I'm studying for my exam, and unsure about my answers for these 3 practise questions:
My answers are as follow:
Q1) Independent, as can have HTH which satisfies B but not A.
Q2) Dependent, as the outcome of A affects B. Example if A is one head, then B is true.
Q3) Dependent, example: If B and C are satisfied, then A is not. Making A depend on B and C.
Overall, I don't think I understand the logic how to see if they are dependent or independent.
The exam will be multiple choice, so I don't need to prove or give formal reasoning, are there any tricks to figure out if events are dependent or independent?
Best Answer
For question 1: $A:=\{THH, HHT, HHH\}, B:=\{HTH, THH, HHT, TTT\}, P(A|B)=P(\{THH,HHT\}|\{HTH, THH, HHT, TTT\})=\frac{2}{4} \neq P(A)=3/8$
so (1) is not independent, as A becomes more likely if we know B is true.
You just need to find one counterexample to the rule given by MPW above: P(A|B)=P(A) or $P(A\cap B)=P(A)P(B)$