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I'm UTG Pre-flop with two opponents, both all in, one is RICHIDIOT (3-6s) in the big blind, and the other PLAYER2 (UTG+2), probably has a made hand (10-10) and goes all in, triggering RICHIDIOT to make the pot effectively $300, I need to bring my $3 up to $150 to call and make the pot $450.

I'm holding AQ offsuit, knowing that I've probably two overcards on PLAYER2 and a 47.84% chance of pairing with the board, strongly indicating a positive EV situation.

Now my question relates to PLAYER2 pairing the board (17%) or catching a set (19%), as well as calculating my pot equity if I call.

(I use the accurate number above even though I inaccurately would calculate pairing the board as (1-3/49)(1-3/48)(1-3/47)*(1-3/47)= 23%)

  1. How do I structure the equation using 47.84%, 17%, 19% to represent my actual chance of winning the hand. (Not including the chance either he or I catches a straight, flush, or I catch a set, etc.)

  2. What is my pot equity from calling?

  3. What would be the calculation to show my true chance of winning the hand, including chances either of us catch straights, flushes, etc.

I didn't call as I had already determined that this was my last hand of the night, but after thinking about it being what I thought to be clearly a positive EV situation, I later regretted not staying in. I would like to make a better decision in the future by understanding fully what my decision here should have been based upon. And yes, a Queen came out on the turn, but that isn't what got me to ask this question.

chovesh
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  • Use a poker calculator. There are so many possibilities that you just have to run all the deals. It gets to be so many deals they have to use statistical probability http://poker.stackexchange.com/questions/6636/chance-formula-for-texas-holdem/6638#6638 – paparazzo Jan 27 '16 at 22:50
  • Bad Answer. I'm asking how to calculate and how to structure the equation to derive the answer. – chovesh Jan 27 '16 at 23:19
  • I don't see why frisbee's comment is a bad one. Unless you know exactly what cards your opponents hold, the calculations are an unnecessarily complicated hassle. An equity calculator like pokerstove is perfect for this kind of thing. – Chris Farmer Jan 28 '16 at 20:26
  • The correct answer is: X chance of my catching another A or Q, AGAINST his Y chance of NOT catching the set. Close approximation is 47.84% * 81% = 38.75% Pot ($450) equity is then calculated using 38.75-33.33 = 4.52% (of $450) is $20.34. As pot equity is positive, I should have called. Chances of other players or myself catching a flush or straight balance out, with RICHIDIOT (2.5%) getting a flush The 17% chance of PLAYER2 pairing the board is irrelevant, because if I catch a pair, I will still beat him. Question was mostly about how you set up the math. – chovesh Jan 28 '16 at 22:09
  • Correct? 81% not make set is wrong. Can you show show the calculation on any of those numbers? Flush or straight wash out - how? – paparazzo Jan 29 '16 at 02:37
  • We both have 2 cards, so 1st card is a 2/48 probability of him catching a third 10, 2nd card 2/47. Therefore (1-2/48)(1-2/47)(1-2/46)(1-2/45)(1-2/44) = 80.05% of him not getting a set. I have a slightly better chance at a straight than him (AQ vs 10-10), and RICHIDIOT has a 36s, so better than the rest of us for having either. But I'm was asking how to set up my the math (considering probability against PLAYER2 who I conclude either has a made hand has or good cards, but worse than AQ). Remember: This situation includes knowledge of the players styles and is not a straight random hand. – chovesh Jan 29 '16 at 05:49
  • OK but that simplification of AK versus TT way off. Calculator is 43% – paparazzo Jan 29 '16 at 13:15

2 Answers2

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  1. There are far too many situations you'd need to work out to calculate by hand.

1a. Just for AQo vs P1 & P2 range, the permutations of hand matchups is already huge. 

    P1 range: 40% x 1326 = 530 hands
    P2 range: 10% x 1326 = 133 hands
    530x133 = 70,490 match ups

1b. And each one of those would still be too difficult to calculate for the permutations of 5 card boards.

   50x49x48x47x46 = 254,000,000 boards

1c. Ignoring V1 is already over simplifying the hand.

    63s: 28.8%
    10%: 36.7%
    AQo: 34.5%

  1. Equity for call vs ranges: 40% range that includes 63s and 10% range for V2:

    P1 40%: 28.3%
P2 10%: 36.9%
P3 AQo: 34.8%

  2. Odds for each hand winning:

    63s: 19.3%
TTp: 44.5%
AQo: 36.2%

Results were from an equity calculator.

BTW, They don't actually calculate using formulas either. They simulate results by dealing millions of hands for a Monty-Carlo sim or dealing out all possible permutations for a complete solution.

TakenItEasy
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Your simplification is over simplified. If you want actual odds uses a calculator

At the table it is simple. If you are getting 2:1 on your money AQ is only not getting pot odds against AA, KK, AK, and QQ. It is pretty easy to figure out the hands you are 2:1 dog to. KQ is actually close. The quick safe bet is any two cards Q or better. You are better than 2:1 against any lower pair. If you put them on JJ or less then you had odds.

AK you are only not getting odds against AA and KK. And you have blocks on both AA KK. AQ is much more exposed.

paparazzo
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  • Alas, my question is indeed one asking a statistics question of how to "set up" the math equation to do the calculation myself. I've read ahwile ago "The Math Of Poker" but don't own a copy. Before I knew their hands, I knew my AQ were probably overcards to PLAYER2. But even before I knew he had 10-10, I had to figure he had a made hand (of a pair, JJ at best). So against him having a pair below AQ, I had been trying to figure out how to do the math, which is why I came here to get "Non-answers" from people who try to answer with "use calculator" instead of using one to answer me. LOL – chovesh Jan 29 '16 at 14:46
  • As I said in my first comment the true outs based math is too complex. TT overlaps AK straight and could get a full house. Plus another straight and flush going on. Your simple calculation was off by a full 10% (43/38). The question statement is vague at best. "I would like to make a better decision in the future by understanding fully what my decision here should have been based upon." This is a valid answer to that question. At the table you don't have a chance of running the *true* math. – paparazzo Jan 29 '16 at 15:04
  • 1) "Your simple calculation was off by a full 10% (43/38)." and I would indeed appreciate it if you could show me the correct calculation. 2) "The question statement is vague at best." probably true as I was having a hard time articulating my question as I was still grappling with how to set up the problem accurately. 3) Thank you for taking the time to work on this with your responses, even though it isn't exactly what I'm looking for as an answer. 4) I'm going to have another look at "The Math Of Poker" it has been too long since I last read it. – chovesh Jan 29 '16 at 19:42
  • Dude, I tell you the real calculation is too hard and your response is show me the real calculation. For the third time - use a poker calculate. Go to Google and search on Poker Calculator. – paparazzo Jan 29 '16 at 20:06