Post History

As mentioned in the answers you referenced in earlier versions of your question, $(-\mid-)$ is not standalone notation in usual probability theory notation.^[I have seen it used, e.g. in Jaynes' "Probability Theory: The Logic of Science", as standalone notation, but not in a way such that $P(A\mid B)$ meant $P((A\mid B))$ where $P$ was some unconditional probability measure.] Instead, $P(A\mid B)$ is just an unusual notation for a binary function. A common definition of this binary function *in terms of a **separate** unary, unconditional probabilty function also called $P$* and the one I believe Blitzstein is using is $$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$

While I don't know how clear Blitzstein makes this elsewhere, the way to read what's being written is we're defining a new unary, unconditional probability function $\tilde P$ via $\tilde P(A) = P(A \mid M_1)$. The *binary* conditional probability function $\tilde P(A \mid B)$ is still defined as usual, i.e. $$\tilde P(A \mid B) = \frac{\tilde P(A \cap B)}{\tilde P(B)}$$ **NOT** as the meaningless $P((A \mid B) \mid M_1)$. If we want to expand $\tilde P(A \mid B)$ in terms of $P$, we get $$\tilde P(A \mid B) = \frac{P(A \cap B \mid M_1)}{P(B \mid M_1)} = \frac{P(A \cap B \cap M\_1)}{P(B \cap M\_1)}$$ which doesn't present any issues. The answer to (c) follows readily.