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.