Criminal X and Criminal Y are both 20 percent likely to commit a certain crime, and they are both 50 percent likely to be near the site of the crime at a given time. As a result of the investigation, it is revealed that Criminal X was near the site at the time of the crime, but Y was not. What are the posterior probabilities of committing the crime for X and Y?
In this video, $\frac{1}{2}$ is plugged in for $\Pr(X$ near site$)$ in the denominator of Bayes' Theorem.
Similarly for $Y$, it is implied that the denominator of Bayes' Theorem is not $0$ (otherwise limits would be required and the fraction would not become $0$).
Why are the outdated unconditional probabilities used in Bayes' Theorem, instead of the new ones generated by the investigation?
Because the investigation is only one observation/outcome of the probability distribution. In fact, the investigation does not generate any new probabilities, the probabilities are already defined.
Trying to find out the probability distribution from the outcomes, is basically statistics, and not applicable in this case.
For P(x near site), by definition, is always 0.5.
Imagine if you rolled a 3 on a dice, you wouldn't then claim that P(rolling a 3) is 1.