Another Ace on the turn

[uphigh_downlow]

Ther has been a small discussion on this topic in the SNG forum.

http://forumserver.twoplustwo.com/showfl...e=0#Post6155615

The argument being that

there is an ace on the flop
ACTION takes place
another ace hits on the turn

Without considering the action that took place, you can assume that there is a lower chance that your opponent has an ace, because of the turn.

I'm not entirely sure, although I lean towards disagreeing with that sentiment. Course I'm asking here because I could find myself arguing for it as well.

Mebbe I'm just having trouble summarising my thoughts clearly to myself. Uh, the worst kinda trouble. [img]/images/graemlins/laugh.gif[/img]

Any opinions here??

[Tom1975]

Ignoring the action in the hand, the fact an Ace hit the turn makes it less likely your opponent holds an Ace. After the flop, there were 47 unknown cards, 3 of which were Aces. After the turn, there were 46 unknown cards, 2 of which were Aces. The people who are arguing that he already has his cards so the probability can't change are dead wrong. In the thread you referenced, Shillix does a good job of explaining why this is.

[uphigh_downlow]

I buy your point.

Its quite clear to me that of all the times a second ace comes out, the number of times your opponent started out with one ace is less than when the second ace does not come out. This can simply be seen by enumerating the cases.
-------
BUT...
Does the fact that this is a single instance matter?
What has happened has happened. there is no probability anymore.


In that case, the opponent starts with what he has.

The fact that another ace came out on the turn does not reflect a lower chance of having an ace than on the flop.


Damn this is a hard pill to swallow both ways. I'll take the blue pill.

[SamIAm]

[ QUOTE ]
In that case, the opponent starts with what he has.

[/ QUOTE ]Nobody's saying it changes his cards; the idea is that it's improving the best guess of what his cards are, now that you have more information. For instance, before the A[img]/images/graemlins/spade.gif[/img] came on the turn, you thought he could have the A[img]/images/graemlins/spade.gif[/img], right? When the card came, you realized that he obviously didn't have it as a hole card. It didn't "change his cards", though.
-Sam

[pzhon]

[ QUOTE ]

Does the fact that this is a single instance matter?
What has happened has happened. there is no probability anymore.

[/ QUOTE ]
No, that you interpret this as a single instance doesn't matter.

Some explanations of probability ask you to think about situations that are repeated many times. However, that is just a pedagogical tool. Probability makes sense when you only have one trial.

[Tom1975]

[ QUOTE ]

BUT...
Does the fact that this is a single instance matter?
What has happened has happened. there is no probability anymore.
In that case, the opponent starts with what he has.



[/ QUOTE ]

Ok, let's play a game. Forget poker. Say I have bag containing 1000 marbles. One is white and 999 are black. We have an independent third party select one marble and hide it where neither of us can see it. I then foolishly decide to bet 10 dollars against your 100 that the marble is white. You immediately agree since you’re laying 10-1 odds on a 1000-1 shot. At this point the odds that the marble chosen is white are .0001. I now blindly remove 998 marbles from the bag, all of which turn out to be black. I then offer you the chance to retract the bet. Do you accept? According to your theory you shouldn’t because “What has happened has happened. there is no probability anymore.”

[AaronBrown]

The probability question is controversial and requires a lot of hair-splitting to answer. What definitely changes is your subjective belief that another player has an Ace.

To make it unambiguous, change the problem to:

(1) Two hold'em starting hands are dealt. You look at one and it has no Ace. The other is left face down on the table, no one looks at it. The probability that it contains at least one Ace is 1 - (46/50)*(45/49) = 0.1551. Some people would insist tiresomely that the probability is either 0 or 1, you just don't know which. But we often use the word "probability" in this context to allow answers like 0.1551 which really means either (a) that's your subjective belief (Bayesian), (b) that's what would happen if we repeated this experiment many times (frequentist), or (c) something more complicated.

(2) A flop is dealt with exactly one Ace. Now the probability that the unseen hand has at least one Ace is 1 - (44/47)*(43/46) = 0.1249.

(3) The turn is an Ace. Now the probability that the unseen hand has at least one Ace is 1 - (44/46)*(43/45) = 0.0860.

The only difference in your example is that someone has seen the hand, and you can make additional inferences based on her actions. For example, the fact that she saw the flop raises the chance she has an Ace to maybe 40% (depending on the game and the player, of course) because people are more likely to stay in with Aces than other cards.

If she looked happy and raised after the flop, that's more evidence for having an Ace. It might even cause your belief she has an Ace to increase from 40%, despite the fact that with an unseen hand the Ace on the flop caused the probability of an Ace in the hand to go down.

If she laughs and claps, "A set! I'm all in!" when the turn card comes, that would also influence your estimate (which way depends on how big an idiot the player is).

[uphigh_downlow]

[ QUOTE ]

Ok, let's play a game. Forget poker. Say I have bag containing 1000 marbles. One is white and 999 are black. We have an independent third party select one marble and hide it where neither of us can see it. I then foolishly decide to bet 10 dollars against your 100 that the marble is white. You immediately agree since you’re laying 10-1 odds on a 1000-1 shot. At this point the odds that the marble chosen is white are .0001. I now blindly remove 998 marbles from the bag, all of which turn out to be black. I then offer you the chance to retract the bet. Do you accept? According to your theory you shouldn’t because “What has happened has happened. there is no probability anymore.”

[/ QUOTE ]

I like the idea of this game. It definitely helps in simplifying things.

Allow me to propose an alternate game.

There are 6 marbles in a box 3 white and 3 black. You take out one marble (___LOOK AT IT___) and hide it and offer me 15 to my 10 that its black.

I accept the bet. Now you draw a random marble from the box, and place it outside. Both of us see that it is black. Now you make an offer of doubling the stakes or retracting the bet.

What shud I do? How shud I deal with the new information that has been presented given that you have already seen the hidden marble, and accept/decline any alteration in our original bet??

is this new information relevant at all??

[uphigh_downlow]

[ QUOTE ]
The probability question is controversial and requires a lot of hair-splitting to answer. What definitely changes is your subjective belief that another player has an Ace.


[/ QUOTE ]

Its no surprise that you ave effectively summarised my dilemna.

I feel that our subjective belief that opponent has an ace might increase/decrease in different cases, as a result of the ace, and the consequent action.

Wud it be fair to claim that without information of the resultant action, we cannot claim any change in our subjective beliefs??

--Is this a question that actually belongs somewhere in the psychology or mebbe poker theory forum?

[AaronBrown]

[ QUOTE ]
Allow me to propose an alternate game.

There are 6 marbles in a box 3 white and 3 black. You take out one marble (___LOOK AT IT___) and hide it and offer me 15 to my 10 that its black.

I accept the bet. Now you draw a random marble from the box, and place it outside. Both of us see that it is black. Now you make an offer of doubling the stakes or retracting the bet.

What shud I do? How shud I deal with the new information that has been presented given that you have already seen the hidden marble, and accept/decline any alteration in our original bet??

is this new information relevant at all??

[/ QUOTE ]
Clearly, you should refuse all bets before or after seeing the second marble. I should only offer the bet if I know I win it. There's no new information, because my offering the bet already told you the color of the hidden marble.

If that assumption is not true, if I sometimes offer bets that lose for me, then the new marble could have some information (also, your new offer could have some information). But in order to assess it mathematically, I have to know how you decide which bets to offer.