課程名稱︰機率導論
課程性質︰必修
課程教師︰彭栢堅
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2008/11/11
考試時限（分鐘）：120min (PM 1:20~3:10)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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.
以下是中文試題，內容相同
1.
盒子裡有5個紅球、6個白球和7個藍球。從盒子裡一個一個的隨機取球;每次拿一個球
不放回。
(a)對 k=1,2...18計算前k-1個球都為紅色，而第k球為白色的機率
(b)利用(a)來計算白球比藍球早被取得的機率
2.
考試時有6題是隨機從10題中取得。學生能答對10題中的8題
計算
(a)考試時學生能答對6題的機率
(b)考試時學生能答對至少3題的機率
3.老王要你選擇開三門中之一;一個車門後有汽車，另外兩門後有山羊。
你選擇第一;然後老王又開一個門;結果此門後面有山羊。(老王知道門後有什麼;
如果汽車在第二或第三門後，他就開後面有山羊的那個門;如果汽車在第一門後，
他就開第二或第三門，每個可能有機率1/2)。假設老王開的門是第三個，
計算汽車在第二門後的條件機率。
4.
假設A投擲公正銅板，投n+1次；B也投公正銅板，n次。試證A得到正面次數比B多的機率
是1/2 [建議:用調劍機率，條件是當他們都投了n次之後誰得的正面次數多]
5.
(a)一個秘書每次打一個文章平均會有5個錯誤。
計算她打一個文章有2個錯誤的機率的近似值。
(b)另一個秘書每次打一個文章平均會有3個錯誤。
計算她打一個文章有2個錯誤的機率的近似值。
(c)我們擲公正銅板，如果出現正面則文章是由(a)秘書打的，要不然是(b)秘書打的。
計算文章有兩個錯誤的機率的近似值
(d)若文章中剛好有2個錯誤，試求文章是(a)秘書打的機率。
6.盒子裡有5白球，3黑球2紅球。隨機取4球。如果兩球剛好取2個黑球為止。計算
取n次的機率。(n為正整數)
7.盒子裡有1黑與2白球。從盒子裡一個一個的隨機取球，每次拿一個球放回去，而且再放
一個同顏色的球。這樣下去到取得白球為止。令X為取球的次數。
(a)求P{X>i}, i≧1
(b)試證P{X<∞}=1
(c)E[X]
8.假設你收集n優待券，優待券有兩種類A和B，每次拿到優待券，它為種類A的機率是
1/2。當n≧k≧2時，計算剛好拿到k張A種類的優待券，但不能連拿兩張的機率，
即如果拿到A的時間為i1<i2<...<ik，則i_j-i_(j-1)≧2 fot j=2,...,k。
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