課程名稱︰機率導論
課程性質︰系定必修
課程教師︰彭(木百)堅 教授
開課系所︰數學系
考試時間︰2004.11.9
試題：
1.A box contains five red, six white and seven blue balls. Five balls are
randomly chosen. Calculate the probability that at least one ball of each
colour is chosen.
2.An exam has five problems which are randomly chosen from ten. A student
can answer seven of the ten problems. Calculate
(a) the probability he can answer all questions on the exam correctly;
(b) the probability he can answer at least four questions on the exam
correctly
3.98% of all babies survive delivery. 15% of all births are C sections.
If a C section is done, only 96% of the babies survive. Calculate the
conditional probability a baby survives given that a C section is not done.
4.Two die are rolled repeatedly. Calculate the probability that a seven is
rolled before a ten.
5.Suppose independent trials are performed with probability of success p
(0 < p < 1). Let X be the number of trials needed to get one success.
Show that
E[X] = 1/p
6.(a) A typist averages 3 erroes per article. Approximate the probability
she makes no error in an article.
(b) Another typist averages 4.2 erroes per article. Approximate the
probability she makes no error in an article.
(c) Suppose a fair coin is tossed. If it comes up heads, the typist in (a)
types the article; otherwise the typist in (b) types it. Now calculate
the probability there are no errors in article.
7.A box contains four white and four black balls.We choose four balls
randomly. If exactly two are white, we stop. If not, we replace the four
balls and randomly choose four again. We continue until excatly two white
are choosen. For each positive integer n, calculate the probability we
make n selections.
8.When a coin flipped the probability it comes up heads is p. The coin is
repeatedly flipped until either two heads or two tails appear in succession.
Let X be the number of flips. For each positive integer calculate P{X=n}
and then calculate E[X].
簡答：
1. 865/1224
2. (a) 1/12 (b) 1/2
3. 0.9835
4. 2/3
5. 略
6. (a) 0.0498 (b) 0.0150 (c) 0.0324
7. (17/35)^(n-1)‧(18/35)
8. P{X=n} = q^((n-1)/2), E[X] = (2 + p(1-p))/(1 - p(1-p))
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※ 編輯: monotones 來自: 140.112.250.148 (02/18 21:44)