The fairness debate

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2495

Hi @SoupChicken, I wonder if you have heard of Google? 60 seconds of research makes you look 60% smarter on the internet.

From AA vs KK - odds and probability for the poker hands AA vs KK - Poker Bankroll Blog

The odds of being dealt any specific pocket pair, such as aces, is 220-to-1 but the odds of being dealt a pair of aces and then someone at the same table being dealt pocket kings is slightly less as the two aces have been removed from the 52-card deck. This means that the odds of someone being dealt a pair of kings when you have aces is 205-to-1. However, that only applies when you are heads-up against a single player, against a full table with nine other opponents you will find yourself in an AA vs KK situation once in every 20 times you are dealt aces.

Hope this helps.

Regards,
TA

2496

Analyst

You should take your own advice because you are confusing a “conditional probability” with the probability of an event .

Stating the “chances of AA WHEN I already have a KK” is what is called a “conditional probability” and it is completely different probability calcluation of “the chances of AA and KK being dealt to two players” in any one deal

In my post i was careful to say “chances of AA and KK being dealt to two players”, not the conditional “what are the chances of AA if I already have KK” which is what you have posted and calculated.

I hope I have just made you 60% smarter theanlayst

Hope that helps

2497

@SoupChicken, I really don’t see your argument.

If “A” is that you are dealt AA preflop and “B” is that your opponent, heads up as you stated, is dealt KK, what we want to know is P(B|A). I have answered that question.

Now then, as stated above, the odds of any given player being dealt a specific pocket pair is 1:220 - more often said to be 1:221 but it makes no difference for this discussion. Therefore you, specifically, should expect to be dealt pocket kings once in every 220 hands on average.

We also know, from above, that your opponent (because you stipulated heads-up play) will be dealt pocket aces once for every 20 times that you have pocket kings.

Given that, we expect to see AA vs KK (pocket pairs) once in every 4400 hands. That is a full order of magnitude less than you suggested.

It is important to note that these are long term expected values. I would expect to see a clear trend towards the expected values over a sample of some 10’s of thousands of hands. Your results, over “less than 3000s deals” are not at all remarkable or interesting.

If you want to do some more reading, please have a look at:

The “executive summary” version is that it would be highly suspicious if results such as those that you observed did not happen. We fully expect that these “unlikely” events will occur in groups. You observed such a grouping and now you can sleep knowing full well than the random world is operating entirely as expected.

For the record, I did say that research is the bit that makes one appear smarter. If you care to discuss this further, with reputable citations that support your claims, I am always interested in improving my knowledge.

Understanding random distribution is not at all intuitive and this often leads people, including me, far astray. In fact, it is because I understand it so poorly that I rely on quoting other people.

Regards,
TA

1 Like
2498

Yes you gave the conditional probability P(A|B), but as I said , that’s not the probability I was referring too

I in my post was referring to A AND B together , P(A and B)

Which will be (very) approx 1/221 * 1/221 = 1:40000 deals in a heads up (very approx because they are not independent events)

Regarding the number of deals I used. Yes it’s not enough, which is why i am still recording it and analysiing, and why i put the question out to others, what are your experiences.

However I also gave the example of 15 showdowns where i was 80:20 to win on the turn but only won three of them. The chances of 3 success from 15 with p=0.8 is way less than 1%

2499

I suspect that my English skills are as substandard as my mathematical skills since I, quite obviously, am unable to understand your distinction in the question. I have answered the only question that I can see.

If neither player has aces or kings then this whole discussion is rather silly :slight_smile:

Given that one player has kings (or aces, if you prefer), what are the chances of the other player having aces (or kings)? The answer, my friend, is P(A|B).

That is the answer to Player A has pocket aces AND Player B has pocket kings and vice versa, of course.

Quoting from:

"I’ve had many friends ask me what the odds are and I finally sat down and did it.

By the way, you get KK, on average, once in 221 hands and you will run into aces about once in every 22 times you get KK (in a ring game). So KK vs AA will happen to you about 1 out of 4642 hands you play."

There is a fairly decent discussion there but nothing that isn’t covered elsewhere with more clarity.

You may also wish to read:

This is an interesting discussion of various odds regarding aces in no limit hold 'em.

Regards,
TA

2500

Then I am afraid , analyst, that indeed you don’t understand the difference between a conditional probabiltiy P(A|B) and the probabilitiy of two events happening P(A and B)

P(B|A) is not equal to P(A and B)

Maybe go back to Google like you suggested to me

Conditional probability - Wikipedia

2501

@SoupChicken, this doesn’t seem to be advancing the discussion for the benefit of the community so, with the greatest of respect, I will leave things where they are now.

Regards,
TA

1 Like
2502

Yeah I don’t see it either, though I admit to being 60% dumber because I refuse to Google it.

P(A AND B) can’t be true unless A is true. The real question being asked is, “If I have pocket aces, how often will someone have kings?” (or the other way around)

Either the condition is implicit or the question is meaningless… what is the chance of aces and kings being dealt in the same hand when nobody is dealt aces?

2503

My original post stated that I had seen pockets AA (Event A) AND pockets KKs (Event B) being dealt 3 times in less than 3 thousand hands to two players

So i asked myself what are the chances of this ie what are the chances P(A and B) . Did i see something that is highly unprobable.

And yes I concluded that what i saw was improbable , to do this I calculated P(A and B) in a heads up which will happen about 1 every 30-40K hands. So what i saw was improbable imho

However theanalyst stated that this will happen every 200 deals , but unfortunately he was confusing P(A and B) with P(A|B).

2504

If we agree that the answer will depend on the number of players being dealt in, which seems pretty obvious, then we should agree that we would have to specify the number of players.

Since the number of players is a condition, the question is conditional.

It’s either conditional or not conditional. Adding other conditions doesn’t change that. It’s thus either conditional or meaningless.

2505

I asked the “what is the chance of these two events both happening” in a heads up deal, I didn’t phrase it conditionally.

2506

“heads up” is a condition.

2507

not the conditional that this discussion is about though.

This condition is “I look down and see K-K, given that I have K-K what is the chance that Villan has A-A”, which is indeed around 1 every 200 or so P(A|B) , in heads up.

But it is an entirely different probability to P(A and B)

2508

Also, we obviously can’t multiply the chances because that makes it seem much less likely. If one player gets aces, for example, the second player is now looking for 2 of the 4 kings in the remaining 50 cards, making it more likely, not less.

Take this further and you should see the point. With 20 people in the hand, there would be a much greater chance. If we multiply the 220-1 chances, it would soon become nearly impossible when its clearly more and more likely with a larger number being dealt.

2509

Yes of course the probabilities would be higher the more players. But I wasn’t playing with lots of players

My original post was simply " I think i have seen an highly improbable number of K-K, A-A being dealt together. Here is the number of hands and the number of times it occurred. What are other peoples experience of Online money games"

(because I am new to online money poker)

2510

My point was that, if you can’t simply multiply the odds with 20 players, you can multiply heads up either.

We can make a truth table for A AND B…

A=1: B=0; A AND B = 0
A=0; B=1; A AND B = 0
A=1; B=1; A AND B = 1

But…
A=?; B=0
A=?; B=1

Cannot be evaluated. Only when A=1 (someone is dealt aces) can we proceed with the evaluation. That aces have been dealt is an implicit condition for the question to make sense. If aces have not been dealt A AND B can never be true. Only if we stipulate that aces have been dealt can we calculate the odds of kings being dealt too. I don’t see how we can escape the conditionality of the question and still get an answer.

2511

There is no issue looking at both probabilities. Both can be useful for evaluation.

So, in a heads up I can calculate P(A|B) and i can calculate P(A and B)

The only issue is when people get them confused. P(A and B) happens about every 30-40K hands, whereas P(A|B) happens once every 200 hundred hands when I have K-K

Think about two dice instead. What is the probability of throwing two 6s, and given that the first dice is a 6 what is the probability that the second dice will be a 6

P(A|B) = 1/6, and P(A and B) = 1/6 * 1/6 = 1/36

No problem to calculate both

2512

Yes, I (finally) see your point.

The way I would approach it is like this…

If I have aces, there are 50 cards remaining, of which 4 are kings. So my opponent’s first hole card will be a king 4 in 50 times. There are then 49 remaining cards and 3 kings, so the second will be a king 3 in 49 times.

(4/50) * (3/49) = 0.005 and 0.005 x 100 = 0.5%, which is 1 in 200

But yes, you are right to say this will only happen about once in 40k hands.

The last time I was in Vegas, I was dealt KK in a $100 SnG. Myself and 2 others ended up allin preflop. Both of the other players had AA. It happens, thankfully, not that often!

2513

image

2514

And just to go full circle, my original post was that I saw this happen 3 times in 2 days of online play, which just seemed so improbable

And of course the reason that this particular scenario is interesting is that it usually ends up with someone losing a very big pot at the showdown.

So that and other things just made me a bit suspicious , since I’m new to money poker.

so just wondering what other’s experiences were