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What are the odds...

interrobanginterrobang DramachinePosts: 25Registered User
edited December 2014 in Guides & Poker Strategy
...that my pair of 2s will beat A-K unsuited, with no other players in the game? This number might help me a little bit when trying to extract chips from bingo players.

Thanks!

RobR

22 will win 52.75% of the time. You have a coin flip scenario.

QQ sends you to 57%, still a coin flip.

Comments

  • POkcetAaisNUTSSPOkcetAaisNUTSS Not a Title, but a Star Posts: 1Registered User New to the Forums
    edited July 2011
    Every heads up hand is always 50-50.
  • Johnny_XJohnny_X Delightful Duckling Posts: 12Registered User
    edited July 2011
    Do you have black 2s or red 2s?

    I bet on black, not on red.
  • interrobanginterrobang Dramachine Posts: 25Registered User
    edited July 2011
    Russki Nun wrote: »
    22 will win 52.75% of the time. You have a coin flip scenario.

    QQ sends you to 57%, still a coin flip.

    Isn't that 53% figure against an unknown hand, or is it against A-K unsuited?

    But most of the time when I run into a bingo player, he doesn't go all in with every single hand. I can assume he's got something halfway decent, such as A-K unsuited. So I'm curious about what my chance of winning with 2-2 vs A-K is.

    Let's see. The chance of filling a pair if your hand does not have a pair is 8 out of 50 for the first flop card, or one in 6.5. Muliplied by 5 for the five cards on the table, the chance of ending up with a pair is pretty close to 5 out of 6. On the other hand, for my 2s to win, either he has to miss his pair (1 out of 6 chance), or I have to get another 2 (2/50 * 5, or 1 in 5), or I have to fill a straight or flush from the table (odds too small to make a difference here). So I've got a 1/6 + 1/5 = about 1/3 chance of winning.

    That analysis holds true for any hand where I have a pair and I guess my opponent does not have a pair but both his cards are higher than mine.

    Am I correct?

    RobR
  • Poker0ptimi5ticPoker0ptimi5tic Party Starter Posts: 150Suspended User
    edited July 2011
    Isn't that 53% figure against an unknown hand, or is it against A-K unsuited?

    But most of the time when I run into a bingo player, he doesn't go all in with every single hand. I can assume he's got something halfway decent, such as A-K unsuited. So I'm curious about what my chance of winning with 2-2 vs A-K is.

    Let's see. The chance of filling a pair if your hand does not have a pair is 8 out of 50 for the first flop card, or one in 6.5. Muliplied by 5 for the five cards on the table, the chance of ending up with a pair is pretty close to 5 out of 6. On the other hand, for my 2s to win, either he has to miss his pair (1 out of 6 chance), or I have to get another 2 (2/50 * 5, or 1 in 5), or I have to fill a straight or flush from the table (odds too small to make a difference here). So I've got a 1/6 + 1/5 = about 1/3 chance of winning.

    That analysis holds true for any hand where I have a pair and I guess my opponent does not have a pair but both his cards are higher than mine.

    Am I correct?

    RobR

    Hi there Rob,

    Odds of pocket 2s vs AK is 51.93% to 47.42%. Odds to tie are .65%. If the AK is suited, the hand is almost even. Pocket pair is favored, but essentially a coin flip.
  • EarthshineEarthshine Delightful Duckling Posts: 17Registered User
    edited July 2011
    Interrobang -

    It would be a mistake to call with 22 against a bingo player you suspect is not shoving with any two. 22 is almost exactly 50/50 against a random hand. The more hands you can take out of his range, the worse 22 performs. If he's shoving good Aces, he is almost certainly shoving with good pairs too. Against a range of hands such as: A7 - AK, 66 - AA, the equity of your Deuces drops to 40%.

    Even against someone you believe is shoving any two cards, you can pick better hands than 22. Obviously better pairs such as 66 and 77, but also any two broadway cards Jack or better, are going to be at least 60% against a random hand. If you're happy with 55%, any Ace and any King down to K7 will do. JT and T9, both suited, will also work.

    Good luck at the tables.
  • teetdogsteetdogs Loose Cannon Posts: 5Registered User Not a Title, but a Star
    edited July 2011
    Every heads up hand is always 50-50.
    Johnny_X wrote: »
    Do you have black 2s or red 2s?

    I bet on black, not on red.

    I am intrigued by your theories and would like to subscribe to your newsletter.
  • interrobanginterrobang Dramachine Posts: 25Registered User
    edited July 2011
  • interrobanginterrobang Dramachine Posts: 25Registered User
    edited July 2011
    PokerOptimi5tik gives my pocket 2s a slightly better than even chance and very specific numbers, but Earthshine gives them 40%, which roughly agrees with me. Who's right?

    RobR
  • Mike McDermottMike McDermott Loose Cannon Posts: 6Registered User Not a Title, but a Star
    edited July 2011
    You have 53.040% equity 22 vs AKo

    The thing is they're not always going to have AK, if you widen their range to something like; 66-22,ATs+,AsKh,AQo-ATo you'll have 41.275% equity vs their range.

    I've only played like 50 hands of Zynga poker so I don't really know what they're thinking, maybe they shove tighter or looser IDK

    You're pretty much never going to be ahead, try and play actual poker instead of just calling shoves pre.
  • interrobanginterrobang Dramachine Posts: 25Registered User
    edited July 2011
    OK, now I'm moving from the area of poker to the area of mathematical curiosity. We've got three numbers. PokerOptimi5tic gave 22 a 51.93% chance of beating AKo. Mike McDermott goves it a 53.04% chance of winning. I gave it about 35%. So, the following questions come up:

    1. Mike and PokerOptmi5tic, what is the source of your numbers?
    2. Where is the error in my analysis?
    3. It should be possible to write a program that would compare a hand consisting of 22 and 5 random cards against AKo and 5 random cards repeatedly and keeping track of the wins vs number of hands (a Monte Carlo test). I am a software developer, and I'm sure I could write such a program. But like any good software developer, I'd rather use something someone else already wrote. Can anyone point me to a web site where I could download a program that does this?

    Thanks again!

    RobR

    P.S. The following web site contains a download link for a library of sophisticated poker hand evaluators, in case anyone's curious: http://www.codingthewheel.com/archives/poker-hand-evaluator-roundup#senzee_eval7
  • interrobanginterrobang Dramachine Posts: 25Registered User
    edited July 2011
    You have 53.040% equity 22 vs AKo

    You're pretty much never going to be ahead, try and play actual poker instead of just calling shoves pre.

    Yes, in practice that's true. I find that my bankroll increases as my frequency of trying to punish the bingo-players decreases.

    RobR
  • Russki NunRusski Nun Fisherman Posts: 76Facebook Connect User
    edited July 2011
    Every heads up hand is always 50-50.

    Sorry, not even close to being true.

    AA versus AK os = 91.75% favorite for the Aces. I will win 92 out of 100 hands with this scenario, over the long run.

    AA versus any other hand, heads up is always favored. That is why you want to isolate against one player while holding AA. **** near impossible in here but that is why Zynga seems to be tilted towards crappy hands. There are too many chasers who will chase anything.
  • EarthshineEarthshine Delightful Duckling Posts: 17Registered User
    edited July 2011


    3. It should be possible to write a program that would compare a hand consisting of 22 and 5 random cards against AKo and 5 random cards repeatedly and keeping track of the wins vs number of hands (a Monte Carlo test). I am a software developer, and I'm sure I could write such a program. But like any good software developer, I'd rather use something someone else already wrote. Can anyone point me to a web site where I could download a program that does this?

    PokerStove is a free app that does exactly what you have described, and more. It's where I got my 40% figure, taken against a range of hands A7-AK and 66 - AA.

    You can download it here: http://www.pokerstove.com/
  • interrobanginterrobang Dramachine Posts: 25Registered User
    edited July 2011
    Great! Thanks very much!

    RobR, still curious about what's wrong with his analysis
  • fyreman235fyreman235 Fisherman Posts: 80Registered User Black Sheep of ze Internet
    edited July 2011
    Though I am not a statistician, I have watched enough commentated TV poker to know that a pocket pair has a very tiny advantage over two unsuited overcards, something in the 1 or 2 percent range. That's why that situation is called a "race". It is nearly 50-50. The only exception is that when I am holding the pair, the overcards have a 10 to 1 advantage, and when I have the overcards the pairs are unbeatable:)
  • EarthshineEarthshine Delightful Duckling Posts: 17Registered User
    edited July 2011
    Great! Thanks very much!

    RobR, still curious about what's wrong with his analysis

    What's wrong with whose analysis? Actually, everyone who has posted actual figures (with exception to the 50/50 guy) has been correct. It's just that all of our numbers mean different things. The 52% figure is only if you're against specifically AK. But in practice you'll never know you're against specifically that hand or a similar hand, which is why we do what's called assigning hand ranges. The 40% figure that Mike McD and I came up with, is how 22 plays against a range of hands that someone not playing any two cards is likely to shove.
  • MannabellMannabell Loose Cannon Posts: 7Facebook Connect User Not a Title, but a Star
    edited July 2011
    If it's heads up it's pretty much 50/50 odds...as long as the other player doesn't have pocket pair themselves.
  • interrobanginterrobang Dramachine Posts: 25Registered User
    edited July 2011
    Earthshine wrote: »
    What's wrong with whose analysis? Actually, everyone who has posted actual figures (with exception to the 50/50 guy) has been correct. It's just that all of our numbers mean different things. The 52% figure is only if you're against specifically AK. But in practice you'll never know you're against specifically that hand or a similar hand, which is why we do what's called assigning hand ranges. The 40% figure that Mike McD and I came up with, is how 22 plays against a range of hands that someone not playing any two cards is likely to shove.

    No, my analysis was specifically 2-2 vs AK offsuit. The same analysis applies to 2-2 vs any non-pair hand with cards higher than 2. So, according to me, 2-2 would win less than 40% of the time against 3-4 offsuit, against 5-6 offsuit, and so on. And everybody with exact numbers for 2-2 vs AKo has disagreed with me, including you, Mike, and expert commentators on poker tournaments. Someplace, I'm wrong, and I'd like to know where. I disregarded the chances for straights and flushes, figuring they'd be too rare to affect my approximation. Was that wrong?

    RobR
  • interrobanginterrobang Dramachine Posts: 25Registered User
    edited July 2011
    When I'm playing the way I should, my play is based on simple approximations of odds that I can calculate in my head. If I'm doing those calculations wrong, I'd like to know why.

    RobR
  • PZaffPZaff Mazter Guardian Posts: 159Registered User
    edited July 2011
    I see you like mathematical calculations, unfortunately the methods you are using are way too approximate. I'll show now a better calculation (of 22 vs AKo) which is still a little approximate.
    I'll make this two approximations:
    1. if you get another 2 (so you have a set), you win, so we'll exclude AKo may get a straight/flush/full house/...;
    2. in order to AKo to win, he must have to pair the ace or the king, so we'll exclude the case of two pairs on the board, which would make your pair of 2 useless, and we are also excluding ties.

    I'll calculate the probability of AKo to lose, so this will be the probability of 22 to win.
    AKo loses, with our approximations,
    • when there is at least one 2 on the board
    • or when there aren't As or Ks on the board

    Now the calculations:
    • There are two 2s in the deck that has 48 cards left, so the probability of missing the 2 at the first card dealt is 1-2/48. After missing the 2 in the first card there will be 47 cards left in the deck, so the probability to miss again is 1-2/47. Continue for all the 5 cards and you'll see that the probability of missing the 2 in the whole board is (1-2/48)*(1-2/47)*(1-2/46)*(1-2/45)*(1-2/44)=80%, so the probability of getting at least one 2 is 20%.
    • Now we'll calculate the probability of missing the A, the K, AND THE 2 (this is important, because if there was a 2, it'd be the previous case, so we need to exclude it now in order to correctly add together the probabilities in the end). In the deck there are three As, three Ks and two 2s left out of 48 cards (8 "interesting" cards out of 48). So the probability of missing the A, the K, and the 2 in the whole board is (1-8/48)*(1-8/47)*(1-8/46)*(1-8/45)*(1-8/44)=38%.

    Now the conclusions: the probability of AKo to lose is the probability to have at least one 2 in the board, or, in the case there aren't any 2, to not have any A nor any K on the board, and this is 20%+38% = 58%. So 22 wins against AKo approximately 58% of the times.
    As I said before this calculation is still approximate because AKo has other ways to win (most relevant ones are straights and two pairs on the board). So 22 wins a little less frequently.
    A more accurate calculation would say 22 wins about 52% of the times, with little dipendence on suits. In fact the result may vary from about 51.93% to about 52.75% depending on the suits. That's why Russki Nun said 52.75%, Poker0ptimi5tic said 51.93% and Mike McDermott said 53.04% equity (in this case, equity is calculated as the odds to win plus half the odds to tie, and the odds to tie are about 0.65%).


    The advice I can give to you is to memorize the odds of the main situations preflop (you can't make these long calculations during the hand), and apply the approximate formulas you know after the flop (the approximations are very good after the flop is dealt).
  • teetdogsteetdogs Loose Cannon Posts: 5Registered User Not a Title, but a Star
    edited July 2011
  • tomdwanetomdwane Pumpkin Posts: 8Registered User Loose Cannon
    edited July 2011
    u hafto hit a 2 and u only have an 11 % chance so u are 1/11 aganst ak
  • interrobanginterrobang Dramachine Posts: 25Registered User
    edited July 2011
    tomdwane wrote: »
    u hafto hit a 2 and u only have an 11 % chance so u are 1/11 aganst ak

    Tom,

    See the detailed anaysis above. The chance of hitting a 2 when holding a pair of 2s is almost exactly 20%. And you have to figure the chance of your opponent not getting a pair out of his A-K.

    RobR
  • interrobanginterrobang Dramachine Posts: 25Registered User
    edited July 2011
    Many thanks to you, PZaff, for your detailed and easily understood calculation. I'm surprised that the chance of ending up with a set when holding a pair is as high as 20%!

    RobR
  • PZaffPZaff Mazter Guardian Posts: 159Registered User
    edited July 2011
    Many thanks to you, PZaff, for your detailed and easily understood calculation. I'm surprised that the chance of ending up with a set when holding a pair is as high as 20%!

    RobR

    You're welcome. And yes, you'll end up with a set about 20% of the times you hold a pair, while the probability of getting the set right on the flop is "only" 12%.
  • Johnny_XJohnny_X Delightful Duckling Posts: 12Registered User
    edited July 2011
    teetdogs wrote: »
    LOL

    10char

    LOL 10char outside of quote tags.
  • Lass of EireLass of Eire Forum Meister Posts: 397Facebook Connect User
    edited July 2011
    Many thanks to you, PZaff, for your detailed and easily understood calculation. I'm surprised that the chance of ending up with a set when holding a pair is as high as 20%!

    RobR

    To drift to the side a bit... That is 20% if you chase all the way to the river. When I see a flop I will be seeing 3 over cards negating my 22 as a viable hand. With any bet, other then 1 BB, depending on pot size, I am gone.
  • Blair StephensBlair Stephens Pet Queen Posts: 461Registered User Colossal Titan Slayer
    edited July 2011
    Wow, how did I miss this topic.

    I need to check the boards here more often I guess.
  • BaywolfeBaywolfe Loose Cannon Posts: 8Registered User Loose Cannon
    edited December 2014
    Every heads up hand is always 50-50.

    Not exactly true. If you have a pair and they don't, you don't need to hit any cards to win.
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