Consider the sample space created by two coin flips. We will denote the outcome of first a head and then a tail by \((H,T)\). The even that there is exactly one head will be written \(\{(H,T), (T,H)\}\). Let \(\mathcal{A}\) denote the set of all events on this sample space.
- Write down all of the events in \(\mathcal{A}\). How many are there ?
- Pick a few elements of \(\mathcal{A}\) and show that the rules of being an algebra are in fact satisfied. Namely, if \(A,B \in \mathcal{A}\) then \( A \cup B \in \mathcal{A}\) and \(A^c \in \mathcal{A}\). (Here \(A^c\) is the compliment of \(A\) which is simply every event which is not is \(A\).)
- Why did I not ask you to verify by hand this property for all of the sets in \(\mathcal{A}\) ? Hint: can you tell me a lower bound on the number of such conditions you would have to check ?
- (*) Using the definition of an algebra, show that the empty set \(\emptyset\) is in \(\mathcal{A}\).
- (*) Using the definition of an algebra, show that if \(A, B \in\mathcal{A}\) then so must be \(A \cap B\).
- (*) Argue that if \(A_1, A_2,\dots,A_k \in\mathcal{A}\) then so are
\[ A_1\cup A_2\cup\cdots\cup A_k=\bigcup_{j=1}^k A_j\]
\[ A_1\cap A_2\cap\cdots\cap A_k=\bigcap_{j=1}^k A_j\]
