The point there is that there is no contradiction because the informational content is different. “Which is the baseline” is up to the person writing the problem to answer. You’ve asserted that the baseline is A vs B; then you’ve added information that A is actually A1 and A2.
The issue here is entirely semantic ambiguity.
Observe what happens when we remove the semantic ambiguity:
You’ve been observing a looping computer program for a while, and have determined that it shows three videos. The first video portrays a coin showing tails. The second video portrays two coins; the left coin shows heads, the right coin shows tails. The third video also portrays two coins; the left coin shows heads, the right coin shows heads.
You haven’t been paying attention to the frequency, but now, having determined there are three videos you can see, you want to figure out how frequently each video shows up. What are your prior odds for each video?
33/33/33 seems reasonable. I’ve specified that you’re watching videos; the event is which video you are watching, not the events that unfold within the video.
Now, consider an alternative framing: You are watching somebody as they repeat a series of events. You have determined the events unfold in three distinct ways; all three begin the same way, with a coin being flipped. If the coin shows heads, it is flipped again. If the coin shows tails, it is not. What are your prior odds for each sequence of events?
25/25/50 seems reasonable.
Now, consider yet another framing: You are shown something on a looping computer screen. You have determined the visuals unfold in three distinct ways; all three begin the same way, with a coin being flipped. If the coin shows heads, it is flipped again. If the coin shows tails, it is not. What are your prior odds here?
Both 25/25/50 and 33/33/33 are reasonable. Why? Because it is unclear whether or not you are watching a simulation of coin flips, or something like prerecorded videos; it is unclear whether or not you should treat the events within what you are watching as events, or whether you should treat the visuals themselves you are watching as the event.
Because it is unclear, I’d lean towards treating the visuals you are watching as the event—that is, assume independence. However, it would be perfectly fair to treat the coin tosses as events also. Or you could split the difference. Prior probabilities are just your best guess given the information you have available—and given that I don’t have access to all the information you have available, both options are fair.
Now, the semantic ambiguity you have introduced, in the context of this, is like this:
You’re told you are going to watch a computer program run, and what you see will begin with a coin being flipped, showing heads or tails. What are your probabilities that it will show heads or tails?
Okay, 50⁄50. Now, if you see the coin shows heads, you will see that it is flipped again; we now have three possibilities, HT, HH, and TT. What are your probabilities for each event?
Notice: You didn’t specify enough to know what the relevant events we’re assigning probabilities to even are! We’re in the third scenario; we don’t know if it’s a video, in which case the relevant event is “Which video we are watching”, or if it is a simulation, in which case the relevant event is “The outcome of each coin toss.” Either answer works, or you can split the difference, because at this point a large part of the probability-space is devoted, not to the events unfolding, but towards the ambiguity in what events we’re even evaluating.
The point there is that there is no contradiction because the informational content is different. “Which is the baseline” is up to the person writing the problem to answer. You’ve asserted that the baseline is A vs B; then you’ve added information that A is actually A1 and A2.
The issue here is entirely semantic ambiguity.
Observe what happens when we remove the semantic ambiguity:
You’ve been observing a looping computer program for a while, and have determined that it shows three videos. The first video portrays a coin showing tails. The second video portrays two coins; the left coin shows heads, the right coin shows tails. The third video also portrays two coins; the left coin shows heads, the right coin shows heads.
You haven’t been paying attention to the frequency, but now, having determined there are three videos you can see, you want to figure out how frequently each video shows up. What are your prior odds for each video?
33/33/33 seems reasonable. I’ve specified that you’re watching videos; the event is which video you are watching, not the events that unfold within the video.
Now, consider an alternative framing: You are watching somebody as they repeat a series of events. You have determined the events unfold in three distinct ways; all three begin the same way, with a coin being flipped. If the coin shows heads, it is flipped again. If the coin shows tails, it is not. What are your prior odds for each sequence of events?
25/25/50 seems reasonable.
Now, consider yet another framing: You are shown something on a looping computer screen. You have determined the visuals unfold in three distinct ways; all three begin the same way, with a coin being flipped. If the coin shows heads, it is flipped again. If the coin shows tails, it is not. What are your prior odds here?
Both 25/25/50 and 33/33/33 are reasonable. Why? Because it is unclear whether or not you are watching a simulation of coin flips, or something like prerecorded videos; it is unclear whether or not you should treat the events within what you are watching as events, or whether you should treat the visuals themselves you are watching as the event.
Because it is unclear, I’d lean towards treating the visuals you are watching as the event—that is, assume independence. However, it would be perfectly fair to treat the coin tosses as events also. Or you could split the difference. Prior probabilities are just your best guess given the information you have available—and given that I don’t have access to all the information you have available, both options are fair.
Now, the semantic ambiguity you have introduced, in the context of this, is like this:
You’re told you are going to watch a computer program run, and what you see will begin with a coin being flipped, showing heads or tails. What are your probabilities that it will show heads or tails?
Okay, 50⁄50. Now, if you see the coin shows heads, you will see that it is flipped again; we now have three possibilities, HT, HH, and TT. What are your probabilities for each event?
Notice: You didn’t specify enough to know what the relevant events we’re assigning probabilities to even are! We’re in the third scenario; we don’t know if it’s a video, in which case the relevant event is “Which video we are watching”, or if it is a simulation, in which case the relevant event is “The outcome of each coin toss.” Either answer works, or you can split the difference, because at this point a large part of the probability-space is devoted, not to the events unfolding, but towards the ambiguity in what events we’re even evaluating.