My teacher doesn't give answers to exercises. I am not sure about 4th question. Could you tell me please if it is right or not?
Let A and B be two events, with P(A ∩ B) = 0.2 and the probability of B, P(B), = 0.7.
a. What is P(A|B)?
0.2/0.7 = 0.3
b. What is P(A) if the two events are independent?
P(A|B) = P(A) = 0.3
c. Can events A and B be exhaustive?
Yes, because: P(A) + P(B) = S = 1
P(A) + 0.7 = 1.
if P(A) = 0.3 then events A and B are exhaustive.
d. What are the minimum and maximum value of P(A)?
Max.value of P(A) is 0.3
Min. value of P(A) is 0.
Best Answer
For d) the maximum value is $0.5$ and the minimum is $0.2$. For an example where $P(A)=0.5$ partition the probability space into three parts $C,D,E$ with $P(C)=0.3, P(D)=0.2$ and $P(E)=0.5$. Take $A=C\cup D$ and $B=D\cup E$. For the mimimum consider an example with $A \subset B$.