Consider a deck with three cards number 1, 2,  and 3. Furthermore, assume that 1 and 2 cards are colored red and the 3 card is colored black. Two of the cards are drawn with out replacement. Let \(D_1\)  be the first card drawn and \(D_2\) be the second card drawn. Let \(T\) be the sum of the two cards drawn and let \(N\) be the number of red cards drawn.

1. Write down the algebra of all possible event on this probability space.
2. What is the algebra of events generated by \(T\), which we denote will \(\mathcal{A}(T)\) ?
3. What is the algebra of events generated by \(N\) , which we denote will \(\mathcal{A}(N)\) ?
4. Is \(T\) adapted to  \(\mathcal{A}(N)\) ? explain in terms of the above algebras.
5. Is \(N\) adapted to  \(\mathcal{A}(T)\) ? explain in terms of the above algebras.
6. What is \[ \mathbf{E} [ N \,|\, \mathcal{A}(T)] ? \]
7. What is \[ \mathbf{E} [ T \,|\, \mathcal{A}(N)] ? \]