课程名称︰机率导论 课程性质︰数学系必修 课程教师︰郑明燕 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2017/3/9 考试时限（分钟）：30分钟 试题 : Quiz 1 (2017/3/9) ∞ 1. Consider a sample space Ω with event space A. Let {A } be a sequence n n=1 in A. (a) (15%) State the three axioms of probability for a probability function P(‧) defined on A. ∞ (b) (15%) Show that if P(A ) = 1 for all n ∈ N, then P( ∩ A ) = 1. n n=1 n (c) (20%) If lim A exists, show that P( lim A ) = lim P( A ). n→∞ n n→∞ n n→∞ n 2.(20%) Let T denote the number of different partitions of {1,2,...n}. Derive n an equation for T in terms of T ,...,T and find T . n+1 1 n 5 3.(a) (15%) A 5-card hand is dealt from a well-shuffled deck of 52 playing cards. What is the probability that the hand contains at least one card from each of the four suits? (b) (15%) A pair of dice is rolled until a sum of either 5 or 7 appears. Find the probability that a 5 occurs first. --

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