而魚池 B 有 18 條橙色金魚和 26 條黑色金魚。若分別從魚池 A 和魚池 B 隨意抽出一條金魚，問從哪一個魚池抽出黑色金魚的概率較大？
5.There are only orange and black goldfish in fishponds A and B. In fishpond A, there are 8 orange goldfish and 13 black goldfish. In fishpond B, there are 18 orange goldfish and 26 black goldfish. If a goldfish is drawn from each fishpond at random, which fishpond has a higher probability of getting a black goldfish?
6.過去一小時，有 328 名遊客進入海洋 公園。若從該批遊客中隨意選出一人，選中北京遊客的概率是
8
1
。問在過去一小時內有多少名北京遊客進入 海洋公園？
6.Over the past hour, 328 tourists have entered Ocean Park. If one of them is chosen at random, the probability of
choosing a tourist from Beijing is
8
1
.
How many tourists from Beijing have entered Ocean Park over the past hour?
7.中三甲班有 24 名學生戴眼鏡。若從該班中隨意抽出一名學生，該名學生是有戴眼鏡的概率是
13
8
，求中三甲班的學生人數。
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
13
8
, find the number of students in S3A.
程度二
8.某公司舉行的聖誕派對中，有 24 名員工參加，其中 4 名屬人事部，3 名屬電腦部，12 名屬銷售部，其餘屬倉務部。若該公司從參加的員工中隨意抽出一人可得獎，問抽中的員工屬下列部門的概率是多少？
(a)電腦部或人事部。
(b)既不是人事部，也不是倉務部。
8.A Christmas party is held by a company with 24 staff 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.某報攤有 27 名顧客訂購報紙，其中 15 名顧客只訂購一份中文報紙，9 名顧客只訂購一份英文報紙，餘下的則同時訂購中文報紙和英文報紙。若其中一名顧客致電報攤，
(a)問該顧客同時訂購了中文報紙和英文報紙的概率是多少？
(b)問該顧客只訂購一份報紙的概率是多少？
9.There are 27 customers 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.下圖所示為 100 位學生數學科考試成績的累積頻數多邊形。已知得到 80 分或以上的學生可獲得甲等成績。若從該100 位學生隨意選出一位，問選中獲得甲等成績的學生之概率是多少？
10.The following cumulative frequency polygon shows the results of 100 students in 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.
4
0
3
0
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0
1
0
0
1
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0
位
學
生
的
數
學
科
考
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成
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h
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a
M
a
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m
a
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x
a
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o
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數
S
c
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r
e
5
0
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0
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9
0
1
0
0
1
0
2
0
4
0
3
0
6
0
5
0
7
0
8
0
9
0
1
0
0
累
積
頻
數
C
u
m
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l
a
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v
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f
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u
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c
y
11.在一箱玩具火車中，有 x 件由機器 A 製造，餘下的則由機器 B 製造。若從 該箱玩具火車中隨意抽出一件，抽中由機器 A 製造的概率是
8
5
。試以 x 表示由機器 B 製造的玩具火車數目。
11.In a box of toy trains, x of them are made by machine A and the rest are made by machine B. If one toy train is chosen at random, the probability of getting a toy train made by machine A is
8
5
. Express the number of toy trains made by machine B in terms of x.
12.在 100 張汽水獎券中，有 10 張獎券的獎品是公仔一個。問應增添多少張送贈公仔的獎券使隨意抽出一張獎券，其獎品是公仔的概率是
7
1
？
12.Among 100 lucky 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
7
1
, how many tickets with prizes of a doll each should be added?
13.桌上有盒裝檸檬茶和蘋果汁兩款飲品，其中檸檬茶較蘋果汁多 5 盒。若從桌上隨意取走一盒飲品，該盒飲品是檸檬茶的概率是
7
4
，問該桌上原有多少盒檸檬茶？
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
7
4
. How many cartons of lemon tea are there on the table originally?
14.一盒內有若干紅色和藍色原子筆，其中藍色原子筆較紅色原子筆多 3 枝。若從盒中隨意取出一枝原子筆，抽得紅色原子筆的概率是 0.4，求該盒內原有的原子筆總數。
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.
((((((((((((((( 高級組合 (((((((((((((((
程度一
1.已知一合唱團有 24 名成員，當中有 3 名成員的年齡小於 16 歲。若合唱團團長隨意選出一名成員，問選中 16 歲以下的成員之概率是多少？
1.There are 24 members 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.某音樂會有 600 張贈券供市民申請索取，且每人限索取一張。若申請人數超額，主辦機構會隨機把贈券分配給申請人。已知主辦機構收到 1 764 個申請，寶怡是其中一個，求寶怡獲得贈券的概率。
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 1 764 applicants and Rachel is one of them, find the probability for her to obtain a ticket.
3.文傑有以下撲克牌。若他隨意抽出一張牌，求抽得「J」的概率。
3.Robert has the following cards. If he selects a card randomly, find the probability of getting a ‘J’.
4.在 40 名學生中，2 人的年齡小於 14 歲，37 人的年齡小於 16 歲，餘下的則大於或等於 16 歲。若從該 40 名學生中隨意抽出一名學生，求抽中年齡大於或等於 16 歲的學生之概率。
4.In a group of 40 students, 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.某大廈有 280 個住宅單位，其中 36 個是二人家庭，80 個是三人家庭，64 個是四人家庭，40 個是五人家庭，28 個是六人或以上的家庭，其餘是獨居人士。若隨意選出一個住宅單位，求該單位少於四名家庭成員的概率。
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 people living alone. If a flat is chosen at random, find the probability that the flat is occupied by less than four family members.
6.數學學會有 42 名會員是男生，12 名會員是女生。物理學會有 30 名會員是男生，8 名會員是女生。若從該兩個學會內各隨意抽出一名會員，問從哪個學會抽出女生的概率較大？
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.某飛機上有 204 名乘客。若隨意抽出一名乘客，該乘客是美國人的概率是
17
2
。問該飛機上有多少名乘客是美國人？
7.There are 204 passengers in an aeroplane. If one of the passengers is selected at random, the probability of selecting an American is
17
2
. How many American passengers are there in the aeroplane?
8.一盒雜果糖內有檸檬味、葡萄味和香橙味三款糖果，其中香橙味糖果有 9 包。若從盒內隨意抽出一包糖果，該包是香橙味糖果的概率是
8
3
，問該盒內有多少包糖果？
8.There are three types of candies, lemon flavour, grape flavour and orange flavour, in a box, where 9 packs of them are orange flavour. If a pack of candies is chosen at random, the probability of choosing a pack of orange flavour candies is
8
3
, how many packs of candies are there in the box?
程度二
9.某音樂中心有 56 名學生。24 名學生只學習鋼琴，22 名學生只學習小提琴，其餘學生同時學習鋼琴和小提琴。若隨意抽出一名學生，
(a)求該學生同時學習鋼琴和小提琴的概率。
(b)求該學生只學習一種樂器的概率。
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.中三乙班有 36 名學生，其中每名學生只能參加一個學會。已知 6 名學生參加體育學會，8 名學生參加電腦學會，及12 名學生參加數學學會。若從該班隨意抽出一名學生，求下列事件的概率。
(a)該名學生是電腦學會或數學學會的會員。
(b)該名學生既不是電腦學會也不是體育學會的會員。
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.以下累積頻數曲線為一組學生的英國語文科考試成績。已知分數低於 50 便不及格。若隨意抽出一名學生，求抽得一名取得及格的學生之概率。
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.
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12.某工廠有 A、B 和 C 三條生產線負責生產電燈泡。若隨意抽出一個電燈泡，該電燈泡是由生產線 A 和生產線 B 生產的概率分別是
12
5
和
5
2
。
(a)若生產線 A 生產了 x 個電燈泡，試以x 表示生產線 B 生產的電燈泡數目。
(b)若生產線 C 生產了 440 個電燈泡，求生產線 A 生產的電燈泡數目。
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 are
12
5
and
5
2
respectively.
(a)If production line A produced 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.某批獎券中獎的概率為
30
1
。若加入 240 張沒有獎品的獎券，中獎的概率變為
90
1
。求原本獎券的數目。
13.For a batch of lottery tickets, the probability of getting one with a prize is
30
1
. If 240 tickets without prizes are added, the probability of getting one with a prize becomes
90
1
. Find the original number of lottery tickets.
14.已知架上有若干數量的 VCD、DVD 和CD，其中 VCD 比 DVD 多 10 張，且比CD 少 4 張。若隨意選出一張光碟，選中VCD 的概率是
27
10
。求架上 VCD 的數目。
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
27
10
. Find the number of VCDs on the shelf.
15.某雪櫃內有 4 件朱古力蛋糕，5 件芝士蛋糕和 1 件芒果蛋糕。淑潔、業成和 芷珊順序從雪櫃內各隨意拿取 1 件蛋糕。
(a)求淑潔拿取了 1 件朱古力蛋糕的概率。
(b)若淑潔拿取了 1 件朱古力蛋糕，求業成拿取朱古力蛋糕的概率。
(c)若淑潔拿取了 1 件朱古力蛋糕，而 業成拿取了 1 件芒果蛋糕，求芷珊拿取芒果蛋糕的概率。
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.
練習7B
(((((((((((((((( 初級組合 ((((((((((((((((
程度一
1.在過去 40 個上課日，卓文有兩日遲到。求他上學遲到日數的相對頻數。
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.某朱古力製造商檢查了 100 包所生產的一款朱古力糖，所得的結果如下：
2.A chocolate manufacturer inspected 100 packets of chocolates and obtained the following results.
每包內的朱古力糖數量 (粒)
Number of chocolates in a packet
40
41
42
43
44
45
46
頻數
Frequency
14
10
12
15
16
14
19
(a)求取得一包朱古力糖內有 42 粒的實驗概率。
(b)若每包少於 42 粒朱古力糖便屬於不合乎標準，求抽出一包合乎標準的朱古力糖之實驗概率。
(a)Find the experimental probability of getting a packet with 42 chocolates.
(b)If any packet with less than 42 chocolates is below standard, find the experimental probability of getting a standard packet.
3.下表是某學生在四月份每天乘搭巴士的次數：
3.The following frequency distribution table shows the number of daily bus trip of a student in April.
每天乘搭巴士的次數
Number of daily bus trip
0
1
2
3
4
頻數
Frequency
8
10
6
4
2
(a)求該學生於一天內沒有乘搭巴士的相對頻數。
(b)求該學生於一天內有乘搭巴士的相對頻數。
(a)For this student, find the relative frequency of not travelling by bus in one day.
(b)For this student, find the relative frequency of travelling by bus in one day.
4.以下是某中學過往三年中三學生的智商分佈。
4.The following table shows the distribution of the IQ of S3 students of a school in the past three years.
智商
IQ
學生人數
Number of students
86 - 90
7
91 - 95
74
96 - 100
203
101 - 105
214
106 - 110
106
111 - 115
1
(a)求該中學中三學生智商在 96 和 105 之間 (包括 96 和 105) 的相對頻數。
(b)求該中學中三學生智商高於 105 的相對頻數。
(a)Find the relative frequency of S3 students with IQ lies between 96 and 105 inclusive.
(b)Find the relative frequency of S3 students in the school with IQ higher than 105.
5.從 1 600 隻雞蛋中隨意抽出 x 隻雞蛋，其中有 4 隻是壞的。
(a)若在抽出的雞蛋中，壞蛋的相對頻數是
8
1
，求 x。
(b)根據上述情況，在 1 600 隻雞蛋中，你估計有多少隻是壞的？
5.x eggs are chosen at random from 1 600 eggs, of which 4 of them are rotten.
(a)If the relative frequency of rotten eggs out of the chosen ones is
8
1
, find x.
(b)Among the 1 600 eggs, how many rotten eggs are you expecting based on the above situation?
程度二
6.以下是投擲一枚硬幣 100 次、1 000 次和10 000 次的結果。你認為這枚硬幣是否均質？試簡單解釋。
6.The following table shows the results of tossing a coin 100 times, 1 000 times and 10 000 times. Do you think that the coin is fair? Explain briefly.
投擲次數
Number of tosses
「公」的次數
Number of heads
「字」的次數
Number of tails
100
64
36
1 000
487
513
10 000
5 014
4 986
7.以下頻數分佈表顯示某月餅製造公司在一次調查中量得 100 件月餅的重量。
7.The following frequency distribution table shows the weights of 100 moon cakes measured by a moon cake manufacturer in a survey.
月餅重量 (g)
Weight of a moon cake (g)
201 - 210
211 - 220
221 - 230
231 - 240
241 - 250
251 - 260
頻數
Frequency
11
13
21
16
18
21
(a)求下列事件的實驗概率。
(i)
月餅的重量介乎 221 g 和 230 g之間 (包括 221 g 和 230 g)。
(ii)
月餅的重量介乎 241 g 和 250 g之間 (包括 241 g 和 250 g)。
(iii)月餅的重量超過 230.5 g。
(b)若有20 000件月餅，試估計重量超過230.5 g的月餅數目。
(a)Find the experimental probabilities of each of the following events.
(i)The weight of a moon cake lies between 221 g and 230 g inclusive.
(ii)The weight of a moon cake lies between 241 g and 250 g inclusive.
(iii)The weight of a moon cake is more than 230.5 g.
(b)If there are 20 000 moon cakes, estimate the number of moon cakes which weigh more than 230.5 g.
8.以下頻數分佈表顯示用電話訪問1 000 名市民關於黃昏時段最常收看的收費電視頻道。
8.The following frequency distribution table shows the result of telephone interviews with 1 000 people about the paid channel they watch most frequently in the evening.
電視頻道
Channel
劇集台
Drama channel
娛樂台
Entertainment channel
新聞台
News channel
電影台
Movie channel
其他
Others
頻數
Frequency
220
250
200
230
100
(a)若隨意抽出一位市民，求下列各事件的實驗概率。
(i)
該市民最常收看新聞台。
(ii)
該市民最常收看劇集台或娛樂台。
(b)如果你打算於黃昏時段在其中一個收費電視頻道登廣告，你會選擇哪一個頻道呢？為甚麼？
(c)已知某天全港有 300 000 人於黃昏時段收看收費電視。試估計當中最常收看新聞台的人數。
(a)If a person is chosen at random, find the experimental probability of each of the following events.
(i)
The person watches the news channel most frequently.
(ii)
The person watches the drama channel 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 300 000 people watch paid channels in one evening, estimate the number of people who watch the news channel most frequently.
((((((((((((((( 高級組合 (((((((((((((((
程度一
1.下表所示為一群日本旅客來香港旅遊的次數。
1.The following table shows the number of times of a group of Japanese tourists visiting Hong Kong.
次數
Number of times
1
2
3
4
5
6或以上
6 or above
頻數
Frequency
644
576
325
195
107
153
一名日本旅客剛抵達香港，
(a)求該名旅客是第一次來香港旅遊的相對頻數。
(b)求該名旅客是第四次來香港旅遊的相對頻數。
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.以下的累積頻數分佈表顯示了嘉威在十一月的每晚睡眠時間。
2.The following cumulative frequency table shows the number of sleeping hours of Derek each night in November.
睡眠時間少於 (小時)
Sleeping hours less than
5
6
7
8
9
10
11
累積頻數
Cumulative frequency
1
4
10
22
29
29
30
(a)求嘉威晚間睡眠少於 8 小時的相對頻數。
(b)求嘉威晚間睡眠多於或等於 7 小時但少於 9 小時的相對頻數。
(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.下表所示為某校中三級學生的身高分佈。
3.The following table shows the distribution of the heights of S3 students in a school.
身高 (cm)
Height (cm)
學生人數
Number of students
少於130
less than 130
2
130 - 139
8
140 - 149
32
150 - 159
84
160 - 169
76
170 - 179
30
180或以上
180 or above
8
(a)求一名中三級學生身高介乎 140 cm 和 169 cm 之間 (包括 140 cm 和 169 cm) 的實驗概率。
(b)求一名中三級學生身高少於 140 cm的實驗概率。
(a)Find the experimental probability that the height of a S3 student lies between 140 cm and 169 cm inclusive.
(b)Find the experimental probability that the height of a S3 student is less than 140 cm.
4.從 2 400 件電子零件中隨意抽出 x 件，當中有 3 件是損壞的。
(a)若在抽出的電子零件中，損壞的電子零件之相對頻數是
75
1
，求 x。
(b)根據上述情況，你估計在該 2 400 件電子零件中，有多少件是損壞的？
4.From 2 400 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
75
1
, find x.
(b)Among the 2 400 pieces of electronic components, estimate the number of defective ones based on the above situation.
5.從 1 200 本新印刷的書中隨意抽出50 本，發現其中 x 本是錯版。
(a)若在抽出的書中，錯版書的相對頻數是
25
1
，求 x 的值。
(b)根據上述情況，在 1 200 本新印刷的書中，你估計有多少本是錯版的？
5.From 1 200 copies of a new book, 50 of them are chosen at random, of which x of them are misprinted,
(a)If the relative frequency of misprinted copies out of the chosen ones is
25
1
, find x.
(b)Among the 1 200 copies of the new book, estimate the number of misprinted copies based on the above situation.
程度二
6.某袋中有若干數目的紅球和黑球，它們的大小和重量皆相同。從袋中隨意抽出一個球後再放回袋中。下表所示為抽球100次、1 000 次和 10 000 次中，抽出紅球和黑球的數目。
抽球1
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, 1 000 draws and 10 000 draws.
抽球次數
Number of draws
紅球數目
Number of red balls
黑球數目
Number of black balls
100
28
72
1 000
358
642
10 000
3 052
6 948
你認為袋中紅球和黑球的數目相同嗎？試簡單解釋。
Do you think that there is the same number of red balls and black balls in the bag? Explain briefly.
7.為了調查某地區的烏鴉數目，科學家捕捉了 100 隻烏鴉，然後在牠們的腳上裝上一個環，再把牠們放走。過了一段時間，在該地區再捕捉 100 隻烏鴉，發現當中有 8 隻裝有該種環。
(a)求該地區內裝有該種環的烏鴉之相對頻數。
(b)試估計該地區內烏鴉的數目。
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.下表所示為一個關於市民最常閱讀的報章之調查結果。
8.The following table shows the result of a survey about the most frequently read newspaper with people.
報章
Newspaper
讀者年齡
Age of reader
南方日報
Southern Daily
月亮報
Moon Daily
橙報
Orange Daily
平報
Ping Daily
潮報
Chiu Daily
30歲或以下
30 or below
225
186
174
167
260
30歲以上
Above 30
288
94
190
294
122
根據該結果，回答下列各題。
(a)求一名市民最常閱讀橙報的概率。
(b)求一名三十歲以上的市民最常閱讀南方日報的概率。
(c)在最常閱讀月亮報的讀者中，求讀者年齡是三十歲或以下的概率。
(d)某公司打算在該五份報章的其中 一份中刊登廣告。在南方日報、 月亮報、橙報、平報和潮報刊登該廣告的費用分別是 $2 400、$1 600、$2 000、$2 200 和 $1 800。若該公司的對象主要是年青人，應選擇在哪一份報章刊登廣告呢？試簡單解釋。
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 Daily most 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 $2 400, $1 600, $2 000, $2 200 and $1 800 respectively. If the company wants to target on youngsters, which newspaper should be chosen for posting the advertisement? Explain briefly.
練習7C
(((((((((((((((( 初級組合 ((((((((((((((((
程度一
1.試列出下列情況的樣本空間。
(a)擲一枚硬幣一次。
(b)擲一枚硬幣兩次。
1.List out the sample space for each of the following.
(a)A coin is tossed once.
(b)A coin is tossed twice.
2.恩娜是一名中三學生。她打算升讀中四時，從中國歷史科和歷史科中選一科修讀，及從電腦科、地理科和經濟科中選一科修讀。問有多少種可能的組合？
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.某袋裡有 1 個黃球和 2 個白球。從該袋中隨意抽出一個球，然後放回袋中，再隨意抽出一個球。
(a)試用樹形圖表示兩次抽球的可能結果。
(b)求兩次均抽出白球的概率。
3.There are 1 yellow ball and 2 white 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.已知生育男嬰和女嬰的機會均等。慧敏有 3 名孩子。
(a)試用樹形圖表示該 3 名孩子性別的可能結果。
(b)在 (a) 小題中，共得出多少個可能結果？
(c)求慧敏有 2 名兒子和 1 名女兒的概率。
4.It is given that the probabilities of giving birth to a baby boy and a baby girl are the same. Vivian has 3 children.
(a)Use a tree diagram to represent the possible outcomes about the sex of the 3 children.
(b)How many possible outcomes are obtained in (a)?
(c)Find the probability that Vivian has 2 sons and 1 daughter.
5.擲兩粒均質的骰子，求下列事件的概率。
(a)兩粒骰子的點數之差是 3。
(b)兩粒骰子的點數之積小於 10。
5.Two fair dice are tossed together, find the probability that
(a)the difference is 3.
(b)the product is less than 10.
6.從「APPLE」和「ORANGE」兩個英文字中各隨意抽出一個字母。
(a)試在下表列出樣本空間。
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
E
A
P
P
L
E
(b)求下列事件的概率。
(i)兩個字母是相同的。
(ii)兩個字母都是元音字母。
(b)Find the probability of each of the following events.
(i)The two letters are the same.
(ii)The two letters are vowels.
程度二
7.某遊戲中，參加者先從 A 箱隨意抽出一張紙幣，然後從 B 箱隨意抽出一張數字卡。參加者贏取的獎金為紙幣的面額與卡上數字之積。已知 A 箱有 $10、$20、$50、$100、$500 和 $1 000 紙幣各一張；B 箱有數字卡 0、0.5、1、5 和 10 各一張。
(a)求獎金為 $0 的概率。
(b)求獎金為 $100 的概率。
(c)求獎金多於 $500 的概率。
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 are six banknotes, $10, $20, $50, $100, $500 and $1 000 in box A, and five number cards, 0, 0.5, 1, 5 and 10 in box B.
(a)Find the probability of obtaining a cash 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.有 4 張分別寫上「F」、「O」、「U」和 「R」的紙牌。現隨意抽出兩張牌，且抽出的牌不放回。
(a)試在下表中列出樣本空間。
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.
第二張牌
2nd card
第一張牌
1st card
F
O
U
R
F
O
U
R
(b)求下列各事件的概率。
(i)抽出兩張相同的牌。
(ii)抽出的兩張牌可組成英文字「OR」。
(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.小美的錢包內有三張 $20 紙幣、兩張 $50 紙幣和一張 $100 紙幣。現從錢包內同時隨意抽出兩張紙幣，求下列事件的概率。
(a)抽出的兩張紙幣面額相同。
(b)抽出的兩張紙幣面額不同。
9.May has three $20 banknotes, two $50 banknotes and one $100 banknote inside her wallet. Two banknotes are chosen together at random. Find the probability of getting
(a)two banknotes with equal face value.
(b)two banknotes with different face values.
10.某超級市場訪問了 100 位顧客，調查他們最常購買的食品類別。
10.A supermarket interviewed with 100 customers to find out the type of food that they bought most frequently.
食品類別
Type of food
水果
Fruit
麵類
Noodle
小食
Snack
飲品
Drink
冷凍食品
Frozen food
頻數
Frequency
18
17
28
32
5
從該 100 位顧客中隨意選出兩位。
(a)試完成以下樹形圖。
Two customers are chosen at random from the 100 customers.
(a)Complete the following tree diagram.
購
買
飲
品
B
u
y
i
n
g
d
r
i
n
k
購
買
其
他
B
u
y
i
n
g
o
t
h
e
r
s
購
買
飲
品
B
u
y
i
n
g
d
r
i
n
k
購
買
飲
品
B
u
y
i
n
g
d
r
i
n
k
購
買
其
他
B
u
y
i
n
g
o
t
h
e
r
s
購
買
其
他
B
u
y
i
n
g
o
t
h
e
r
s
顧
客
數
目
N
u
m
b
e
r
o
f
c
u
s
t
o
m
e
r
s
顧
客
數
目
N
u
m
b
e
r
o
f
c
u
s
t
o
m
e
r
s
1
8
+
1
7
+
2
8
+
5
=
6
8
6
8
-
1
=
6
7
(b)(i)求抽出兩位最常購買飲品的顧客之結果數目。
(ii)求抽出兩位顧客的可能結果總數。
(c)求抽出兩位最常購買飲品的顧客之概率。
(b)(i)Find the number of outcomes of getting two customers who buy drink most frequently.
(ii)Find the total number of possible outcomes of getting two customers.
(c)Find the probability of getting two customers who buy drink most frequently.
((((((((((((((( 高級組合 (((((((((((((((
程度一
1.抽屜內有 2 條紅色頸巾和 3 條白色頸巾。美嘉隨意抽出一條頸巾，再把它放回，然後再隨意抽出一條頸巾。
(a)求兩次均抽中同一條頸巾的概率。
(b)求抽中相同顏色頸巾的概率。
1.There are 2 red scarves and 3 white scarves in a drawer. Mary takes a scarf out at random and then puts it back. Then she takes a scarf out at random again.
(a)Find the probability that both scarves taken are the same scarf.
(b)Find the probability that both scarves taken are in the same colour.
2.有若干個三位數，當中出現的數字不是「2」便是「3」。
(a)列出所有符合以上條件的三位數。
(b)若從 (a) 小題所得的數中隨意抽出一個，求下列各事件的概率。
(i)只有其中一個位的數字是「2」。
(ii)該數是偶數。
2.There are some 3-digit numbers with digits of either ‘2’ or ‘3’.
(a)List out all 3-digit numbers that satisfy the above condition.
(b)If a number is chosen at random from the numbers obtained in (a), find the probability of each of the following events.
(i)Only one of the digits is ‘2’.
(ii)The number is an even number.
3.擲兩粒均質骰子，下列哪件事件發生的機會較高？
事件 A：兩個點數是連續數。
事件 B：兩個點數相同。
3.Two fair dice are tossed. Which of the following events is more likely to occur?
Event A:The two numbers are consecutive numbers.
Event B:The two numbers are the same.
4.從英文字「ABILITY」和「DISABLE」中各隨意抽出一個字母。
(a)試在下表中列出樣本空間。
4.A letter is chosen at random from each of the words ‘ABILITY’ and ‘DISABLE’.
(a)List out the sample space in the following table.
D
I
S
A
B
L
E
A
B
I
L
I
T
Y
(b)求下列各事件的概率。
(i)兩個字母是相同的。
(ii)兩個字母都是輔音字母。
(b)Find the probability of each of the following events.
(i)The two letters are the same.
(ii)The two letters are consonants.
程度二
5.某袋中有2個紅球、2個黑球和3個白球，分別以R1、R2、B1、B2、W1、W2和W3表示。現隨意抽出兩個球且抽出的球不放回袋中。
(a)試在下表中列出樣本空間。
5.There are 2 red balls, 2 black balls and 3 white balls in a bag and they are represented by R1, R2, B1, B2, W1, W2 and W3. Two balls are chosen randomly one by one without replacement.
(a)List out the sample space in the following table.
第二個球
2nd ball
第一個球
1st ball
R1
R2
B1
B2
W1
W2
W3
R1
R2
B1
B2
W1
W2
W3
(b)求下列各事件的概率。
(i)
抽出兩個黑球。
(ii)
抽出兩個白球。
(iii)抽出兩個相同顏色的球。
(iv)抽出一個黑球和一個白球。
(b)Find the probability of each of the following events.
(i)Two black balls are drawn.
(ii)Two white balls are drawn.
(iii)Two balls of the same colour are drawn.
(iv)One black ball and one white ball are drawn.
6.某盒內有 4 張卡，分別寫上 1、3、5 和 7。從該盒內同時隨意抽出兩張卡，求它們之和是 8 的概率。
6.There are 4 cards numbered as 1, 3, 5 and 7 in a box. Two cards are drawn together at random. Find the probability of getting a sum of 8.
7.在 10 份禮物中，有 1 份是公仔、2 份是朱古力、3 份是顏色筆、4 份是故事書。若美賢從該 10 份禮物中同時隨意抽出2 份，問她抽中故事書的概率較沒有抽中故事書的概率高多少？
7.There are 10 prizes, where 1 of them is a doll, 2 of them are chocolate, 3 of them are colour pencils and 4 of them are storybooks. If Macy chooses 2 prizes together at random, how much higher is the probability of getting a storybook than that of not getting a storybook?
8.下表所示為最受歡迎女歌手選舉中，其中 100 人的投票結果。從該 100 人中隨意抽出 2 人。
8.The following table shows the voting result of 100 people for the most popular female singer. Two people are chosen at random from the 100 people.
歌手
Singer
秀明
Jenny
嘉欣
Karen
詠儀
Esther
芷珊
Sophia
少芬
Ada
頻數
Frequency
33
28
18
12
9
(a)試完成以下的樹形圖。
(a)Complete the following tree diagram.
投
秀
明
一
票
V
o
t
e
f
o
r
J
e
n
n
y
不
投
秀
明
一
票
N
o
t
v
o
t
e
f
o
r
J
e
n
n
y
投
秀
明
一
票
V
o
t
e
f
o
r
J
e
n
n
y
投
秀
明
一
票
V
o
t
e
f
o
r
J
e
n
n
y
不
投
秀
明
一
票
N
o
t
v
o
t
e
f
o
r
J
e
n
n
y
不
投
秀
明
一
票
N
o
t
v
o
t
e
f
o
r
J
e
n
n
y
人
數
N
u
m
b
e
r
o
f
p
e
o
p
l
e
人
數
N
u
m
b
e
r
o
f
p
e
o
p
l
e
3
3
3
3
-
1
=
3
2
(b)(i)求抽出兩人都是投秀明一票的可能結果數目。
(ii)求抽出兩人的可能結果總數。
(c)求抽出的兩人都是投秀明一票的概率。
(b)(i)Find the number of possible outcomes of getting two people who vote for Jenny.
(ii)Find the total number of possible outcomes of getting two people.
(c)Find the probability of getting two people who vote for Jenny.
9.某袋中有 1 個紅球和 2 個白球。隨意抽出一球後，把 1 個紅球和 1 個白球加入袋中，然後再隨意抽出一球。
(a)求抽出兩個不同顏色的球之概率。
(b)求第一次抽出白球，第二次抽出紅球的概率。
9.There are 1 red ball and 2 white balls in a bag. After a ball is drawn at random, 1 red ball and 1 white ball are put into the bag, and then a ball is drawn at random again.
(a)Find the probability of getting two balls in different colours.
(b)Find the probability of getting a white ball first and then a red ball.
練習7D
(((((((((((((((( 初級組合 ((((((((((((((((
程度一
1.下圖是一張正方形卡紙，分成 I、II、III和 IV 四個相等的區域。若用大頭釘隨意在卡紙上釘一點，求這點位於區域 I的概率。
1.The figure shows a square card equally divided into four regions I, II, III and IV. If a point is marked on the card at random by a pin, find the probability that the point locates in region I.
I
I
I
I
V
I
I
I
2.下圖為一個直徑 20 cm 的圓形鏢靶，當中有一個圓內接正方形。若一飛鏢隨意擲中鏢靶上且不擲中任何邊界，求擲中正方形區域的概率。(答案以 ( 表示。)
2.A circular dartboard with a diameter of 20 cm is shown with a square inscribed in it. Suppose a dart hits on the circular dartboard randomly without hitting on any boundaries, find the probability that it hits on the square region. (Express your answer in terms of (.)
2
0
c
m
3.一塊長方形木板闊 0.5 m，長 0.8 m。木板上有四個大小相同的圓，每個圓的面積均是 40 cm2。若向木板隨意擲一飛鏢，並擲中木板，
3.The figure shows a rectangular dartboard with the width of 0.5 m and length of 0.8 m. There are four identical circles with an area of 40 cm2 each on the dartboard. If a dart is thrown towards the dartboard at random and it hits the dartboard,
0
.
8
m
0
.
5
m
A
C
D
B
(a)求擲中圓 A 的概率。
(b)求擲中其中一個圓的概率。
(a)find the probability that the dart hits on circle A.
(b)find the probability that the dart hits on a circle.
4.下圖所示為一塊正方形瓷磚，其邊長為10 cm，瓷磚中心有一個半徑為 2 cm 的圓。若放下一粒彈珠，並讓它在正方形瓷磚內自由地滾動，直至停下來為止，求彈珠不是停在圓上的概率。(答案以 ( 表示。)
4.The figure shows a square tile with sides of 10 cm each. There is a circular region with a radius of 2 cm on the tile. If a marble is rolling freely on the square tile until it stops, find the probability that it does not stop within the circular region. (Express your answer in terms of (.)
1
0
c
m
1
0
c
m
2
c
m
5.在一場足球比賽中，勝方可得 3 分，賽和各得 1 分，負方則沒有分數。小虎隊在比賽中勝出、賽和及落敗的概率分別是 0.4、0.3 和 0.3。
(a)求該隊每場比賽得分的期望值。
(b)試估計該隊進行了 20 場比賽後的總得分。
5.The winning team scores 3 points and the losing team scores nothing in a football match. If two teams draw, each of them scores 1 point. The probabilities for Tiger Team to win, draw and lose in a match are 0.4, 0.3 and 0.3 respectively.
(a)Find the expected value of the scores obtained by the team in each match.
(b)Estimate the total scores obtained by the team after 20 matches.
程度二
6.下圖為一直徑 15 cm 的半圓形鏢靶，其中陰影部分為直徑 5 cm 的半圓。現隨意擲一飛鏢至鏢靶，擲中陰影部分得 5 分，擲中白色部分得 2 分。
6.The figure shows a semi-circular dartboard with the diameter of 15 cm. The shaded region is a semi-circle with the diameter of 5 cm. Now a dart is thrown and hits on the dartboard randomly, 5 marks will be obtained if it hits on the shaded region, and 2 marks will be obtained if it hits on the white region.
1
5
c
m
5
c
m
(a)求擲中陰影部分的概率。
(b)求得 2 分的概率。
(c)求每次投擲所得分數的期望值。
(a)Find the probability of hitting the shaded region.
(b)Find the probability of getting 2 marks.
(c)Find the expected value of each throw.
7.聯歡會進行抽獎活動，共有 500 張抽獎券，其中頭獎 1 名，可得獎金 $5 000；二獎 3 名，各得獎金 $1 000；三獎 5 名，各得獎金 $500；安慰獎 10 名，各得獎金 $100。若小生持有一張抽獎券，
(a)求他獲得頭獎的概率。
(b)求他獲獎的概率。
(c)求他的獎券之期望值。
7.There are 500 lucky draw tickets distributed in a party with one first prize of $5 000 in cash. There are two second prizes of $1 000 in cash each, five third prizes of $500 in cash each, and ten consolation prizes of $100 in cash each. If Stanley has a lucky draw ticket,
(a)find the probability that he will win the first prize.
(b)find the probability that he will win a prize.
(c)find the expected value of his lucky draw ticket.
8.在一份考試卷中，有 40 條多項選擇題。每條題目均有 4 個選項，當中只有一個是正確答案。學生每答對一條，可得 x 分，答錯會扣 y 分。若某學生全部題目均隨意選取答案，得分的期望值是 0 分。求 x : y。
8.There are 40 multiple choice questions in an examination paper. Each question has 4 options in which only 1 of them is correct. x marks will be scored for each correct answer, and y marks will be deducted for an incorrect one. If a student answers all the questions by guessing randomly, the expected value of his score will be 0. Find x : y.
9.張先生上班可選乘巴士、地鐵或小巴，所需時間分別為 48 分鐘、30 分鐘和 40 分鐘。過去 20 個工作天中，他有 7 天乘搭巴士，10 天乘搭地鐵及 3 天乘搭小巴。
(a)求張先生上班所需時間的期望值。
(b)若每程巴士、地鐵和小巴的車費分別為 $8.5、$12 和 $10，求張先生上班所需車費的期望值。
9.Mr. Cheung can go to work by bus, MTR or minibus using 48 minutes, 30 minutes and 40 minutes respectively. Over the past 20 working days, he travelled by bus, MTR and minibus in 7 days, 10 days and 3 days respectively.
(a)Find the expected value of the travelling time to work of Mr. Cheung.
(b)If the fares of bus, MTR and minibus are $8.5, $12 and $10 respectively, find the expected value of the fare to work of Mr. Cheung.
10.某遊戲中，每回合參加者需同時擲兩枚均質硬幣。若擲得兩個「公」，參加者便可得 10 分。若擲得兩個「字」，參加者便被扣 20 分。若擲得一「公」一「字」，便可得到 5 分。求十個回合後所得分數的期望值。
10.In a game, a player requires to toss two fair coins in each round. If two heads are shown, the player will get 10 points. If two tails are shown, 20 points will be deducted. If one head and one tail are shown, the player will get 5 points. Find the expected value of the result obtained after ten rounds.
((((((((((((((( 高級組合 (((((((((((((((
程度一
1.下圖所示為一張被分為六等份的正六邊形紙板。若在該紙板上隨意畫上一點，且該點不會畫在邊界上，求該點畫在陰影部分上的概率。
1.The figure shows a regular hexagonal paper divided into six equal regions. If a point is marked on the paper at random, without touching any boundaries, find the probability that the point is marked on the shaded region.
2.下圖所示為一圓形鏢靶，其中一個大小為 40 cm ( 80 cm 的長方形內接於該鏢靶內。若隨意擲出一枚飛鏢並擊中鏢靶，且飛鏢不會落在邊界上，求擊中陰影部分的概率。
2.A circular dartboard is shown in the figure. A rectangle with dimensions 40 cm ( 80 cm is inscribed in the dartboard. If a dart is thrown and hits on the circular dartboard randomly without hitting on any boundaries, find the probability that it hits on the shaded region.
8
0
c
m
4
0
c
m
3.下圖是一幅大小為 50 cm ( 80 cm 的圖畫。圖畫上有 4 個圖案，分別標示為I、II、III 和 IV。圖案 I 是一個等腰直角三角形，其中兩條相等邊的長度均是15 cm。圖案 II 的面積是圖案 I 的兩倍。圖案 III 是一個邊長 10 cm 的正方形。圖案 IV 是一個直徑 30 cm 的半圓形。一隻昆蟲隨意落在該幅畫上，且不落在任何邊界。
3.A rectangular picture with dimensions 50 cm ( 80 cm is shown. There are 4 figures in the picture, labelled I, II, III and IV. Figure I is an isosceles right-angled triangle, where the two equal sides are 15 cm each. The area of figure II is twice as large as that of figure I. Figure III is a square with sides of 10 cm each. Figure IV is a semi-circle with a diameter of 30 cm. It is given that a bug rests on the picture at random without touching any boundaries.
8
0
c
m
5
0
c
m
I
I
I
I
I
I
I
V
(a)求昆蟲落在圖案 II 或圖案 IV 的概率。
(b)求昆蟲不會落在圖案 I、II、III 和 IV中任何一個圖案的概率。
(答案以 ( 表示。)
(a)Find the probability that the bug rests on either figure II or IV.
(b)Find the probability that the bug rests on neither figures I, II, III nor IV.
(Express your answers in terms of (.)
4.永倫的錢包內有三張 $20 紙幣、四張 $50 紙幣和兩張 $100 紙幣。若他隨意取出一張紙幣，求取出紙幣的金額之期望值。
4.Alan has three $20 banknotes, four $50 banknotes and two $100 banknotes in his wallet. If he picks out a banknote at random, find the expected value of the amount of the banknote.
程度二
5.下圖中的圓形鏢靶，是由兩個半徑分別為 8 cm 和 16 cm 的同心圓組成。若飛鏢擲中區域 I 或 II，可得 20 分。若飛鏢擲中區域 III 或 IV，可得 5 分。擲中白色區域則不獲任何分數。若隨意擲出一枚飛鏢且擲中鏢靶，而飛鏢不會擲中邊界，
5.The figure shows a circular dartboard formed by two concentric circles with radii of 8 cm and 16 cm. If a dart hits on region I or II, 20 points will be scored. If a dart hits on region III or IV, 5 points will be scored. No points will be scored if the dart hits on the white regions. If a dart is thrown and hits on the dartboard randomly without hitting on any boundaries,
I
V
I
I
I
I
I
I
(a)分別求擲中區域 I、II、III 和 IV 的概率。
(b)求所得分數的期望值。
(a)find the respective probabilities of hitting regions I, II , III and IV.
(b)find the expected value of the points scored.
6.每天早上，嘉儀會以一個售價 $6 的麵包、三件共售 $14 的熱香餅或一碗售價 $20 的麥皮作為她的早餐。過去 30 天的早上，她有 6 天吃麵包、15 天吃熱香餅和9 天吃麥皮作早餐。
(a)求嘉儀每天花在早餐的金額之期望值。
(b)若嘉儀花在吃一個麵包、三件熱香餅和一碗麥皮的時間分別是 5 分鐘、12 分鐘和 15 分鐘，求她每天吃早餐的時間之期望值。
6.Every morning Denise buys either a bread costing $6, 3 pancakes costing $14, or a bowl of cereal costing $20 for breakfast. Over the past 30 days, she had bread for 6 days, pancakes for 15 days and cereal for 9 days.
(a)Find the expected value of the amount spent by Denise on breakfast each day.
(b)If Denise spent 5 minutes, 12 minutes and 15 minutes to finish a bread, 3 pancakes and a bowl of cereal respectively, find the expected value of the time spent by Denise on breakfast each day.
7.某抽獎遊戲需要參加者付 $20。根據遊戲規則，參加者須在一個放有 1 個紅球和 3 個白球的盒裡隨意抽出一個球後放回箱裡再隨意抽第二個。若兩個球均是紅色，可贏得 $100；若兩個球顏色不同，可贏得 $20；若兩個球均是白色，則沒有獎金。
(a)求抽出兩個紅球的概率。
(b)求抽出不同顏色的球的概率。
(c)求抽出兩個白球的概率。
(d)求所得獎金的期望值。
(e)你會參加該遊戲嗎？為甚麼？
7.The fee for joining a lucky draw is $20. According to the rule, a player should draw a ball at random from a bag containing 1 red ball and 3 white balls, put the ball back into the bag and draw a ball at random again. If two red balls are drawn, the player obtains $100. If two balls in different colours are drawn, the player obtains $20. No cash prize is given for drawing two white balls.
(a)Find the probability of getting two red balls.
(b)Find the probability of getting two balls in different colours.
(c)Find the probability of getting two white balls.
(d)Find the expected value of the cash prize obtained.
(e)Will you join the lucky draw? Why?
8.下圖中的圓形鏢靶，是由三個半徑為x cm、12 cm 和 z cm 的同心圓組成，其中 x ( 12 ( z。若飛鏢擲中區域 I，可得1 分。若飛鏢擲中區域 III，扣 4 分。擲中區域 II 不會得到或扣除任何分數。若隨意擲出一枚飛鏢且擲中鏢靶，而飛鏢不會擲中邊界，則所得分數的期望值是0 分。
8.The figure shows a circular dartboard formed by three concentric circles with radii of x cm, 12 cm and z cm, where x ( 12 ( z. If a dart hits on region I, 1 point will be scored. If a dart hits on region III, 4 points will be deducted. No points will be scored or deducted when a dart hits on region II. If a dart is thrown and hits on the dartboard at random without hitting on any boundaries, the expected value of the points scored is 0.
I
I
I
I
I
I
(a)試以 z 表示 x。
(b)求所有 x 和 z 的可能整數值。
(a)Express x in terms of z.
(b)Find all the possible integral values of x and z.
9.某火車站於繁忙時段，每隔兩分鐘便有一班列車到站，且每班列車會在站內停留 20 秒。下圖所示為該站繁忙時段內的其中 8 分鐘，陰影部分代表列車停留在站內時間。
9.During the peak hours, a train arrives at a railway station every 2 minutes and each train stays in the station for 20 seconds. A period of 8 minutes during the peak hours is shown in the following figure in which the shaded region represents the duration of a train staying in the station.
0
1
2
3
4
5
6
7
8
時
間
(
分
鐘
)
T
i
m
e
(
m
i
n
u
t
e
)
(a)試在圖中把列車停在站內的時段塗上陰影。
【附錄已提供上圖的複本。】
(b)偉文在不知列車班次的情況下於繁忙時段抵達月台。求他抵達月台時剛巧有列車停在站內的概率。
(a)In the figure, shade the regions representing the duration of trains staying in the station.
[ A copy of the figure is provided in the Appendix. ]
(b)Raymond arrives at the platform of the station during the peak hours without knowing the timetable of trains. Find the probability that a train stays in the station when he arrives.
本章測驗
(時限：1小時)
甲部 (1) [每題3分]
1.在 12 張顏色卡紙中，有 3 張是紅色， 5 張是藍色，其餘是白色。現隨意抽一張卡紙，問抽到紅色卡紙的概率是多少？
1.Among 12 coloured cards, 3 of them are red, 5 are blue and the rest are white. A card is chosen at random. What is the probability of getting a red card?
2.已知生育男嬰和女嬰的機會均等，慧珊有兩名孩子，求兩名均是男孩子的概率。
2.It is given that the probabilities of giving birth to baby boys and baby girls are equal. Flora has two children. Find the probability that both of them are boys.
3.從整數 20 至 40 中 (包括 20 和 40) 隨意抽一個數，求該數不能被 4 整除的概率。
3.A number is chosen at random from integers 20 to 40 inclusive. Find the probability that the number is not divisible by 4.
4.下圖為一個圓形鏢靶，分為 A 和 B 兩部分，其中 O 是圓形鏢靶的圓心。若一飛鏢隨意擲中鏢靶，且飛鏢不會擲在邊界上，求擲中 A 部分的概率。
4.The figure shows a dartboard divided into two regions, A and B. O is the centre of the dartboard. If a dart is thrown and hits on the dartboard randomly without hitting on any boundaries, find the probability that it hits on region A.
3
x
°
x
°
A
B
O
5.下表所示為六箱橙中壞橙的數量，其中每箱有 50 個橙。把該六箱橙混合，然後從中隨意抽出一個橙，求抽出壞橙的概率。
5.The following table shows the number of rotten oranges in six boxes each contains 50 oranges. If an orange is chosen at random after mixing up the six boxes of oranges, find the probability of getting a rotten orange.
箱
Box
A
B
C
D
E
F
頻數
Frequency
6
2
1
1
3
7
6.下表為一組學生在某次測驗的得分：
6.The following table shows the distribution of scores of a group of students in a test.
得分
Score
49或以下
49 or below
50 - 59
60 - 69
70 - 79
80 - 89
90或以上
90 or above
頻數
Frequency
12
10
8
5
4
1
已知及格分數是 60 分。若從該組學生中隨意抽出一位，求該學生是取得及格成績的概率。
Given that 60 is the passing score, if a student is chosen at random from the group, find the probability that the student passed the test.
甲部 (2) [每題6分]
7.從「HAPPY」和「SAD」兩個英文字中各隨意抽出一個英文字母。求抽到最少一個字母「A」的概率。
7.A letter is chosen at random from each of the words ‘HAPPY’ and ‘SAD’. Find the probability of getting at least one letter ‘A’.
8.某士多的雪櫃內有四款飲品，價錢及數量如下：
8.There are four types of drink in the refrigerator of a store with their prices and quantities as shown in the table.
飲品
Drink
橙汁
Orange juice
綠茶
Green tea
檸檬茶
Lemon tea
咖啡
Coffee
售價 ($)
Price ($)
8.4
6.3
5.6
9.4
數量 (罐)
Quantity (can)
12
14
30
24
(a)若從中隨意抽出一罐飲品，求抽得一罐橙汁的概率。
(b)若隨意抽出一罐飲品，求飲品售價的期望值。
(a)If a can of drink is chosen from the refrigerator at random, find the probability that it is a can of orange juice.
(b)If a can of drink is chosen at random, find the expected value of the price of the drink.
9.某次最受歡迎歌星的選舉中，點算了640 張選票，得到以下結果。
9.The following pie chart shows the result of 640 votes obtained from an election about the most popular singer.
1
2
6
°
4
5
°
5
4
°
歌
星
A
S
i
n
g
e
r
A
歌
星
B
S
i
n
g
e
r
B
歌
星
C
S
i
n
g
e
r
C
歌
星
D
S
i
n
g
e
r
D
(a)若從選票當中抽出一張，求抽得的選票是選歌星 C 的概率。
(b)已知還有 80 張選票尚未點算。根據以上結果，估計所有選票中選歌星 C的選票數量。
(a)If a vote is chosen at random, find the probability that it votes for singer C.
(b)If 80 votes are yet to be counted, estimate the total number of votes for singer C according to the above result.
10.下圖為一個直徑為 10 cm 的圓形鏢靶，鏢靶中心有一個直徑 1 cm 的圓形 3 分區域。鏢靶餘下面積分為 8 等份，每份寫上一個分數。若隨意擲一支飛鏢擲中鏢靶，且飛鏢不會擲中邊界上，
10.The figure shows a circular dartboard with the diameter of 10 cm. A 3-point circular region with the diameter of 1 cm is located at the centre. The rest of the dartboard is equally divided into 8 regions marked with a score each as shown in the figure. If a dart is thrown and hits on the dartboard at random without hitting on any boundaries,
2
2
2
1
1
1
1
1
(a)求擲中 3 分區域的概率。
(b)求擲中 2 分區域的概率。
(a)find the probability that it hits on the 3-point region.
(b)find the probability that it hits on a 2-point region.
乙部
11.在一組 40 人的學生中，有 30 名是數學學會會員，18 名是英文學會會員，而所有學生最少要參加上述其中一個學會。
(a)問有多少名學生同時是兩個學會的會員？(2 分)
(b)從這班中隨意選出 1 名學生，求他不是數學學會會員的概率。(5 分)
(c)5 名原本只參加數學學會的學生，上星期亦加入英文學會。若現從該班中隨意選出 1 名學生，問他只參加上述其中一個學會的概率是多少？ (6 分)
11.In a group of 40 students, 30 of them are members of the Mathematics club, 18 of them are members of the English club, and all students must join at least one of the clubs as mentioned above.
(a)How many students are members of both clubs?(2 marks)
(b)If a student is chosen at random, find the probability that the student is not a member of the Mathematics club.(5 marks)
(c)Five members of the Mathematics club also joined the English club last week. If a student is chosen at random, what is the probability that the student is a member of either one of the clubs as mentioned?(6 marks)
多項選擇題 [每題3分]
12.下列何者是不可能事件？
I.擲一枚硬幣兩次，兩次均是「公」。
II.從三對藍色、白色和黑色襪子中抽出兩隻襪子，得到一隻白色和一隻藍色襪子。
III.擲兩粒骰子，所得的點數之和是 13。
A.只有 I
B.只有 III
C.只有 I 和 III
D.只有 II 和 III□
12.Which of the following is an impossible event / are impossible events?
I.Obtaining two heads in tossing a coin twice.
II.Obtaining a white sock and a blue sock when two socks are drawn from 3 pairs of socks in blue, white and black each.
III.Getting a sum of 13 in tossing two dice.
A.I only
B.III only
C.I and III only
D.II and III only□
13.下列何者是正確的？
I.對於任何事件 E，0 ( P(E) ( 1。
II.事件的實驗概率
數
次
試
嘗
總
數
次
試
嘗
的
件
事
合
符
=
III.
目
數
果
結
的
件
事
合
符
數
總
的
果
結
能
可
)
(
E
E
P
=
A.只有 I
B.只有 II
C.只有 I 和 II
D.只有 II 和 III□
13.Which of the following is / are true?
I.For an event E, 0 ( P(E) ( 1.
II.Experimental probability of an event
(
Number of trials favourable to the event
Total number of trials
III.P(E)
(
Total number of possible outcomes
Number of outcomes favourable to event E
A.I only
B.II only
C.I and II only
D.II and III only□
14.某小巴上有乘客 6 人，其中 4 人是男性，2 人是女性。當該小巴到達某車站時，其中 2 人下車，求該 2 人是一男一女的概率。
A.
4
1
B.
15
4
C.
2
1
D.
15
8
□
14.There are 6 passengers on a minibus, 4 of them are males and 2 of them are females. When the minibus arrives at a stop, 2 passengers get off. Find the probability that one of them is a male and the other is a female.
A.
4
1
B.
15
4
C.
2
1
D.
15
8
□
15.從四張分別記有數字 2、3、4 和 5 的卡中同時隨意抽出兩張。以下哪一件事件發生的概率最高？
A.抽出兩張卡的數字之積是偶數。
B.抽出兩張卡的數字均是偶數。
C.抽出兩張卡的數字均是質數。
D.抽出兩張卡的數字之積是質數。
□
15.Two cards are drawn together at random from four cards numbered 2, 3, 4 and 5. Which of the following events will happen with the highest probability?
A.The product of the numbers of the two cards is an even number.
B.The numbers of the two cards are even numbers.
C.The numbers of the two cards are prime numbers.
D.The product of the numbers of the two cards is a prime number.□
16.已知 A、B 和 C 三件貨品的價錢分別為 $10、$8 和 $6。建明欲購買其中一件貨品，且他購買 A、B 和 C 的概率分別為0.2、0.5 和 0.3。求建明購買一件貨品的期望值。
A.$8.2
B.$8
C.$7.8
D.$7.6□
16.The prices of items A, B and C are $10, $8 and $6 respectively. Adams wants to buy one of the three items and the probabilities for him to buy A, B and C are 0.2, 0.5 and 0.3 respectively. Find the expected value of buying an item by Adams.
A.$8.2
B.$8
C.$7.8
D.$7.6□
17.某投資中，已知六個月後賺取 $2 400 的概率是 0.4，虧蝕 $1 800 的概率是 0.6。問六個月後投資回報的期望值是多少？
A.$600
B.$120
C.$0
D.($120□
17.In an investment, the probability of gaining $2 400 after 6 months is 0.4, and the probability of losing $1 800 after 6 months is 0.6. What is the expected value of the return after 6 months?
A.$600
B.$120
C.$0
D.($120□
18.某測驗有兩條是非題，答對一題得 1 分，答錯一題扣 1 分。若隨意作答兩題，求取得 1 分或以上的概率。
A.
4
3
B.
2
1
C.
4
1
D.
8
1
□
18.There are two true or false questions in a test. 1 mark will be obtained for a correct answer and 1 mark will be deducted for a wrong answer. If two questions are answered randomly, find the probability of obtaining 1 mark or above.
A.
4
3
B.
2
1
C.
4
1
D.
8
1
□
19.某袋中有白球和黑球共 30 個。若隨意抽出一個白球的概率是 0.4，求該袋中白球的數目。
A.4
B.6
C.12
D.28□
19.There are a total of 30 black balls and white balls in a bag. If the probability of getting a white ball at random is 0.4, find the number of white balls in the bag.
A.4
B.6
C.12
D.28□
20.下表顯示一組學生中，戴眼鏡的男女生人數分佈。若從這組學生中隨意選出一人，求該學生有戴眼鏡的概率。
20.The following table shows the distribution of the number of students wearing glasses in a group. If a student is chosen at random from the group, find the probability that the student is wearing glasses.
戴眼鏡
Wearing glasses
沒有戴眼鏡
Without wearing glasses
男生
Boy
35
15
女生
Girl
30
20
A.
5
1
B.
10
3
C.
20
7
D.
20
13
□
A.
5
1
B.
10
3
C.
20
7
D.
20
13
□
21.擲兩枚均質骰子，求所得點數之和大於10 的概率。
A.
36
1
B.
18
1
C.
12
1
D.
6
1
□
21.Find the probability of getting a sum greater than 10 in tossing two fair dice.
A.
36
1
B.
18
1
C.
12
1
D.
6
1
□
22.從 1 至 100 的整數中 (包括 1 和 100) 隨意抽出一個數，求該數可被 5 整除的概率。
A.
2
1
B.
5
1
C.
10
1
D.
20
1
□
22.If a number is picked from integers 1 to 100 inclusive, find the probability that the number is divisible by 5.
A.
2
1
B.
5
1
C.
10
1
D.
20
1
□
23.在一半徑為 2 cm 的圓內隨意選一點，求這點距離圓心不多於 1 cm 的概率。
A.0.25
B.0.5
C.1
D.2□
23.If a point is chosen at random in a circle with the radius of 2 cm, find the probability that the distance between the point and the centre is not more than 1 cm.
A.0.25
B.0.5
C.1
D.2□
24.某袋內有 3 個紅球、3 個白球和 n 個黑球。若從該袋中隨意抽出1個球，抽中黑球的概率較抽中白球的概率高
4
1
。求該袋中球的總數。
A.12 個
B.9 個
C.6 個
D.3 個□
24.There are 3 red balls, 3 white balls and n black balls in a bag. If a ball is drawn at random, the probability of getting a black ball is
4
1
higher than that of white ball. Find the total number of balls in the bag.
A.12
B.9
C.6
D.3□
25.某袋內有 1 個紅球和 3 個藍球。若隨意先後抽出兩個球，且每次抽出球後不放回袋內，求第二次抽出紅球的概率。
A.
24
1
B.
12
1
C.
4
1
D.
3
1
□
25.There are 1 red ball and 3 blue balls in a bag. If two balls are drawn successively at random without replacement, find the probability that the second ball drawn is a red ball.
A.
24
1
B.
12
1
C.
4
1
D.
3
1
□
26.在一個派對中，有 56 名男賓客和 15 名女賓客。詩琪是其中一位賓客。已知在賓客中，有
4
3
的男賓客和
7
3
的女賓客是她的舊同學。若詩琪隨意與一位賓客交談，求該名賓客是她的舊同學之概率。
A.
35
24
B.
56
33
C.
2
1
D.
28
9
□
26.There are 56 male guests and 15 female guests in a party. Kiki is one of the guests. It is known that among the guests,
4
3
of the males and
7
3
of the females are her ex-classmates. If Kiki talks with one of the guests at random, find the probability that she talks with her ex-classmate.
A.
35
24
B.
56
33
C.
2
1
D.
28
9
□
練習7C 高級組合
練習7B 高級組合
練習7B 高級組合
練習7A 高級組合
練習7A 高級組合
練習7D 高級組合
練習7D 高級組合
練習7D 高級組合
練習7C 高級組合
練習7C 高級組合
練習7B 高級組合
練習7B 高級組合
練習7A 高級組合
練習7A 高級組合
練習7A 高級組合
練習7D 初級組合
練習7C 高級組合
練習7D 初級組合
練習7C 初級組合
練習7C 初級組合
練習7C 初級組合
練習7B 初級組合
練習7B 初級組合
練習7A 初級組合
練習7A 初級組合
練習7D 初級組合
練習7D 初級組合
練習7A 初級組合
練習7A 初級組合
練習7B 初級組合
練習7B 初級組合
練習7A 初級組合
練習7C 初級組合
練習7C 初級組合
練習7C 高級組合
練習7D 高級組合
練習7D 高級組合