Probability question

[MathBabe]

Absolutely incorrect reasoning.

But I'm too tired to figure out a better way to explain right now.

A demain!

[Talking Poker]

I'll be looking forward to your attempts at "math bending" tomorrow. Cause, well, it's just math, sister. And math doesn't lie.

{This isn't to say that I haven't made an error in my computations somewhere along the way - that's quite possible. But the point I'm trying to make is valid: If there are three Aces on the board, the chances of someone holding AA are 0%. If the 4d is on the board, the chances of someone holding the 4d are 0%. Trying to prove otherwise will be... well... interesting, if nothing else.}

[GTDawg]

I think you skipped right to post-flop while she is talking about pre-flop still.

You guys keep going around in circles talking about different things.

[Talking Poker]

How could she be talking about preflop when she is saying things like: "regardless of that flop Ace, since their cards were dealt first."



I'm pretty sure we're talking about the same thing here.

[GTDawg]

The point that she made that comment is in reference to calculating the chances of the aces being in play before the flop...in which case, it wouldn't matter what the flop was, because it hasn't happened yet.

=========

In an attempt to clarify...

She is referring to the chances of dealing out the aces preflop. Someone getting AA, or two people getting an A...or whatever.

Sure, after the flop, you can eliminate some choices or teh probabilites can change (which is what you are referencing)...but that's not what she is talking about.

[Talking Poker]

Actually, that's exactly what I'm saying. Maybe I'm misunderstanding her, but rereading what she has written, I don't think so. She has very clearly stated that the order of events matters and that AFTER the flop, regardless of the board, the probability of one of our opponents holding X (whatever) does not change... When in reality, it does.

Maybe you're right and I'm misinterpreting what she's saying, but if that's the case, she needs to be a lot more clear when choosing her words.

[GTDawg]

But when you originally calculate the probability of the events when dealing cards...the flop doesn't matter.

And, when you see an ace on the board, it doesn't change the probability of someone being dealt those cards...it changes the probability of someone having those cards.

Two different things.

[MathBabe]

I always pick out the piece I am responding to, so don't think I'm arguing about something I'm not. I was not arguing with this comment:

I am arguing with this:

Just because there are three Aces on the board, you can't go back and re-calculate the probabilities of someone getting an Ace for a hole card. All you can do is eliminate a category of event.

Look at it this way. With 10 players, here's how the scenarios break down:
A: 13% probability that nobody has a hole card Ace
B: 37% probability there is one hole card Ace
C: 35% probability there are two hole cards Ace (may or may not be AA, we don't care)
D: 13% probability there are three hole card Aces
E: 2% probability there are four hole card Aces

If you see AAA on the flop, you know you are in the 50% of the time that there are none or one Aces in the hole cards. But that doesn't make the probability that there is a fourth Ace in play 20%, as you calculated - it's still 37%.

In fact, you could argue that since you've narrowed it down to those scenarios, and their relative probability is the same, there's actually a 26% chance you are in scenario A and a 74% chance you are in scenario B!

I think this is worth niggling about, because if you see three cards of a rank and you think there is a 4:1 chance of something having the fourth card, it's really more like 2:1. It may not happen often enough to affect your winnings, but over hundreds of thousands of hands, it you assume that fourth card isn't out there, you're going to be wrong more than you should.

[Quint]

Before the flop you know what 2 of the 52 cards are (your hole cards). After the flop you know what 5 of the 52 cards are. You've gained information, certainly you can, and should, recalculate the probability of the other Ace being dealt.

Using your logic, if the fourth ace comes on the turn is their still a 37% chance someone has an ace?

[Talking Poker]

I believe MB's response to this will be that we would need to rule out Case B (as we did on the flop with Cases C, D, and E), but she will incorrectly state that Case A remains true - that there is still only a "13% probability that nobody has a hole card Ace" when in reality, we know that this number is 100%.

[GTDawg]

Well, I am sufficiently confused as to what this thread is asking. I've tried to read original posts a few times and I am completely lost.

I will thus conclude that everybody is right because I really don't know if we're talking about the same thing or not.

Time for golf.

[Talking Poker]

Well then, your conclusion is wrong.

I'm actually surprised you find this so confusing. I think MathBabe and I understand exactly what the other is saying. Just read it all carefully, one post at a time.

[New Guy]

0%

[GTDawg]

Ok...I've read the first few posts and again..I'm at the same point I was a few hours ago.

She is talking about the chances of someone being dealt a hand in poker. The probability of someone being dealt a hand in poker does not depend on what may or may not come on the flop...because you don't calculate future events when figuring out probabilities.

However, you are talking about the probability of someone having a certain hand given the information that a flop would give you.

Again, two different things.

I think you are reading her statement of "regardless of the flop ace" differently than I am.

I'm reading it as "it doesn't matter because it hasn't happened yet when you are calculating what people were dealt"

[Talking Poker]

I'm not sure how I can explain this any better... but what you are saying is just wrong. I thought the examples I've written would help you see the light, but so far, no such luck. Maybe I should think about this some more and post again when I can come up with a different way of explaining this.

Ok, how about this............:

1. Let's simplify this: Let's just talk about the Ac. I'm confident that with 20 cards being dealt to 10 players preflop, you will agree that the odds of the Ac being in play is 20/52=38%. Right? Now let's substitute that information into this quote of yours, fixing it to match our new scenario (I'll bold what I'm changing, and I'm cutting out the middle part that is irrelevant):

Old scenario:
New scenario:
CLEARLY, our new scenario is incorrect. I mean, the Ac is ON THE BOARD, so we know it's not in someone's hand, but using your logic, you would have me believe that there is a 38% chance that someone is also holding it. And that's obviously wrong. Agreed?

[Talking Poker]

Actually, let's focus on this. If I am reading this right, you are telling me that when three of a rank flop at a 10 handed table, 74% of the time, someone will have flopped quads and 26% of the time, no one will be holding the case card. Is that correct?

Because if it is, I'm pretty confident we can disprove that by running a simulation, and that should put an end to this argument right now.

I too think this is worth niggling about, btw - well, not for me so much, as I'm confident that I'm right - but for you and anyone else who is buying your logic, because I too agree that you are going to cost yourself a lot of money in the long run if you don't understand that the current probabilities of your opponents hole cards change as you gain more information.

Actually, here's yet another example. All in preflop on the WPT: AA vs KK. Normally, they would say the KK is about a 4:1 dog, but when someone at the table says "I folded a King," everyone understands that the KK guy now only has one out instead of two, making him a much bigger underdog. This "new information" has changed the probability that he will make the best hand with 5 cards left to come. I realize this isn't the same scenario as what we are talking about, but the logic is the same: We need to use all of the information available to us (all known cards) when determining probabilities (not probabilities for the last hand or the next hand or at a different street of this hand, but the probabilities right now).

[Talking Poker]

While I'm at it, I'm going to test this, as well:

To test this, I'm going to run 100,000 hands (10 players each) with the AAA flop and simply see how many times someone has flopped quads (meaning they were dealt the case Ace). According to MB, the expected answer here should be either 37% or 74%. I'm not sure - please clarify, MB.

According to me, it should be 20%, as per this:

Whooops. Looking at that again now, I think I made a (big) mistake. This would be if each player was only holding one hold card. Because each player is holding 2 hole cards, I think we need to multiple that all by 2: 1/49*20=40.8%. But that can't be right, because to do that with 0 Aces on the flop, we'd have a greater than 100% chance of someone holding an Ace.

Ok, I've definitely confused myself here. But dammit, I still want to run some simulations!!!

[bunny]

You're both correct in your own way. I initially disagreed with MB's first post. TP's post made more sense to me. But down the line, MB's post made sense too. You both confused me so much so, I had no choice but to look it up. Here's what I found. Hope this helps.

TalkingPoker = Bayesian.

MathBabe = Frequentist.

"There is a history of antagonism between Bayesians(TP) and frequentists(MB), with the latter often rejecting the Bayesian interpretation as ill-grounded. The groups have also disagreed about which of the two senses reflects what is commonly meant by the term 'probable'."

Bayesian statisticians believe that Bayesian inference uses aspects of the scientific method, which involves collecting evidence that is meant to be consistent or inconsistent with a given hypothesis. As evidence accumulates, the degree of belief in a hypothesis changes. WHAT TP WAS TRYING TO EXPLAIN With enough evidence, it will often become very high or very low. Bayesian statisticians also believe that Bayesian inference is a suitable logical basis to discriminate between conflicting hypotheses. Hypotheses with a very high degree of belief should be accepted as true; those with a very low degree of belief should be rejected as false.

Bayesian inference uses a numerical estimate of the degree of belief in a hypothesis before evidence has been observed and calculates a numerical estimate of the degree of belief in the hypothesis after evidence has been observed.

Bayesianism is more popular among decision theorists. Frequentists can't assign probabilities to things outside the scope of their definition. In particular, frequentists attribute probabilities only to events while Bayesians apply probabilities to arbitrary statements.


Frequentists talk about probabilities only when dealing with well-defined random experiments. The set of all possible outcomes of a random experiment is called the sample space of the experiment. An event is defined as a particular subset of the sample space that you want to consider. For any event only two things can happen; it occurs or it occurs not. The relative frequency of occurrence of an event, in a number of repetitions of the experiment, is a measure of the probability of that event.

(thankyou wiki )


Now you have no choice but to agree to disagree. So, shake hands, make up and let's move on.

With love...bunny



(now... back to clearing that "oh, so ever exhausting bonus" at UB)

[MathBabe]

How are you planning to force the flops? If you deal the flop first, I don't think that's a reasonable simulation. The best way to simulate would be to deal some huge number of hands, and only choose the ones where AAA came on the flop.

I think in the end we've arrived at the same answer. I said 40% when I used the conditional probability formula. Your mistake, which I have to admit I didn't catch, when fixed comes up with 40% as well.

That is correct. With more then one card, though, you can't use the easy 4/49*20 calculation - that's where you have to get into the combinations.

[Talking Poker]

Q: If I deal a gazillion hands and throw away all the ones that don't have exactly an AAA flop (keeping only those 100,000), what is the difference between doing that and dealing 100,000 flops with AAA and then distributing the rest of the hole cards?

A: Absolutely nothing! The only difference is the insane amount of time it would take to deal all those hands only to throw 99.9%+ of them away.

That logic right there alone should prove that there is absolutely no merit to your "it matters what happens first" line of thinking. If all the non-AAA (or whatever we are testing) cases are thrown away, why even bother doing them? The subset of hands we will be left with will be exactly the same as if we had put the AAA aside, dealt the hole cards, and then flopped the AAA.

And I'm still confused... did you not also say 74%? Where did that come from then?

And more importantly, how about the "real life scenario" that you have yet to address, where I say we have KK and the flop comes _ _ _ (multiple scenarios) and I want to know how likely it is that someone has a pair of Aces for each of them? Do you agree with me that the AA9 flop is safer for us than the A92 flop, for example?

[GTDawg]

In one scenario, you are distributing the hole cards...and then waiting for a AAA flop. You have four aces when dealing the pockets. The second scenario you already used three aces and you only have one ace when dealing the pockets.

No?

[Talking Poker]

Yes. But if we are throwing away all of the hands that don't have AAA flops, what's the difference, other than the amount of time it will take to complete the simulation?

In other words, we have two choices:

Choice 1:
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? YES!
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
etc, etc, etc for a long, long time to get 100,000 AAA flops.

Choice 2:
Just do this 100,000 times:
Deal AAA flop (set cards aside, whatever). Deal 20 hole cards.

In the end, you'll have the exact same 100,000 hands of AAA flops either way.