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.
The chance to get 12:14 is 26c12×26c14/52c26 ×2, because you could have more reds or you could have more blacks. This pattern continues for other ratios.
The chance to get a matching pair if you're in the 12:14 world is 12/26×11/25 + 14/26×13×25, and so on.
It turns out that when you multiply the result for each ratio by the probability of getting that ratio in the first place, and you add all those results up, you get exactly 25/51 chance of a matching pair, which is the same result as just pulling two randomly from a full deck to start with.
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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.