本次澳洲代写是概率学的一个限时测试，以下是作业具体内容：
Instructions:
- Read the questions carefully and answer only what is being asked.
- Answer all questions directly on the examination paper; use the last pages if you need
more space, and provide clear pointers to your work. - Show your intermediate work, and write clearly and legibly.
1. (20 points) Consider a probability space and three (jointly) independent events A;B;C
with probabilities P(A) = 1=2; P(B) = 1=3; P(C) = 1=4.
Find the value of P((A [ B) \ C).
2. (20 points) Consider a probability space and three (jointly) independent events A;B;C
with probabilities P(A) = 1=2; P(B) = 1=3; P(C) = 1=4.
Find the value of P((A \ B)c\ C).
3. (20 points) Consider n persons, among them are Tom and Ben, who are arranged ran-
domly in a row (say from left to right). What is the probability that there are exactly
k persons between Tom and Ben? (Assume n >= k + 2.)
4. (20 points) Consider n >= 3 persons, among them are Tom and Ben, who are arranged
randomly in a row (say from left to right). What is the probability that Tom and Ben
are NOT next to each other?
5. (20 points) Consider a medical condition C and two associated symptoms S1 and S2.
The prevalence of this condition in the population is 10%, and any person with the condi-
tion can show none, one, or both symptoms, with probabilities: P(S1jC) = 30%; P(S2jC) =
70%; P(S1 \ S2jC) = 20%. The symptoms can also appear in individuals without the
condition with equal probability P(S1jCc) = P(S2jCc) = 5%, and in this case the symp-
toms are conditionally independent, i.e. P(S1 \ S2jCc) = P(S1jCc) P(S2jCc) .
Find the conditional probability P(CjS1\Sc
2), i.e. the probability of having the condition
if you only show symptom S1, but not S2.