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

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

• – glS
Commented Jun 19, 2023 at 5:58

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 $$\frac12(1+|A|).$$