Question: Consider the experiment of tossing three (fair) coins.

1. State what constitutes a probability space and their definitions.
2. Give an example of a probability space for this experiment excluding two trivial cases where the sigma field is given by either \(\{\emptyset, \Omega\}\) or the collection of all the subsets of \(\Omega\) and verify that your example satisfies all the conditions.
3. Construct two events that are independent and whose probabilities are greater than zero and less than one. Justify your answer.
4. Construct a 2 -dimensional random vector on this sample space that takes four distinct values, and compute the distribution functions for each element in the vector and expectation of the vector.

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