Let \(S\) be the set of all permutations of \(a\), \(b\), \(c\), together with the triples \(aaa\), \(bbb\), and \(ccc\). Imagine we draw an outcome randomly from this set. Define the event \(A_k=\{\text{the kth spot is occupied by the letter a}\}\) for \(k=1,\,2,\,3\). Compute the probabilities \(P(A_k)\), \(P(A_i\cap A_j)\), \(P(A_1\cap A_2\cap A_3)\), and comment on the independence of the three of events;
Imagine we roll two fair, six-sided die. Compute \(P(A)\), \(P(B)\), \(P(C)\), \(P(A\cap B)\), \(P(A\cap C)\), \(P(B\cap C)\), and \(P(A\cap B\cap C)\) for the following three events and comment on their independence:
\[ \begin{aligned} A&=\text{first die is 1, 2, 3}\\ B&=\text{first die is 3, 4, 5}\\ C&=\text{the sum of the two rolls is 9}. \end{aligned} \]