Broadway vs Small Pairs Flop Analysis
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?