lihiue,
According to AI, your analogy is sharp and intuitive.
AI goes on and on…
Exactly. You nailed the core principle: shared information creates dependency, and that fundamentally alters how probabilities behave.
In your coin analogy, the community coin is like the flop in poker—it’s a shared variable that links all players’ outcomes. So while each player’s private coin flip is independent, their final result (two heads) depends on both their own flip and the shared coin. That’s why you can’t just multiply 1/4 × 1/4 for two players—it would wrongly assume the community coin is flipped twice, which it isn’t.
In poker terms:
- The flop is fixed for all players.
- So when calculating the odds of multiple players hitting a boat, you must condition on the flop.
- Once the flop is known, the probability space collapses—everyone’s hand interacts with the same board, so their outcomes are correlated, not independent.
This is a classic mistake in poker probability: assuming independence where there’s shared information. It’s why calculating the odds of anyone flopping a boat is very different from calculating the odds of multiple players flopping boats simultaneously.
Your initial estimate didn’t account for all the permutations—like how many combinations of hole cards could produce a boat given a specific flop, or how many board textures allow it. Once you factor in:
- The number of players
- The number of possible hand combos per player (Omaha gives 4 hole cards!)
- The board texture constraints (paired vs unpaired)
- The requirement for a boat (set + pair or trips + pair)
…it’s easy to see how your original number could be off by 24x. And yeah, 1 in a few million feels about right for some of those ultra-specific scenarios.
