Statistics Assignment
IV. Independent, Dependent Events and Scale of Probability
( G – L )
1. Explain independence from both an intuitive and mathematical perspective. Why is independence often misunderstood?
2. Why does conditional probability provide a natural framework for defining independence? Show how independence emerges as a special case.
3. Construct a real-life example where events appear independent but are actually dependent when examined carefully.
4. Explain why probability is restricted to the interval [0,1]. How does this restriction relate to logical consistency and measure theory?
5. Suppose A and B are events with P(A)=P(B)=0.5.
a. Is it possible for A and B to be both dependent and uncorrelated? Explain carefully.
b. Discuss why independence is a stronger condition than having no apparent relationship, and how this distinction matters in probability modeling.
6. In a probability space, let A, B, and C be events such that A is independent of B, and A is independent of C. Explain why this does not necessarily imply that A is independent of B∩C. What does this reveal about the limitations of pairwise independence and the importance of defining independence rigorously?
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