Updating priors?

[Alan3]

This seems like a pretty simple question, but I can't convince myself that either answer is correct.

Short version:
In Hold'em, if I see two aces on the flop, can I lower the expectation that my opponent has an ace?

Long Version:
If I am playing heads up and don't have an ace, then there is a 15.5% chance my opponent has an ace. If he has an ace (or two) then the chance that an ace (or two) will hit the flop is reduced. But that event is going to happen sometimes, even when the chance is reduced. The three cards that show on the flop cannot change the cards that were dealt previously when my opponent had a 15.5% chance to receive the ace.


On the other hand, if the flop were dealt before the hole cards and two aces came up then there is only a 8.4% chance of my opponent having an ace. I have some trouble believing that the order in which the cards are dealt have any bearing on the odds of certain combinations.

I'm leaning towards the reduced chance of my opponent having aces, but I get these sort of probability questions wrong pretty often.

[Tom1975]

[ QUOTE ]
This seems like a pretty simple question, but I can't convince myself that either answer is correct.

Short version:
In Hold'em, if I see two aces on the flop, can I lower the expectation that my opponent has an ace?


[/ QUOTE ]

The short is Yes. There's a difference between asking "What are the chances that when event A happens, the result will be outcome X?" and "Given that event A has already happened and I have acquired some information which has bearing on the result, what are the chances of outcome X?"

Let's look at an extreme example. Suppose there are two Aces on the flop. Now we deal the turn and river, and they are both Aces. Do you still think that there's a 15.5% chance your opponent holds an Ace?