Computing the advantage when checking PRF

Question

I am reading a pdf on pseudorandom function I found here https://www.cs.utexas.edu/~dwu4/courses/sp21/static/reductions.pdf

My problem/struggle is with the computation of the distinguisher's $B$ advantage.
According to the notes $b=0$ means that $B$ receives a sample from the function of interest, let's call it $F$, whereas $b=1$ means that they receive a sample from a truly random let's call it $f$. Then the advantage is defined as: $$ |\Pr[b'=1|b=0]-\Pr[b'=1|b=1]| $$ The first probabillity $\Pr[b'=1|b=0]$ tells us that $B$ wrongly assumed that sample was from a truly random and the second one tells us that they correctly assumed that the sample was from a truly random.
Now it would make more sense to me if instead we computed the probabillity that they correctly assumed that the sample was from $F$ a.k.a $\Pr[b'=0|b=0]$ .

For example if we look at example 1 in the pdf : $G'(s)= G(s) ||s$ .
Now to my understanding : if $B$ receives $t=G(s)$ , they can think of it as $t= t_1||t_2$ and because they know the length of $s$ and $G(s)$ check if $t_2 = s \wedge G(s) = t_1$ .
So if $b=0$ then $\Pr[b'=0|b=0]=1$.

But in the paper instead $\Pr[b'=1|b=0]=1$ is said to be one. I am confused.

Answer 1

The definition of the distinguishing advantage is a (pseudo)metric that captures the performance of some distinguished $D$ in distinguishing two experiments, say $E_0$ and $E_1$. The larger the value, the better $D$ is doing. Said otherwise, we view $D$ as outputting some a bit after an experiment; this bit somehow encodes the behavior of $D$ in the given experiment. $D$ is a good distinguisher if it always outputs different bit values ($b'$) for different experiments. For instance, the output $0$ in $E_0$ and always $1$ in $E_1$.

What about $|\Pr^{DE_0}[b'=0|b=0]-\Pr^{DE_1}[b'=1|b=1]|$? Well, in this case, our perfect $D$ would have the lowest advantage of $1 - 1 = 0$.

There are other characterizations of security encoded as bit-guessing games where the advantage is then defined by $|\Pr[b' = b] - 1/2|$.