AMC12概 率与统计.docx

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；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. 事件与事件是互斥事件4. 事件与事件可以是互斥事件，也可以不是互斥事件因此在使用上述公式时要根据题意判断事件与事件可以是互斥事件，然后再根据公式进行计算三、 几何概型事件理解为区域的某一子区域，的概率只与子区域的几何度量（长度、面积或体积）成正比，而与的位置和形状无关，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，事件的概率定义为，其中表示区域的几何度量， 表示区域的几何度量五、 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,则样本的算术平均数为；1. （2010A15）A coin is altered so that the probability that it lands on heads is less thanand when the coin is flipped four times, the probability of an equal number of heads and tails is. What is the probability that the coin lands on heads? D(A) (B) (C) (D) (E)2. （2010A16）Bernardo randomly picks 3 distinct numbers from the setand arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the setand also arranges them in descending order to form a 3-digit number. What is the probability that Bernardos number is larger than Silvias number?B(A) (B) (C) (D) (E)3. （2010B11）A palindrome betweenandis chosen at random. What is the probability that it is divisible by? E(A) (B) (C) (D) (E)4. （2010B16）Positive integers , and are randomly and independently selected with replacement from the set . What is the probability that is divisible by ?(A) (B) (C) (D) (E)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 frogs final position is no more than 1 meter from its starting position?(A) (B) (C) (D) (E)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 circles circumference?(A) (B) (C) (D) (E)7. （2011A14）Supposeandare single-digit positive integers chosen independently and at random. What is the probability that the pointlies above the parabola?E(A) (B) (C) (D) (E)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 thedart lands within the center square?(A) (B) (C) (D) (E)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 , and Mel is twice as likely to win as Chelsea. What is the probability that Alexwins three rounds, Mel wins two rounds, and Chelsea wins one round?(A) (B) (C) (D) (E)10. （2012A15）Asquare is partitioned intounit 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 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(A) (B) (C) (D) (E)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 is chosen uniformly at random. What is the probability thatis also a palindrome?(A) (B) (C) (D) (E)12. （2014A5）On an algebra quiz, 10% of the students scored 70 point