課程名稱︰機率導論
課程性質︰數學系必修
課程教師︰鄭明燕
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/3/23
考試時限（分鐘）：50分鐘
試題 :
Quiz 2 (2017/3/23)
1. Consider a probability space (Ω,A,P) and E,F,G ∈ A.
(a) (15%) Show that P(E|E∪F) ≧ P(E|F).
(b) (15%) If E is independent of F, and E is independent of G, and F∩G =ψ,
show that E is independent of F∪G.
(c) (15%) If E , E ,..., E ∈ A and are independent, show that
1 2 n
n
P( E ∪ E ∪...∪ E ) = 1－ Π [1－P(E )].
1 2 n i=1 i
2. (25%) A ball is in any one of n boxes and is in the ith box with pobability
P . If the ball is in box i, a search of that box will uncover it with
i
probability that the ball is in box j, given that a search of box i did not
uncover it.
3. Let S = {1,2,...,n} and suppose that A and B are, independently, equally
n
likely to be any of the 2 subsets of S.
3 n
(a) (20%) Show that P{ A ⊂ B} = (——) .
4
(b) (10%) Find P{ A ∩ B = ψ}.