1. Show from first principles that \( P(a \mid b \wedge a)=1 \).
Ans: The first principles needed here are the definition of conditional probability, \( P(X \mid Y)=P(X \wedge Y) / P(Y) \), and the definitions of the logical connectives. It is not enough to say that if \( B \wedge A \) is given then A must be true! From the definition of conditional probability, and the fact that \( A \wedge A \leftrightarrow A \) and that conjunction is commutative and associative, we have,
\[
P(A \mid B \wedge A)=\frac{P(A \wedge(B \wedge A)}{P(B \wedge A)}=\frac{P(B \wedge A)}{P(B \wedge A)}=1
\]