So the Monty Hall Problem itself is widely known and understood.
Nonetheless, a friend of mine and I were wondering whether the the same strategy could affectively be applied by a participant of Who wants to be a Millionaire? when using the 50/50 Joker.
Let's imagine the following scenario:
The participant P has no clue about the correct answer $ x \in \{A,B,C,D\} $ and wants to use the 50/50 Joker (eliminating two wrong answers). But instead of immediately going for it he first "preselects" one of the answers in his mind. There is no need to tell Quizmaster Q about his "imaginary preselection". Now P tells Q that he wants to use his joker and Q lets the computer eliminate two wrong answers.
(1) In case the answer P had preselected is eliminated he has no choice but to choose between the remaining two answers, effectively leaving him with a 50% chance of success - no magic happening here. (2) But what about the other case when the answer P had preselected survives the elimination? According to the Monty Hall Problemit seems as if changing the selection (i.e. choosing the other remaining option P had not preseleted) seems to give him a 0.75 chance of success.
Nevertheless, I find it hard to believe that this actually holds true, since the so called 50/50 (!) Joker would then not be p(success) = 0.5 after all. Additionally it seems unlikely that making an "imaginary preselection", no one else is told about, actually increases your odds.
I know this problem is not exactly the same as Monty Hall since the the quizmaster does not always eliminate answers only from the ones the participant had not "preselected", meaning that the preselection itself could be eliminated, too, as it happens in (1). Still the second case seems to actually be a just variation of it.
So are we right and making a preselection and then going for the other remaining option is a valid strategy that increases the participant's odds of winning? If not, please help us understand our misconception.
Monty Hall does not give you information about your preselection. Therefore the probability that your first choice is right given that it is still available after the intervention, is not changed. The 50/50 Joker does give information about it (esp. when it gets eliminated). Note that many of the misunderstandings of the Monty Hall problem arise from the missing assumption that the host always willfully opens a goat-door other than the preselected door. If you modify the Monti Hall problem so that the host opens any not preselected door at random, the general misconception that both remaining doors are "equal" becomes correct.