Probability question

[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]

Right, and you are reading it wrong. Everyone is talking about the present. Why would we care what the odds WERE preflop that someone was dealt an Ace when we are on the flop now, trying to play poker and make decisions? Of course we need to use all of the information available to us.

I am saying that the more Aces on the flop, the less likely it is (NOW) that one of our opponent holds one, and MathBabe is disagreeing with me. She is saying that the odds remain the same (NOW) regardless of how many Aces were flopped because the hole cards were dealt first.

If I am misinterpreting what you are saying MathBabe, please clarify... but I can't think for the life of me why anyone would care about completely irrelevant odds (what the odds WERE preflop, discounting all the information we have available to us now after the flop), so I have to assume we're on the same page here.

[MathBabe]

I almost posted the other day and said exactly what you said, but I realized later he's actually right. It's a question of conditional probability.

"What's the probability of the fourth Ace in play GIVEN there are three Aces on the board".

A = three aces on the board
B = exactly one hole card ace

P(B|A) = P(A and B)/P(A)
=1/1353 / 1/553
=553/1353
=1/2.45
which is about 40%.

My calculations assume 10 players, therefore 20 hole cards, and the three Aces being anywhere on the board - not just on the flop.

I guess we'll have to agree to disagree...

[MathBabe]

You need to go read this thread:

[Talking Poker]

Earlier in this thread, you said it was 74%. So, which is it?

I'm not certain I follow your exact math above (I don't know where you came up with those numbers nor do I understand why you changed from the flop to the full board), so let me ask you this: What's the probability of the fourth Ace in play GIVEN there are four Aces on the board? Please use the same "math" you used above to show me how this is also 37%, as I believe you claimed earlier in this thread (you said it's always 37%, regardless of what comes on the flop).

As for agreeing to disagree, I don't see how we could agree to do that. There are right and wrong answers here.

I'm also surprised more people haven't weighed in on this thread. Is it really over everyone else's head, or do they just not care?

[GTDawg]

That's the wrong way to look at probability. You don't put probabilities on events that have already happened. You put probabilities on events that are about to happen given past events.

If you want to ask what the probability of someone having the fourth ace, given AAA flop...sure we can calculate that.

What are the chances of getting any one single hand in poker? Do those probabilities change by what card comes on the flop...turn...river?

The probabilities don't change based on what cards come out afterwards. Why?

[Talking Poker]

No, the PAST probabilities don't change. But I promise you that it is FAR less likely that someone was dealt AA when the flop comes AAA than when the flop comes 234. Yes, when the cards were dealt, the probability for any given player to be dealt AA was 220:1. We all agree with that. But that is completely useless information to us now that we've seen the flop. When putting our opponent on a hand (why else would we care about these probabilities in the first place?), we need to use all of the information available to us, including the flop. To ignore it and think that there is still approximately a 5% chance that someone at the table is holding AA would be asinine.

I really don't get what is so difficult about this. Maybe I'm WAY OFF in my thinking here (that certainly could be the case), but this just seems like common sense to me.

I use the AA hole cards with an AAA flop because I think it is so obvious and simple to understand. A more practical application would be what I posted (and no one responded to) in this post:

[Akverno]

I must have missed that one, anyway, as I said earlier conditional probability. I don't remember formulas, don't remember why or how, just remember a teacher discussing cards in a class and this was a similiar example to some problem we did.

[Talking Poker]

I've decided to settle this once and for all, so I'm spending some time here creating a simulation. What I'm going to test is (approximately) this:

I'm going to force the above three flops and run run 100,000 hands with 10 opponents each for a total of 1 million hands dealt for each scenario. I will then see how many times AA was dealt to each player, and compare the totals.

I believe that AA will be dealt more times when the flop is A92 than when it is AA9 or AAA. Please correct me if I am wrong, but I believe MathBabe thinks that AA will be dealt the same number of times regardless of the flop (well, between the AA9 and A92 flops). Is that correct, MB?

I haven't run this yet, so I don't KNOW the answer. But I'm going to find out and we are going to settle this today.

[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.

[Talking Poker]

It doesn't take all that long to run the simulations, but collecting the data afterwards and putting it all together takes quite a while. This is how much I've done so far, and it exactly supports (proves) my hypothesis.

For 100,000 simulated hands...

AAA flop:
#AA dealt - 0 (0%)
#Ax dealt - 40,674 (40.64%)

AA9 flop:
#AA dealt - 840 (.84%)
#Ax dealt - 77,133 (77.13%)

Please note how with less Aces on the flop, there are now more Aces available to be in people's hole cards (2 instead of 1), and therefore the likelihood of at least one of our opponents holding at least one Ace increases. Exactly like common sense tells us.

Please let me know if you'd like me to continue with A92 and so on... I will, but surely this is enough FINALLY to prove my point, yes?