The Pokémon TCG Pocket game has a mechanic where you are shown 5 cards face down, in which there is usually only one desirable card, and let you pick one at random to keep
That's all well-and-good, but recently they added an event where you get to "peak" at one of the cards before you make your choice.
Now - if the one you peak is correct, you can just pick it, so that math ends there. However, if the card you pick is incorrect, does the logic of the Monty Hall problem apply?
I, and I think most others, mentally decide a card to pick before it's even time (I always pick the bottom right). If I reveal a different card and it's incorrect, is it statistically probable for me to forsake my mental guess and pick one of the other 3 cards? This feels wrong, the game didn't know that was my choice, it feels like it should now be no less likely than the other 3 cards.
However, wouldn't the logic of the Monty Hall problem apply to this, and say it is incorrect? That, logically speaking, my initial probability doesn't change from 1/5 despite the fact another one was eliminate (this is, to be clear, under the theoretical that the 'peaked' card is wrong, as the peaked card being correct ends the scenario). If there was a 1/5 chance my initial guess was correct, there is a 4/5 chance it was wrong. If a card is revealed, there is still a 4/5 chance I was initially wrong, but if there are only 3 possible cards to switch to, they split that 4/5 3-ways, making them each 26.66% likely to be correct (as opposed to my 20%), no?
This is my way of understanding the Monty Hall problem but practically speaking I don't feel like that can be incorrect? The game doesn't even know my initial 'mental' pick, so how could there be a statistical difference if I choose it or swap it.
And if any of the 3 swaps are really 26.66% likely in that scenario, wouldn't that mean mentally envisioning one and then swapping it is actually (very slightly) more likely than peaking at one at random and then picking one of the remaining 4 (25%)? Again, in both these scenarios just not factoring in the possibility that the peak is correct, which should apply to both scenarios equally anyway and not change the end result.
I don't know what the flaw in my logic is but I can't imagine that 26.66% correct. If the 1/5 chance the "peaked" card was incorrect is distributed to all 4 cards then it's 25%, but wouldn't that mean the Monty Hall problem results in a 50% chance instead of a 66% chance?
Can anyone help me break down the probability in this scenario? Is there a flaw in my understanding of the probabiltiy or do you really increase your chances by mentally choosing one and then refusing to take it?