(Solved): Let \( \Omega=\{H H, H T, T H, T T\} \) be the sample space corresponding to two tosses of a coin. ...

Let \( \Omega=\{H H, H T, T H, T T\} \) be the sample space corresponding to two tosses of a coin. Let us define \[ X(H H)=X(H T)=1, \quad X(T H)=X(T T)=0 \] to be the number of heads on the first toss, and define \[ Y(H H)=Y(T H)=1, \quad Y(H T)=Y(T T)=0 \] to be the number of heads on the second toss. (i) List all the sets in \( \sigma(X) \), the \( \sigma \)-algebra generated by \( X \) (the smallest \( \sigma \)-algebra such that \( X \) is measurable). (ii) List all the sets in \( \sigma(Y) \), the \( \sigma \)-algebra generated by \( Y \).