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AMC12概 率与统计


一、 古典概型 1. 基本事件: 一次试验中所有可能的结果都是随机事件, 这类随机事件称为基本事件. 2. 基本事件的特点: (1)任何两个基本事件是互斥的; (2)任何事件(除不可能事件)都可以表示成基本事件的和. 我们将具有这两个特点的概率模型称为古典概率模型,其特征是: (1)有限性,即在一次试验中所有可能出现的基本事件只有有限个. (2) 等可能性, 每个基本事件发生的可能性是均等的; 称这样的试验为古典概型. 二、 古典概型计算公式 1.如果一次试验中可能出现的结果有 n 个,而且所有结果出现的可能性都相等,那么每 一个基本事件的概率都是

1 ; n

2.如果某个事件 A 包括的结果有 m 个,那么事件 A 的概率 P(A)= 3.事件 A 与事件 B 是互斥事件 P ? A

m . n

B ? ? P ? A? ? P ? B ?

4.事 件 A 与 事 件 B 可 以 是 互 斥 事 件 , 也 可 以 不 是 互 斥 事 件

P ? A B ? ? P ? A? ? P ? B ? ? P ? A B ? .
因此在使用上述公式时要根据题意判断事件 A 与事件 B 可以是互斥事件,然后再根据 公式进行计算. 三、 几何概型 事件 A 理解为区域 ? 的某一子区域 A , A 的概率只与子区域 A 的几何度量(长度、面 积或体积)成正比,而与 A 的位置和形状无关,满足此条件的试验称为几何概型. [注意]古典概型使用于所有试验结果是有限且结果是等可能出现的情况, 而几何概型则 适用于试验结果无穷多的情况; 四、 几何概型的计算 几何概型中, 事件 A 的概率定义为 P ( A) ? 示区域 A 的几何度量. 五、 一些统计名词 (1)众数:在样本数据中,出现次数最多的那个数据; (2)中位数:将样本数据按大小顺序排列,若数据的个数为奇数,则最中间的数据为 中位数,若样本数据个数为偶数,则取中间两个数据的平均数作为中位数。 ( 3 ) 平 均 数 : 一 般 地 , 设 样 本 的 数 据 为 x1 , x2 , , xn , 则 样 本 的 算 术 平 均 数 为
x? x1 ? x2 ? n ? xn

?A , 其中 ?? 表示区域 ? 的几何度量, ? A 表 ??



1. (2010A15)A coin is altered so that the probability that it lands on heads is less than
the coin is flipped four times, the probability of an equal number of heads and tails is the probability that the coin lands on heads?

1 and when 2 1 . What is 6

(A)

15 ? 3 6

(B)

6? 6 6 ?2 12

(C)

2 ?1 2

(D)

3? 3 6

(E)

3 ?1 2

2. (2010A16)Bernardo randomly picks 3 distinct numbers from the set ?1,2,3,4,5,6,7,8,9? and
arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set ?1,2,3,4,5,6,7,8? and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number? (A)

47 72

(B)

37 56

(C)

2 3

(D)

49 72

(E)

39 56

3. (2010B11)A palindrome between 1000 and 10000 is chosen at random. What is the probability
that it is divisible by 7 ? (A)

1 10

(B)

1 9

(C)

1 7

(D)

1 6

(E)

1 5

4. ( 2010B16 ) Positive integers a , b and c are randomly and independently selected with
replacement from the set divisible by 3 ? (A)

?1, 2,3,
31 81

, 2010? . What is the probability that abc ? ab ? a is

1 3

(B)

29 81

(C)

(D)

11 27

(E)

13 27

5. (2010B18)A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen
independently and at random. What is the probability the the frog's final position is no more than 1 meter from its starting position? (A)

1 6

(B)

1 5

(C)

1 4

(D)

1 3

(E)

1 2

6. (2011A10) A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled
determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? (A)

1 36

(B)

1 12

(C)

1 6

(D)

1 4

(E)

5 18

7. (2011A14) Suppose a and b are single-digit positive integers chosen independently and at random.
What is the probability that the point (a, b) lies above the parabola y ? ax 2 ? bx ? (A)

11 81

(B)

13 81

(C)

5 27

(D)

17 81

(E)

19 81

8. (2011B12) A dart board is a regular octagon divided into regions as shown. Suppose that a dart
thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square? (A)

2 ?1 2

(B)

1 4

(C)

2? 2 2

(D)

2 4

(E)

19 81

9. (2012A11) Alex, Mel, and Chelsea play a game that has 6 rounds. In each round there is a single
winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is

1 , and Mel is twice as likely to win as Chelsea. What is the probability that Alex 2

wins three rounds, Mel wins two rounds, and Chelsea wins one round? (A)

5 72

(B)

5 36

(C)

1 6

(D)

1 3

(E) 1

10. (2012A15) A 3 ? 3 square is partitioned into 9 unit squares. Each unit square is painted either white
or black with each color being equally likely, chosen independently and at random. The square is the rotated 90 ? clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black? (A)

49 512

(B)

7 64

(C)

121 1024

(D)

81 512

(E)

9 32

11. (2013A22) A palindrome is a nonnegative integer number that reads the same forwards and
backwards when written in base 10 with no leading zeros. A 6-digit palindrome n is chosen uniformly at random. What is the probability that (A)

n is also a palindrome? 11
9 25
(E)

8 25

(B)

33 100

(C)

7 20

(D)

11 30

12. (2014A5)On an algebra quiz, 10% of the students scored 70 points, 35% scored 80 points, 30%
scored 90 points, and the rest scored 100 points. What is the difference between the mean and the median score of the students’ scores on this quez? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5



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