课程名称︰机率导论 课程性质︰必修 课程教师︰彭栢坚 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2008/11/11 考试时限（分钟）：120min (PM 1:20~3:10) 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : The examination begins at 13.20 and ends at 15.10 Calculators are permitted. All working is to be shown. If you use theorems or results which have not been taught, you must give the proofs. 1.[13points] A box contains 5 red, 6 white and 7 blue balls. Balls are randomly chosen one after the other without replacement. (a) For k=1,...18, calculate the probability that the 1st k-1 balls chosen are red and the kth is white. (b) Use (a)to calculate the probability that a white ball is chosen before a blue. 2.[12points] An exam has 6 problems which are randomly chosen from 10. A student can answer 8 of the 10 problems. Calculate: (a)the probability he can answer all questions on the exam correctly. (b)the probability he can answer at least three questions on the exam correctly. 3.[12 points] Suppose Wang gives you the choice of opening 3 doors: behind one door is a car and behind the others are goats. You choose door no.1 and Wang opens another door which has a goat behind it. (Wang knows what is behind the doors: If the car is behind door no.2 or no.3, he opens the door with the goat behind it; if the car is behind door no.1, h3 opens door no.2 or no.3 each with probability 1/2) Assuming Wang opens door no.3, calculate the conditional probability the car is behind door no.2. 4.[13 points] If A flips n+1 and B flips n fair coins, show that the probability that A gets more heads than B is 1/2 (Hint: condituin on which player has more heads after each has flipped n coins.) 5.[12 points] (a)A typist averages 5 errorsper article. Approximate the probability she makes exactly 2 errors in an article. (b)Another typist averages 3 errors per article. Approximate theprobability she makes exactly 2 errors in an article. (c)Suppose a fair coin is tossed. If it comes up heads, the typist in (a) types the article; otherwose the typist in (b) types it. Now calculate the probabilitythere are exactly 2 errors in the article. (d)If there are exactly 2 errors in the article, what is the probability the article was typed by the typist in (a)? 6.[12points] A box contains 5 white, 3 balck and 2 red balls. We select 4 balls randomly. If exactly 2 are black, we stop. If not, we replace the 4 balls and randomly choose 4 again. We continue until exactly 2 black are chosen. For each positive integer n, calculate the probability we make n selections. 7.[13 points] An urn contains 1 black and 2 white balls. At each step a ball is randomly selected and then replaced along with another one of the same color. We stop when the ball selected is white. Let X be the number of selections. (a) Find P{X>i} for i≧1 (b) Show that P{X<∞}=1 (c) Find E[X} 8.[13 points] Suppose ypu collect n cupons of which there are 2 different types, A and B. Suppose each time you collect a cupon it is type A with probability 1/2. If n≧k≧2, calculate the probability you get exactly k type A coupons but not 2 in a row, that is , if i1<i2<...<ik are the times you recieve coupon A, then i_j-i_(j-1)≧2 fot j=2,...,k. 以下是中文试题，内容相同 1. 盒子里有5个红球、6个白球和7个蓝球。从盒子里一个一个的随机取球;每次拿一个球 不放回。 (a)对 k=1,2...18计算前k-1个球都为红色，而第k球为白色的机率 (b)利用(a)来计算白球比蓝球早被取得的机率 2. 考试时有6题是随机从10题中取得。学生能答对10题中的8题 计算 (a)考试时学生能答对6题的机率 (b)考试时学生能答对至少3题的机率 3.老王要你选择开三门中之一;一个车门後有汽车，另外两门後有山羊。 你选择第一;然後老王又开一个门;结果此门後面有山羊。(老王知道门後有什麽; 如果汽车在第二或第三门後，他就开後面有山羊的那个门;如果汽车在第一门後， 他就开第二或第三门，每个可能有机率1/2)。假设老王开的门是第三个， 计算汽车在第二门後的条件机率。 4. 假设A投掷公正铜板，投n+1次；B也投公正铜板，n次。试证A得到正面次数比B多的机率 是1/2 [建议:用调剑机率，条件是当他们都投了n次之後谁得的正面次数多] 5. (a)一个秘书每次打一个文章平均会有5个错误。 计算她打一个文章有2个错误的机率的近似值。 (b)另一个秘书每次打一个文章平均会有3个错误。 计算她打一个文章有2个错误的机率的近似值。 (c)我们掷公正铜板，如果出现正面则文章是由(a)秘书打的，要不然是(b)秘书打的。 计算文章有两个错误的机率的近似值 (d)若文章中刚好有2个错误，试求文章是(a)秘书打的机率。 6.盒子里有5白球，3黑球2红球。随机取4球。如果两球刚好取2个黑球为止。计算 取n次的机率。(n为正整数) 7.盒子里有1黑与2白球。从盒子里一个一个的随机取球，每次拿一个球放回去，而且再放 一个同颜色的球。这样下去到取得白球为止。令X为取球的次数。 (a)求P{X>i}, i≧1 (b)试证P{X<∞}=1 (c)E[X] 8.假设你收集n优待券，优待券有两种类A和B，每次拿到优待券，它为种类A的机率是 1/2。当n≧k≧2时，计算刚好拿到k张A种类的优待券，但不能连拿两张的机率， 即如果拿到A的时间为i1<i2<...<ik，则i_j-i_(j-1)≧2 fot j=2,...,k。 --

※ 发信站: 批踢踢实业坊(ptt.cc) ◆ From: 140.112.250.117