问题1.将硬币按以下顺序抛掷1000次：
头：455，尾巴：545
计算每个事件的概率。
解决方案：
According to the given question,
Coin is tossed 1000 times, that means number of trials are 1000
Let us assume that, event of getting head and event
of getting tail be H and T respectively.
Number of trials in which the H happens = 455
Probability of H = (Number of heads) / (Total number of trials)
P(H) = 455/1000 = 0.455
Similarly,
Number of trials in which the T happens = 545
Probability of T = (Number of trials) / (Total number of trials)
Probability of the event getting a tail, P(T) = 545/1000 = 0.545
问题2：将两枚硬币同时投掷500次，并出现以下不同结果的频率：
两个脑袋：95次
一条尾巴：290次
无头：115次
找出这些事件中每一个发生的概率。
解决方案：
According to the formula,
Probability of any event = (Number of favorable outcome) / (Total number of trials)
Total number of trials = 95 + 290 + 115 = 500 -(given)
P(Getting two heads) = 95/500 = 0.19
P(Getting one tail) = 290/500 = 0.58
P(Getting no head) = 115/500 = 0.23
问题3：将三枚硬币同时投掷100次，并出现以下不同结果的频率：
Outcome |
No Head |
One Head |
Two Head |
Three Head |
Frequency |
14 |
38 |
36 |
12 |
如果同时抛弃这三个硬币，请计算以下几率：
(i)抬起2个头
(ii)3个头抬起
(iii)至少有一个头抬起
(iv)头多于尾
(v)尾巴多于头
解决方案：
According to the formula,
Probability of an event = (Number of Favorable outcomes) / (Total Numbers of outcomes)
Given: Total numbers of outcomes = 100
(i) Probability of 2 Heads coming up = 36/100 = 0.36
(ii) Probability of 3 Heads coming up = 12/100 = 0.12
(iii) Probability of at least one head coming up = (38+36+12) / 100 = 86/100 = 0.86
(iv) Probability of getting more Heads than Tails = (36+12)/100 = 48/100 = 0.48
(v) Probability of getting more tails than heads = (14+38) / 100 = 52/100 = 0.52
问题4.随机选择1500个有2个孩子的家庭，并记录以下数据：
No of the girls in the family |
0 |
1 |
2 |
No of girls |
211 |
814 |
475 |
如果随机选择一个家庭，请计算其具有的概率：
(i)没有女孩
(ii)1个女孩
(iii)2个女孩
(iv)最多一个女孩
(v)女孩比男孩多
解决方案：
According to the formula,
Probability of an event = (Number of Favorable outcomes) / (Total Numbers of outcomes)
Total numbers of outcomes = 211 + 814 + 475 = 1500
(Here, we are assuming that total numbers of outcomes = total number of families)
(i) Probability of having no girl = 211/1500 = 0.1406
(ii) Probability of having 1 girl = 814/1500 = 0.5426
(iii) Probability of having 2 girls = 475/1500 = 0.3166
(iv) Probability of having at the most one girl = (211+814) /1500 = 1025/1500 = 0.6833
(v) Probability of having more girls than boys = 475/1500 = 0.31
问题5：在板球比赛中，击球手打出的30个球中有6次命中边界。找出一个球上打球的概率：
(i)他到达了边界
(ii)他没有达到界限。
解决方案：
Total number of balls played by a player = 30 -(According to the question)
Number of times he hits a boundary = 6
Number of times he does not hit a boundary = 30 – 6 = 24
According to the formula,
Probability of an event = (Number of Favorable outcomes) / (Total Numbers of outcomes)
(i) Probability (he hits boundary) = (Number of times he hit a boundary) / (Total number of balls he played)
= 6/30 = 1/5
(ii) Probability that the batsman does not hit a boundary = 24/30 = 4/5
问题6.学生在每月单元测试中获得的分数百分比如下：
Unit Test |
I |
II |
III |
IV |
V |
Percentage of Mark obtained |
69 |
71 |
73 |
68 |
76 |
找出学生获得的可能性
(i)超过70％的分数
(ii)少于70％的分数
(iii)区别
解决方案：
Total number of unit tests taken = 5
According to the formula,
Probability of an event = (Number of Favorable outcomes) / (Total Numbers of outcomes)
(i) Number of times student got more than 70% = 3
Probability (Getting more than 70%) = 3/5 = 0.6
(ii) Number of times student got less than 70% = 2
Probability (Getting less than 70%) = 2/5 = 0.4
(iii) Number of times student got a distinction = 1
[Marks more than 75%]
Probability (Getting a distinction) = 1/5 = 0.2
问题7.为了了解学生对数学的看法，对200名学生进行了调查。数据记录在下表中：
Opinion |
Like |
Dislike |
Number of students |
135 |
65 |
找出学生随机选择的概率：
(i)喜欢数学
(ii)不喜欢。
解决方案：
Total number of students = 200
Students like mathematics = 135
Students dislike Mathematics = 65
According to the formula,
Probability of an event = (Number of Favorable outcomes) / (Total Numbers of outcomes)
(i) Probability (Student likes mathematics) = 135/200 = 0.675
(ii) Probability (Student does not like mathematics) = 65/200 = 0.325
问题8.记录了第三十班第三十班学生的血型：
A，B，O，O，AB，O，A，O，B，A，O，B，A，O，O，A，AB，O，A，A，A，O，O，AB，B，A， O，B，A，B，O
从献血中随机选择一名学生。查找所选学生的血型为：
(i)a(ii)b(iii)ab(iv)o
解决方案：
(i) Probability of a student of blood group Favorable outcome A
Total outcome = (Number of Favorable outcomes) / (Total Numbers of outcomes)
= 9/30 = 0.3
(ii) Probability of a student of blood group Favorable outcome B
Total outcome = (Number of Favorable outcomes) / (Total Numbers of outcomes)
= 6/30 = 0.2
(iii) The probability of a student of blood group
Probability of an event = (Number of Favorable outcomes) / (Total Numbers of outcomes)
= 3/30 =0.1
(iv) The probability of a student of blood group
Probability of an event = (Number of Favorable outcomes) / (Total Numbers of outcomes)
= 12/30 = 0.4
问题9. 11袋小麦粉，每个标记为5公斤，实际上包含以下重量的面粉(以公斤为单位)：
4.97、5.05、5.08、5.03、5.00、5.06、5.08、4.98、5.04、5.07、5.00
找出随机选择的这些袋子中的任何一个包含超过5千克面粉的可能性。
解决方案：
Given bag of wheat flour in the question
4.97, 5.05, 5.05, 5.03, 5.00, 5.06, 5.08, 4.98, 5.04, 5.07, 5.00
The total number of wheat flour bags =11.
The number of wheat flour bags contain more than 5 Kg are 7.
Then probability of bags chosen at random =
(The number of wheat flour bags contain more than 5 Kg) / (The total number of wheat flour bags)
= 7/11
问题10.下表显示了IX班40名学生的出生月份。
Jan |
Feb |
March |
April |
May |
June |
July |
Aug |
Sept |
Oct |
Nov |
Dec |
3 |
4 |
2 |
2 |
5 |
1 |
2 |
5 |
3 |
4 |
4 |
4 |
查找学生八月份出生的概率。
解决方案：
Probability (Students was born in August) Favorable outcome
Total outcome = (Favorable outcome)/(Total outcome)
= 6/40 = 3/20