If I know that it’s P(1), and I know that it was randomly selected from all the generated problems (rather than being shown to me because it’s the first one), then yes.
If I’m shown a single randomly selected problem from the list of generated problems without being told which problem number it is, it doesn’t make W=0 more likely than W=1 or W=2.
If I know that it’s P(1), and I know that it was randomly selected from all the generated problems (rather than being shown to me because it’s the first one), then yes.
If I’m shown a single randomly selected problem from the list of generated problems without being told which problem number it is, it doesn’t make W=0 more likely than W=1 or W=2.