April 3, 2010 – 10:41 am | 7 Comments

I’ve compiled a short (just 7-pages) e-book, an introduction to the mathematics of poker. It’s basically covers how to calculate your expected value in a certain spot – starting with explaining what EV is, all …

Read the full story »
Theoretical

Discussions about the theory behind the game.

Opponent Modelling

Ideas about modelling opponent’s play, and how to best do that with mathematical tools.

Hand Analysis

Analyses of past hands (my own or otherwise), using math as much as possible to determine how well the hand was played.

Mental Processes

Thoughts about the mental/psychological side of poker – dealing with tilt, etc.

Miscellaneous

Other information – book reviews, contests, announcements, or random spew.

Home » Featured, Theoretical

Broadway vs Small Pairs Flop Analysis

Submitted by on December 22, 2009 – 10:13 am6 Comments

An interesting question came up on the 2+2 forums. The poster had heard the statement “it is easier to play a hand like AJ OOP then a small pocket pair because you will flop TPTK much more often then you will flop a set” on a training video, and questioned the validity of that. I responded on the thread, but thought it’d be an interesting thing to bring to this blog.

So, taking the example of AJ. How often will you flop Top Pair, Top Kicker (TPTK) with the AJ? There are two cases for this. Firstly, if the J pairs and the flop doesn’t have a K or a Q. Secondly, if the A pairs and none of your opponents have a AK or AQ.

Let’s look at the first case. How often will you flop a J (and one J only)? There are 3 other Jacks in the deck, so there are 3 ways for that to happen. For the other two cards in the flop, you don’t want to hit a J. You know the 2 cards in your hand, and the 2nd J (that pairs yours) on the board, so there are 49 more unknown cards. To hit exactly TPTK, there are 2 more Js which you don’t want to hit, as well as 12 overs, so there are 35 possibilities for the second card and 34 for the 3rd card of the flop. So in total, there are 3*35*34 ways of hitting exactly one J and no overs. The order of the cards do not matter, though. So you can multiply that by 3. There are 50 * 49 * 48 possible flops, leaving you a total of (3 * 35 * 34 * 3) / (50 * 49 * 48) = 9.11% chance of hitting just one J with no overs (giving you TPTK).

What about if the A pairs? Again, 16.55% chance of that happening. A lot of whether your opponents have a AQ, AK or AA, however, depends on your read of your opponents, and I haven’t yet figured out how to quantify that.

So we’ll skip that for now, and use the 9.11% as the odds of you getting TPTK.

What if you hold a low pocket pair? How often will you flop a set? You have 2 outs to a set, so by the same reasoning as above, you have (2 * 48 * 47 * 3) / (50 * 49 * 48) = 11.51% chance of flopping a set.

So at a glance, it would seem like the video was right with that statement. However, that isn’t the full story, is it? There are better hands than TPTK. You could flop a flush, or a straight, or two pair. Or even a full house.

What are the odds of these happening (ignoring the cases when the board pairs; if the board pairs there’s the risk of trips which makes it a lot harder to play)?

With AJ suited:

  • (6/50 * 5/49 * 48/48) + (6/50 * 44/49 * 5/48) + (44/50 * 6/49 * 5/48) = 3.47% of the time, you’ll hit at least 2 of your outs (giving you two pair, trips, 4 of a kind or a full house)
  • (11 * 10 * 9) / (50 * 49 * 48) = 0.842% of the time, you’ll flop a flush
  • (12 * 8 * 4) / (50 * 49 * 48) = 0.327% of the time, you’ll flop a straight

You therefore have a 9.11% + 3.47% + 0.842% + 0.327% = 13.749% chance of improving your hand on the flop.

With a pocket pair, the only way to improve your hand is to trips or 4 of a kind. Trips has a probability of 11.51% as stated earlier, 4 of a kind has odds of (2 * 1 * 48 * 3) / (50 * 49 * 48 ) = 0.245%. You also have a (2 * 48 * 3 * 3) / (50 * 49 * 48) = 0.735% chance of hitting a full house, by hitting trips and having the board pair (we ignore the full house from the board tripling up for the same reason we ignore the paired board in the AJ example, it makes things hard because you’re potentially drawing dead against a 4 of a kind). This gives a total of 12.49% chance of improving your hand with a low pair.

So,  if you count all the draws, AJ (suited) has a more likely chance of hitting the flop. 13.749% to 12.49%, to be exact.

Where does the value from a low two pair come in, though? When you hit a set and your opponent hits two pair, or TPTK,  you have the potential to extract a lot of value and maybe even double your stack. I might look more into that (and how flop betting comes into the picture here) in a future post, but for now, I’ll leave the analysis as is.

My conclusion? I think it’s more or less equal, to be honest. The numbers show that AJ has a higher chance of hitting the flop. But a pocket pair can win unimproved, and has more potential for large pots. I think it depends on your style of play, I think AJ will allow you to win more small pots, whereas low/mid pocket pairs will allow you to win fewer, but larger ones.

What do you think – do you prefer playing a hand like AJ or do you prefer a low/mid pocket pair? Why?

6 Comments »

  • [...] back at my post on Broadway vs Small Pairs, I realize I made a mistake in one of the calculations. In the post, I said this: There are 2 more [...]

  • Matt says:

    In calculating the 13.749% chance of AJ hitting the flop, you've ignored all the cases where the flop comes Axx. (You mentioned that you weren't sure how to quantify the times the opponent has AQ or AK, but then you didn't include any figure in the sum for the Axx case).

    Also it is possible for AJ suited to hit the flop in other ways. For example if the flop comes T72 with 2 of the suit then you have the best hand (if the opponent didn't flop a set). Also there are flops like QT9.

    I think if you include the Axx cases and the big draws you're going to find that you “hit” the flop with AJs somewhere around 45% of the time. (or a bit less once you discount the opponent AQ and AK cases).

    I also think you should not be too quick to discount flops like QJ5 ; as we are still ahead of a lot of the opponent's range on a flop like this.

  • Derrick Kwa says:

    Yeah, I think the point I was getting at was the idea that (as stated in the first paragraph) a hand like AJ is easier to play than a pocket pair.

    Generally, on most flops, you tend to be (or at least to feel) quite safe with TPTK or better. Yes, you can still win if you don't actually have a made hand, but it becomes a marginal situation, which doesn't make it “easier to play”, in a sense. So I'm looking more for times when we can be confident in our made hand.

  • Derrick Kwa says:

    Yes, definitely. But basically, I'm looking at what hands are “easier” to play. In the sense of, when you can feel confident of your hand. There are always going to be a lot of cases where each hand is ahead. If you look at it from the other point of view, a mid pair of 8-8 could still be ahead without flopping a set. But it makes it a much more marginal hand and harder to play.

    The premis was which hand is easier to play, and so I don't really want to be looking at more marginal situations like a flop of QJ5.

  • [...] say I have a hand like AJ suited. I then have more than 13.75% chance of hitting TPTK or better, as discussed here. If we use that 13.75% (which, as one of the commentors pointed out, is in actual fact, a low [...]

  • Lexine says:

    Didn’t know the forum rules allowed such briallnit posts.

Leave a comment!

Add your comment below, or trackback from your own site. You can also subscribe to these comments via RSS.

Be nice. Keep it clean. Stay on topic. No spam.

You can use these tags:
<a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong>

This is a Gravatar-enabled weblog. To get your own globally-recognized-avatar, please register at Gravatar.