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问题 1
让 A;B;C 是事件。
(1-i) 我们定义一个新的操作
A B = (AnB) [ (BnA):
A [ B = (A \ B) [ (A B):

(1-ii) 证明
P(A B) = P(A) + P(B) 2P(A \ B):

(1-iii) 证明
P(A\B) > P(A) + P(B) 1

P(A\B\C) > P(A) + P(B) + P(C) 2:

(2) 反复抛硬币。令 An 表示第 n 次抛掷结果为正面的事件。
使用 An 和合适的集合操作来表达 \a Head 最终
出现但不在前 100 次投掷中”。

假设 A:如果一个人被感染,则他/她被检测为阳性的概率为 0:98。
假设 B:如果一个人是健康的,那么他/她被检测为阳性的概率为 0:02。
(i) 随机选择的人检测呈阳性的概率是多少?

Question 1
Let A;B;C be events.
(1-i) We de ne a new operation by
AB = (AnB) [ (BnA):
Use the distributive laws to show that
A [ B = (A \ B) [ (AB):

(1-ii) Show that
P(AB) = P(A) + P(B) 2P(A \ B):

(1-iii) Show that
P(A \ B) > P(A) + P(B) 1
P(A \ B \ C) > P(A) + P(B) + P(C) 2:
[Hint: Use the Addition Theorem.]

(2) Toss a coin repeatedly. Let An denote the event that the n-th toss results in a Head.
Use the An’s and suitable set operations to express the event that \a Head eventually
appears but not within the rst 100 tosses”.

Question 2
There is a pandemic of a particular disease. According to recent statistics, one in two
hundred people in the population is infected. A test has been created accordingly but
it is not completely accurate as suggested by the following information.
Assumption A: If a person is infected, with probability 0:98 he/she will be tested positive.
Assumption B: If a person is healthy, with probability 0:02 he/she will be tested positive.
(i) What is the probability that a randomly chosen person is tested positive?

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