[jeremysalwen]

Following your arguments throughout this thread, I think the piece that is confusing you is the framing of the problem as a game-show host, which primes you to think of the host being "fair" by default.

To understand how the framing might change how you interpret the problem, consider the following scenario: You are in a game of poker, and you have a flush with king high. Your opponent reveals all but one card from their hand, which shows they have 4 hearts, and they also reveal that their last card is an ace, but they don't reveal its suit. It's your turn to bet. Do you bet, or do you fold?

Now you could treat this as a simple statistics problem -- there are four possible aces they could have in their hand, and only one is a heart, so only a 1/4 chance they will beat you. But is the solution to this problem that there is a 3/4 chance of winning the pot? In the problem text, we haven't specified under what conditions your opponent will reveal which cards in their hands. But somehow, by saying it's a game of poker makes you think that they probably are more likely to reveal their hand if they are bluffing, so the true probability is not 3/4.

We are primed by this description of this person as your "opponent" to think about them making the decision adversarially. What if instead we say that that game of poker is part of a game show and your opponent is the host of the game show? Depending on the assumptions you make about your opponent's motivations, you must calculate the odds differently, and simply saying "3/4" is not unambiguously correct.