# Nerding out with math (% preflop QJs vs AKs)

• Basic
Joined: 28.04.2012
So I wanted to see, how can you tell matematicly, what are your advantage if you got A K vs player 2 Q J if you bolth go all-in preflop.

So.. That is how I got so far (somehow not correct). There are 52 cards in the deck. We know 4 of them. So. Lets start with the chance, that my opponent gets pair, whyle my hand doesen't improve in the whole street thus making my opponent win.

6 outs for the player 2.

6 / 48 = 0.125 => 12.5 % chance to get one of those outs if we flop over 1 card from the street. Because we open 5 cards in the whole street, the chance of opening one of those cards improves 5 times => 12.5 * 5 = 62.5 %\

But I know, that is incorrect, because you got higher chance to win with AK, because, if the opponent doesn't improve his hand, you win. I am so confused.

Can someone, please tell me what are the % for player 1 vs player 2 by writing here mathematically how you got those % (Not forgeting about the chance when I can have 1 pair, but opponent 2 pair, or the possibilaty about straight or making a 3 of a kind e.t.c.)

I'm just stuck and can't rap my head around this scenario.

• 15 replies
• Bronze
Joined: 15.06.2009
That calculation would be way too lengthy to carry out by hand (or describe here). Use the Equilab.

/Johan =
• Silver
Joined: 24.03.2008
Why do you want to know how to work it out when you know that it's approx 60/40 in favor of AK?
• Basic
Joined: 28.04.2012
I want to know this because I am interested in the math behind it. I want to know, how to calculate the possibilities, that can come in handy in so dynamic game as no limit hold'em.
• Bronze
Joined: 17.01.2011
that's not exactly accurate. if we opened 10 cards, would we have an over 100% chance?

the easiest way to go about your first question is indeed to see the odds of not improving - but the way that goes is 42/48, then 42/47 (we took one card out), then 42/46 etc...
and then you multiply these probabilities for each other

42/48 * 42/47 * 42/46 * 42/45 * 42/44
which is ~0.636 - 63.6% which is not that far away from the actual figure (~66%)

then you have to factor in when both hit a pair and such... it's all kind of more of the same, but you'll have to calculate a whole damn lot of probabilities
• Basic
Joined: 28.04.2012
Yeah, if you put it that way about those 100%, I feel kind of reatarded.

I took closer look to your example. I don't have the gratest mathematical mind, but by simple logic, I think you made a little mistake, BUT correct me if I am wrong, because that is how I learn.

So.. If I understood corectly, you calculated the advantage of the A K (42/48)

We have 52 cards. We know 48 of them. The opponent have 6 outs (the 1 pair scenario).

Opening the first card of the street:

48 - 6 = 42 (the player2 non-outs)

42/48 = 0.875

Opening the second card from the street:

48 - 1 = 47 (we took one card from non-outs out)
The player still have 6 outs, thus 47 - 6 = 41

41 / 47 = 0.872

Opening third, fourth and fith cards:

40 / 46 = 0.869
39 / 45 = 0.866
38 / 44 = 0.863

Than we take the avarage precentage => 0.875 * 0.872 * 0.869 * 0.866 * 0.863 = 0.4955.. ~ 50%

And on paper this is believable, because if we take the formula from calculating pot odds => (unknown cards - outs) / outs

So lets take a look at the probability for player2 to improve in turn and/or river.

we have 45 unknown cards ( 52 - 2 (your cards) - 2 (known opponents cards) - 3 (the flop, where the player2 didn't improve) = 45 )

(45 - 6) / 6 = 6.5:1 = approx. 15.3 %

Because we open 2 cards the probability is 2 times higher, thus 15.3 * 2 = 30.6 %.

But this still feels so wrong, because it seems so unlogicall, that in that scenario, there is 50:50 chace, that player1 with A K will win player2, who holds Q J by high card (player2 not improving), so than its just a coinflip? Can that be?

Where did I go wrong in this scenario? (please prove me wrong mathematically, because the learning process is what I am the most interested in).
• Bronze
Joined: 07.04.2010
Did you lose a big all in pre-flop and are now obsessed if you made the right decision?

Unless you are up against a range.But if his PFAI range contains QJ,then it's a huge range,which AK is a favorite against.
These calculations are totally useless.

If you are thinking further,post-flop,AK versus QJ of the same suit is hugely profitable,when you flop/turn/river a flush,when you get it in OTF versus his TPTK and dominated flush draws,you have blockers to his straight draw,so many good things going for you.Of course,it depends on the opponents,but I think the discussion is useless.

Shake it off and play on.
• Basic
Joined: 28.04.2012
Alldoagh, I had this hand and I did lose prestige "Sit & Go" in a scenario like this. I know I made a right choise, but this is little off the side. I just think this is good example. I know how to take a loss. Probabily is tricky subject anyway.

I am interested in the mathematical formulas behind all of this. I just want to understand all the statistical side of the game and how the statistics are made.
• Bronze
Joined: 17.01.2011
OMG
yeah, you are right, and it's actually a HUGE mistake, 10%+ i felt that something was wrong because my previous result didn't justify why AK vs pocket pair is a "coinflip".

so well... the main thing is that when AK hits it's quite difficult for QJ to catch up.
AK can win by no one improving OR by improving OR by both improving, while QJ has to improve AND have AK not improve (not counting twopair+ possibilities)

and as for that i don't really remember (or maybe i don't even know) how to calculate it, i may or may not give this another look later
• Bronze
Joined: 15.06.2009
You more or less correctly calculated the probability that your opponent does not make a pair (about 50%). Thats a start. If he makes a pair he still hasn't won because you can make pairs too. Like i said in the second post, this is all way too messy to calculate exactly by hand.

If you are interested in the math (and not only the exact numbers), try to figure out the probability that both of you make a flush. [Hint: Board cards 123, 124, ..., 345 or 1234, 1345, ..., 2345, or 12345 must be hearts. The board must not contain Th9h8h because then he has a straight flush.] This will give you a feeling for the math involved.

/Johan =
• Bronze
Joined: 07.04.2010
If he makes two-pair,guess what,you have a gutshot.

MONEEEEEEEY!!!!!!!
• Bronze
Joined: 07.04.2010
and when he hits,it's so easy for you to fold if you absolutely miss.
• Super Moderator
Super Moderator
Joined: 02.09.2010
I want to know this because I am interested in the math behind it. I want to know, how to calculate the possibilities, that can come in handy in so dynamic game as no limit hold'em.
There is a reasonable article on Wikipedia about poker math.

I don't know if it will help with your exact situation, but it will certainly lay the foundation for whatever you want to do:

http://en.wikipedia.org/wiki/Poker_probability_%28Texas_hold_%27em%29

Enjoy,
--VS
• Bronze
Joined: 29.01.2012
There a bunch of other factors you must consider.

You may hit one of your six outs yet he may also hit an A/K.
You can win with straight/straight flush comboes.

These situations cannot be accurately quantified mathetically using a single formula or method. There are simply too many variables.

Equilab is the best solution.
• Bronze
Joined: 30.12.2009
you should bet small to medium bet for ak... if you hit the flop bet again... if not fold if your opponent bet...
• Bronze
Joined: 07.04.2010
Originally posted by bettingstation
you should bet small to medium bet for ak... if you hit the flop bet again... if not fold if your opponent bet...
best strategy post ever