Newb Question: AQo vs 83o

So, I just watched AQo lose to 83o and I thought what are the odds of that. I was a bit paranoid because I had just lost with QQ to KK and then I watched these same two hands dealt to two other players no more than 5 hands later. The queens lost, of course.

So, I ran it through Poker Stove:

equity win tie pots won pots tied
Hand 0: 67.456% 67.22% 00.23% 165748356 578574.00 { AQo }
Hand 1: 32.544% 32.31% 00.23% 79666272 578574.00 { 83o }

weird, i thought... so i tried another one....

equity win tie pots won pots tied
Hand 0: 68.009% 67.76% 00.25% 167086296 605412.00 { AQo }
Hand 1: 31.991% 31.75% 00.25% 78274656 605412.00 { 72o }

and another...

equity win tie pots won pots tied
Hand 0: 66.069% 65.78% 00.29% 162201672 707004.00 { AQo }
Hand 1: 33.931% 33.64% 00.29% 82956096 707004.00 { 32o }


These results are completely counter to my intuition. Can anyone explain them? Does this mean I will win a little less than 1/3rd of the time with 72o against AQo? What if anything should I take away from these statistics?

That you will win (all in preflop) 67.456% of the time with AQo vs 83o.

these numbers are based on if these hands were all in preflop. as in 67% of the time AQo will beat 83o aipf. the numbers do not reflect postflop where skill and poor decisions are factors.

why do you find this weird?

Try this.

Find out how often any unpaired hand will make at least one pair by the river.

^^ hint: to calculate this, first calculate how often an unpaired hand will not make a pair by the river

rough odds of pairing on the:
flop 1st card: ~ 1/8
flop 2nd card: ~ 1/8
flop 3rd card: 1/8
turn: ~ 1/8
river: ~ 1/8
total: ~ 5/8

so you would not make a pair about 3/8ths of the time or 37.5% of the time.

so the AQ wins by default about 37.5% of the time. the 83o wins by making a pair 37.5% of the time and the AQ wins by making a pair 37.5% of the time. so AQ wins something like 71% of the time and 83o 37.5% by my admittedly very rough calculations.

While figuring this out I started to think about KK vs QQ. You might think the results would be similar to AQo vs 83o, after all KK wins by default or by making a set and QQ wins if it makes a set.

Poker stove says:

equity win tie pots won pots tied
Hand 0: 81.935% 81.70% 00.23% 50363748 143280.00 { KK }
Hand 1: 18.065% 17.83% 00.23% 10992636 143280.00 { QQ }

After several hours of intense meditation it occurred to me that the disparity is so much larger because the odds against making a set are much greater than for pairing a card. Therefore, KK will win by default in an all in situation a much larger percentage of the time. It will also lose a much larger percentage of the time if there is a Q on the flop. However, in a normal non-all-in scenario a person holding KK shouldn't fear a Q on the flop because the odds of being dealt a pocket pair are fairly low and the odds of being dealt specifically QQ with a Q showing on the flop are even lower.

Your calcs are wrong. First calculate how often you will NOT make a pair by the river (assuming your opponent's cards are unknown). There are 50 unknown cards out there, and 44 of them are non A or Q.
- probability of not pairing on the first board card: 44/50
- probability of not pairing on the second board card knowing that you did not pair on the first board card: 43/49
- probability of not pairing on the third board card knowing that you did not pair on the first two board cards: 42/48
- 4th board card: 41/47
- 5th board card: 40/46

Probability of all these events happening: 44/50*43/49*42/48*41/47*40/46=51.25%

So the probability that you do NOT make a pair by the river is 51.25%, and so the probability that you DO make a pair by the river is 100-51.25=48.75%

Your opponent has the same odds of making a pair with his 83o.

So now, how can he win?
1) mostly, if you do not make a pair and he does make at least a pair
2) sometimes, if you do make a pair, but he does make two pairs, trips, a straight or a flush and you do not

1) How often does he NOT make a pair knowing that you do not either and you hold AQ? There are 36 non 8 or 3 cards in the 42 cards deck (we removed all the A and Q and one 8 and one 3 from the 52 cards deck), so again:
- chance of not pairing on the first board card: 36/42
- second board card, knowing the first one did not pair: 35/41
- etc
- so the probability he does not make a pair or better when you do not is 36/42*35/41*34/40*33/39*32/38=44%
- so the probability he does make a pair or better knowing that you do not is 100-44=56%
- so the probability he wins like this is 0.4875*0.56=27.3%

2) This is vastly more complicated to calculate, but we can see it does not happen all that often, so we will just give it a few %, say around 5%. You can check this with Pokerstove: run AQo vs 83o on a board that has a single As card: you find that 83o has about 9% chance to win. Since your chance to make a pair is 51.75%, the chance that you make a pair and he wins is about 0.09*0.5175=4.7%

27.3% + 4.7% = 32%

Headache yet? use Pokerstove

It is also useful to know by heart the rough odds of common preflop all in hands, especially when playing all in or fold game in tournaments. For example high pocket pair vs low pocket pair ~ 80/20, pocket pair vs two higher cards ~ 52/48, etc.

Wanna learn the math? Read Sklansky's "No Limit Theory and Practice", and read the posts by Spoonitnow on this board. Lots of good links here:
http://www.flopturnriver.com/pokerfo...ds-186705.html
and here:
http://www.flopturnriver.com/pokerfo...re-123008.html

hm...it seems like you're right...

everything i've found online with one exception recommends using the reverse method you used. and we all know a pocket pair is a slight favorite over unpaired cards which would seem to reinforce your results.

this page makes it look as if i am right: http://www.ohrt.com/odds/index.php?t[]=50&d[]=5&o[]=6&c[]=1&&s=any&p=9 but i am probably misusing it.

i would still like to know why i can't calculate the odds correctly without using the reverse method. i have a feeling that it has something to do with my probability distribution not summing to one.

I'll keep working on it and post here if I figure it out. I'd really like to be able to "check my work".

thanks for the links...

A pp is not always a slight favorite over unpaired cards. (Going to assume you mean vs. overcards, like 77 vs. KQ. A pocket pair will obviously be a huge favorite over just 1 overcard -- 77 vs. A6 for instance -- or a hand that has no overcards).

AKs vs. 22 is essentially 50-50, whereas 33 is about a 51% favorite, 55 is a 52% favorite. The reason is the increased chances of 22 being counterfeited by the board (99335 for example).

Also many middle suited connectors are favorites over small pairs. JTs is a favorite over 22-77.

Not sure about your poker odds calculator, seems to me your input is correct, but the result is not. As you said yourself, we all know that a pp vs two higher cards is about 50/50. The vast majority of this 50% chance for the two high cards comes from making a pair that beats the pocket pair.

Just checked and there is a bug in that online calculator you linked to. If you enter 50 cards, 5 drawn cards, 50 outs, and 1 out needed it gives you... zero probability! Should obviously be 100%. With the same parameters, if I vary the number of outs, it starts decreasing your probability to draw at 11 outs (less chance to draw with 11 outs than with 10!!). I wrote to the guy to tell him.

You cannot use what you call the direct method. Here is a simple example: you bet on tails and you flip a coin twice. If tails comes at least one time, you win. What are you odds of winning? According to you, it would be 1/2 + 1/2 = 1 -> 100% chance to win. This is of course not the case because the coin could fall on heads twice. So the proper way is to calculate how often you loose: how often does the coin fall on heads twice? 1/2 * 1/2 = 0.25 -> you have 25% chance of loosing -> you have 75% chance of winning.

Similarly, with your method, if there were 9 board cards instead of 5 in your example above, we would have 9/8 chances to win (112.5%)!?

Originally Posted by daviddem

...

Just checked and there is a bug in that online calculator you linked to. If you enter 50 cards, 5 drawn cards, 50 outs, and 1 out needed it gives you... zero probability! Should obviously be 100%. With the same parameters, if I vary the number of outs, it starts decreasing your probability to draw at 11 outs (less chance to draw with 11 outs than with 10!!). I wrote to the guy to tell him.

....

Let me see, you have 50 cards in a deck, 50 outs, you are drawing 5 cards and you need only 1 out? Obviously 0% chance that will happen, since you will never draw 1 out but 5 outs. Try typing 5 in required outs and see what happens.


Also, about bdestefanis pairing method, not sure where did 1/8 number come from? Chances to get your card paired are roughly (1:17.3) * 5, or to be exact 3/50 + 3/49 + 3/48 + 3/47 + 3/46.

I remember finding those numbers a little weird when I first learned them as well. The injustice of a hand as sexy as AQ being only a 2:1 favorite vs trash, amirite?

But when you remember that your hole cards are only 2/7 of the cards used to make the final hand, it makes sense.

Originally Posted by Nekrogovner

Let me see, you have 50 cards in a deck, 50 outs, you are drawing 5 cards and you need only 1 out? Obviously 0% chance that will happen, since you will never draw 1 out but 5 outs. Try typing 5 in required outs and see what happens.


Also, about bdestefanis pairing method, not sure where did 1/8 number come from? Chances to get your card paired are roughly (1:17.3) * 5, or to be exact 3/50 + 3/49 + 3/48 + 3/47 + 3/46.

Yeah apparently this thing actually calculates the probability of hitting EXACTLY the number of required outs you indicate in the input. I was misinterpreting "required number of outs" as "minimum required number of outs". So if you want the probability to hit one or more of your outs, you have to run the calculator for one required out, then two required outs, then three required outs, etc and add up all the results.

In my opinion in general we are more interested in the probability of hitting at least one out though.

note: I was speaking with the author about this by email, and now he changed his calculator to be "at least x outs"

pre flop and post flop we are never playing vs one hand, we are always playing vs a range of hands and you shoud base your decisions on the equity vs that range, but even so, even AA is 85-15 favourite vs a moron that shove/calls AI with any hand preflop, and your AQo is still 64-36 preflop.

postflop depends on the board offcourse , but even so you will find AA getting cracked on Axy rainbow by a hand like 83o, the chance is 0,8% but it's possible, and sometimes you will get busted being 95% or better 3-4-5 times in the same day, but keep in mind that poker is a lomg term game and every time you find a 83o moron, just say ty to him and look him up and play him on as many tables as possible cause he just made you rich

always, but always, play tables with bad players when you wanna make money and play tables with great players when you wanna improve your game.

Originally Posted by bdestefanis

i would still like to know why i can't calculate the odds correctly without using the reverse method

I always find that if i'm struggling with the calcs, it nearly always helps to write the event space out in full and then analyse each row.

simplified example - say you hold KK preflop and want to know the probability of an Ace hitting the flop

50 cards, 4 aces, so therefore 46 misses

AAA 4x3x2 = 24
AAX 4x3x46 = 552
AXA 4x46x3 = 552
AXX 4x46x45 = 8,280
XAA 46x4x3 = 552
XAX 46x4x45 = 8,280
XXA 46x45x4 = 8,280
XXX 46x45x44 = 91,080
Total 117,600

you can then work out the total for each relevant row as a %age and sum them.

can be a bit long-winded, but simple enough if you've got something like Excel, and i find being able to see it as a table makes it easier to see if you've gone wrong

hope this helps...

davidem:
I checked out the new and improved Poker Odds Calculator and it looks like he did fix the bug. It is irritating to find errors like this when the author could have performed the simple test you performed and found the bug.


Nekrogovner:
i came up with 1/8th by taking the number of cards that make at least one pair (6 outs) and dividing it by the number of cards remaining that will not give me a pair on each street. so...

flop 1: 6/50 = 2/16.66 = 1/8.33
flop 2: 6/49 = 2/16.33 = 1/8.17
flop 3: 6/48 = 2/16 = 1/8
turn: 6/47 = 2/15.67 = 1/7.83
river: 6/46 = 2/15.33 = 1/7.67

So roughly 1/8th or 1 in 8 chance on each street with the odds getting slightly better as you approach the river where there are less cards to pull from. On each street we assume that you did not pair either your A or your Q so there are still 3 Aces and 3 Queens among the unknown cards. Your probability for each street is outs divided by cards remaining.

I still don't really see what's wrong with these figures. I think I am making an error when I attempt to figure the total odds, the odds of pairing at least one card by the river. I tried summing the odds for each street but this sum ~5/8ths or 62.5% did not make sense.

To be fair it was not a bug, but it was confusing because he was calculating the probability to hit exactly the number of required outs as opposed to at least the number of required outs without making it clear.

DoubleJ: thanks for the pointer. This does make things more intuitive.

Razvan729: Thanks for this:

always, but always, play tables with bad players when you wanna make money and play tables with great players when you wanna improve your game.

It has been my, admittedly limited experience on Cake poker, that the super-donkeys come out after work on weekdays, and on the weekend during daylight hours. Later at night it's nothing but tough players with non-US players being in the majority.

baudib: thanks for the insight on pocket pairs and JTs. I had also wondered why 22 fared less well than 33 or 66 against AKs when I ran it through poker stove. now i know...

davidem: i am still reading your posts. Thanks again for sorting out my math and that pesky poker calculator.