1. With AK, what are the odds that one of the 9 opponents are holding a KK or AA?
2. With AK, what are the odds that one of the 9 opponents are holding a AA, KK, or QQ?
Thank you in advance.
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1. With AK, what are the odds that one of the 9 opponents are holding a KK or AA?
2. With AK, what are the odds that one of the 9 opponents are holding a AA, KK, or QQ?
Thank you in advance.
Ok...
When we have AK, there's three aces and three kings left out of 50 cards. Probability that an individual villain gets a king or ace first card is 6 in 50, then the chance of him getting another of what he just got is 2 in 49, so the probability is 6/50*2/49, which is approx 0.0048, or 0.5%, 1 in 200.
Since there are four queens, there's twice as many combos of QQ than there are KK or AA... or to put that another way, there's the same number of QQ combos as KK and AA... so the probability of him having QQ+ compared to KK+ is double... so almost 1%.
Now multiply that by how many opponents there are. In the case of 9 opponents, it's approx 4.5% for KK+, and approx 9% for QQ+
I took a shortcut with the queens. It's tricky to calculate the same way, since there's an extra queen, so the probability of him hitting a pocket pair with his second card is awkward... it depends if his first card was a Q or a K/A. I confess I don't actually know how to approach this using multiplication.
However, there's six different ways to make QQ... QsQh QsQd QsQc QhQd QhQc QdQc
(these are the different combos)
Let's say we hold AKss... now there's three ways to make KK... KhKd KhKc KdKc... and the same with AA
So from there we can see that the probability of villain getting QQ when we have AK is twice the probability of KK or AA, since there's twice as many ways to make the hand.
I think I figured it out. Let's see.Quote:
I confess I don't actually know how to approach this using multiplication.
First of all, there's four queens, three aces, and three kings, so ten cards out of 50.
Now for the awkward bit... there's three queens (if he already got a queen), two kings (if he got a king), and two aces (if he got an ace)... I suspect we can use the average, which is 2.3333. That's how many cards in the deck he has, on average, to make a pocket pair, when his first card was a Q, K or A.
So... (10/50)*(2.333/49)
Which is... drumroll please... 0.0095
Bingo! It pleases me when I win by logic.
Worth looking at articles on combinatorics. Think there's a couple on here.
Haven't looked at Ong's workings and I won't give you a perfect answer, but let's say we hold AhKh. Villains can only hold:
AdAc, AdAs, AcAs = 3 combos
KdKc, KdKs, KcKs = 3 combos
All combos of QQ = 6 combos
There are 1326 starting combinations of hole cards in poker, equal to (52*51)/2. There are 12 combos above, so the chance any one player holding QQ-AA is close to 1% i.e. 12/1326. Chance if there are 9 players left behind us to act is approx. 9%.
This isn't perfect because the 1326 includes combos we block with our AK, ignores that if the first player folds he likely doesn't have QQ+, etc, but you get the idea.
Was there a mistake somewhere. 9%?Quote:
Originally Posted by The Bean Counter
I have played for many years and very often see AK heads up vs JJ, AQ, or worse hands.
I used to play 20 hours every week and I think I have only seen AK vs QQ or better like 50 times.
Are you sure there was no mistake somewhere? 9% seems extremely high considering I have not gotten any pocket AAs in the last 14 hours. On most nights, I only get a pocket Q or better twice over 7 hour period.Quote:
Originally Posted by OngBonga
Your calculations does seem more trustworthy than my estimates though lol.
You can't determine probabilities by looking at what's happened in the last 14 hours. What you're experiencing there is variance. 14 hours is not a lot in poker, it's basically like saying if I flip a coin five times and it's heads 4 tails 1, the probability of it being heads is more than the probability of it being tails. That is clearly wrong and is nothing more than an anomoly due to small sample size.
The fact myself and bean have both come to the same answer by different methods should tell you that we're both likely to be right.
The probability of YOU getting QQ+ is a little over 1%, but let's say it's precisely 1% If you go 500 hands without seeing them, then get them six times in the next 100 hands, we're on average. That's variance. The first 500 hands is negative variance (we might call this "card dead"). The next 100 hands is positive variance (we might call this "running hot"). Over the entire 600 hands, it's average.
Variance is a HUGE part of poker, and it's exactly why we have to ensure that we are properly rolled for our stakes, because otherwise negative variance wipes out your bankroll.