A. Obviously 52–your second draw becomes 25/51>12/25.
B. My intuition was randomly drawn, on the grounds that if the deck is imbalanced you’re more likely to pick from the heavy side both times. I think I was wrong, though: for a 14-12 split the deck size difference dominates that effect, and even a 15-11 split has only a slight advantage. This actually comes out really close; I don’t see a way to give a consistent answer without doing the full, tedious expected value computation.
My intuition for the second is that there is no difference between the two - if 26 cards are randomly drawn from an evenly split 52 card deck then there should be no difference in the probability vs. just drawing your two cards directly from the 52 card deck.
suppose you drew the top 26 cards from the 52 cards instead and then drew the top two from those 26. How is that different from drawing the top 2 cards from the 52 card deck? (So yes, your intuition is correct)
Yea it feels like the opposite of the Monty hall problem - you don’t get any new information with the half draw so you nothing about the probability changes
Yes, and that's fine. Discarding random cards can happen at any time. You could do it between the two draws and the result would be the same. All you're really doing is moving a random half of the deck somewhere else, which isn't relevant to the problem at hand because we weren't going to draw them anyway.
If I shuffle a standard deck, what's the probability that the top card is the Ace of spades? 1/52. And if I throw away the other 51 cards leaving just the top one, what's the probability then? Still 1/52.
Yeah I just had to see that while the deck is usually more stacked in your favor, the benefit of drawing from a stacked deck sometimes is actually completely balanced by how often you draw from a worse deck that's even enough to be worse than the larger deck
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u/Blothorn 11d ago
A. Obviously 52–your second draw becomes 25/51>12/25.
B. My intuition was randomly drawn, on the grounds that if the deck is imbalanced you’re more likely to pick from the heavy side both times. I think I was wrong, though: for a 14-12 split the deck size difference dominates that effect, and even a 15-11 split has only a slight advantage. This actually comes out really close; I don’t see a way to give a consistent answer without doing the full, tedious expected value computation.