How to understand the result of Scenario 3.1 in John Watrous' book?

Question

In scenario 3.1, Bob's goal is to correctly determine the value stored in $\textbf{Y} $ using only the information from the observation of $\textbf{X}$. How to understand the claim that "The probability that Bob correctly identifies the value stored in Y using this strategy can be understood by first considering the probability he is correct minus the probability he is incorrect." This probability in probabilities is represented in Eq. (3.2): $\sum_{b \in \Sigma} \vert \lambda p_0(b) - (1-\lambda)p_1(b)\vert_1$. From this quantity, how to derive the probability that Bob is correct? This is in Chapter 3 of the book: https://cs.uwaterloo.ca/~watrous/TQI/TQI.3.pdf

Answer 1

This isn't directly the derivation that you need, but hopefully shows you all the elements that you need!

Imagine that for one sample of $X$, with value $b$, you have a probability of getting the two different answers $p_0$ and $p_1$. We know that $$ p_0+p_1=1. $$ We can choose to define $$ A=p_0-p_1. $$ So, we can express $p_0$ and $p_1$ in terms of $A$ by solving the simultaneous equations: \begin{align*} p_0&=\frac12(1+A) \\ p_1&=\frac12(1-A). \end{align*} Which one is the correct one? The one with larger probability. This is $p_0$ if $A>0$ and $p_1$ if $A<0$. So, this is the same as $$ \frac12(1+|A|). $$