Categories: First Five, Odds


HU 1st to act

I had just woken up from a nap and decided to play a few HU Open Face Chinese hands on one of my Android apps. I was probably on my 3rd hand when I caught the above starting 5. I set my cards like so:

Looking back at the hand later, I realized I must have been a little foggy from sleep when I made that set. For one thing, I'm not a big believer in the 'two-flush' getting there. Not that it doesn't happen, but the odds are against it, and if I was going to try to flush the back row, why not set up the Q2ss instead of 8-high? Furthermore, if I'm not trying to set a flush in back, why not put the 9c with the 87hh and be 3-up-and-down to the straight? 87 suited obviously can make a straight flush, but hoping to hit with only two to it is, in a word, delusional. In retrospect, I wouldn't say it was my smartest play.

As it turns out, I made a straight in the back anyway and scooped the hand for a total of 8 points. An unexpected win that started from very humble beginnings, to be sure. Getting there with the straight and scooping my opponent led me to wonder about the math behind the scenes. Did I just luckbox this one? Here are the final sets:

So, exactly how lucky did I get?

The first part of that analysis requires an examination of how the villain played his hand. I got there with the straight, but an 8-high straight is a pretty weak sub-hand to hold in the back row of an Open Face set, where flushes and boats routinely reside. In this case I was definitely helped out by an opponent who played very conservatively. Not only did he abandon ship on the back row straight that would have beat mine (opting to pair the 10 -- I can only assume he thought I would pair a lower card or miss completely), but he also managed to set a 32 in front right away, leaving himself no options (in lieu of pairing the middle and then pairing the 2 or 3) to win the front except to high-card it. And with J-high in the middle, that's a dicey proposition.

If the villain had stuck to the straight draw, he would have gotten there, winning the back row. Not much else would have changed, but winning one row of three and cancelling my straight royalty would have resulted in a 1 point win for me instead of my scoop+royalty, or a net 7-point savings for him. That may not sound like a lot, but if you're playing for $10-$25/pt, it ain't chump change. And, as I've mentioned in my post Continuum of Scoring Potential, a penny saved is a penny earned in HU play.


In Part I, you saw the first five and the full runout for both players, with a brief discussion of the merits of the hero's set as well as the villain's. Now I will go over the hand as it played out street-by-street, with strategic and mathematical commentary:

First to act, hero sets first five. As mentioned in part I, perhaps not optimal.

Villain sets first five. Note the weak front row developing, as well as the splitting of the 3-flush. Having seen two dead cards, it makes some sense to stay away from the back row flush. On the deal, 3 to the flush in position with no dead cards (10 outs) is a 63% favorite to get there. With two dead cards (8 outs), chances are reduced to 48% - essentially, a coin flip. However, upon further inspection, two of the straight cards are dead as well, leaving only 6 runner-runner outs versus the 8 for the flush.

3d - pretty bad card for hero, set in middle as a 'throwaway'. The odds of catching an 8 or 7, with 6 left in the deck and 7 cards to come, are ~77%*. Odds to catch a 6 or 9 to be 3-up-and-down are ~71%. Villain catches J and sets it in middle row, apparently keeping back row straight draw alive.

6 comes in for hero. Odds to catch 5 or 9 are now ~67%. Odds to catch a 5, 9, or 6 (in other words, not foul) are ~83%. Villain pairs 10 in back row, which indicates to me that he thinks a pair higher than my 8, 7 or 6 might be good enough to take the back. What if he had stuck with his straight draw? With 6 outs (8's and Q's) and 6 pulls at the deck, he is a 67% favorite to keep the draw alive, as well as a 67% favorite to hit one of the remaining 9's, 10's or J's to pair the back row and not foul.

Ace comes in for hero, set in middle. Since I am still a ~79% favorite to pair the back, I'm not too worried about fouling. There's no real sense in setting it in front, as I am already winning there (and villain will have a hard time picking up that row with 3-high there and J-high in middle). Besides, setting it in front puts me in jeopardy of fouling. At first glance, this doesn't look like a very good card for me, but given the contour of the opponent's hand, A-high is currently a decent looking middle row. And, if my back row straight comes in and the villain ends up pairing the middle, I've got additional outs now in my middle to pair higher than him (Q's and A's).

Villain's 67% comes in with the 8, which is a bit of a useless card for him now. The only function it serves is as a pair-catcher for the middle row, which at this point is only a 26% chance to come in. Setting it in front concedes that row, and setting it in back limits him to a pair of 10's only. If I were to catch two pair in the back, I assume he'd like to keep that last spot open to try and beat my two pair. If he had kept his back row straight draw alive one street earlier (not pairing the 10), he would now be up-and-down in the back row to a superior straight than hero's. 3 7's and 3 Q's remain in the deck, and a 6-outer with 6 pulls is a 62% favorite to hit.

A 5 arrives for hero. A miracle? Not really. Since it was supposed to happen 2 of 3 times, it's not a big surprise. Now, I am ~56% to straighten the back row, ~81% to pair the back row, and a massive ~95% to pair OR straighten. Note that I am currently still winning the front and middle rows, while safely chasing my straight draw in very little danger of fouling.

A 6 comes in for villain. Again a fairly useless card at this point, sharing similar futility of purpose with the previous card.

Ks arrives for hero, fortifying the front row, not that it really needed it. Setting this card in front also helps avoid a foul, just in case a stray 9 or runner-runner pair comes in on late streets. Villain catches 5h and sets it in the middle, adding no value but it's another potential pair-catcher. At this point his odds to pair the 5, 6, or 8 are 48%, or about a coin flip.

4d arrives as scheduled for hero. Lucky? To my opponent, maybe. However, this set strategy has solid math behind it and versatility. If the card had been a brick, I'd still be able to pair my back row to avoid fouling and I'm still ahead in the other two rows. My odds to pair the back with two pulls after a brick are 52%. And, if you look at my compound draw to pair OR straighten the back (catch a 4, 5, 6, 7, 8, or 9), I still have 13 outs with two pulls or a ~69% chance to hit.

Ah arrives for villain, another card that because of the setup has absolutely no value (and his heart flush which he relinquished early on gets there, just to rub some salt in it). Setting in front forces him to catch a 5, 6, 8, or A for the middle row on the river - a 29% chance. Setting in the middle is useless as that concedes the row to hero. Back row is the only spot to put it, and it adds no value there.

Nine arrives for hero, goes into middle. Chance to foul on river: 25%. If I had decided to gamble and put it in front, my odds to hit an A or Q on river were ~18%. 4h arrives for villain, set in front, conceding the row but keeping options open to pair the middle row and win there. Odds to pair the 5, 6, or 8: 21%. Odds to pair J and foul: 7%.

In the end, the straight came in, one step at a time. And, even if it hadn't, the strategy holds up because it kept the draw alive while being relatively foul-proof. Any luck I caught on this hand really came from the other side of the board. My play was aided by an extremely conservative opponent who gave up his superior straight draw and relinquished control of the hand at a very early stage.

*Based on Open Face Odds' in-position HU odds chart. Out of position chart is on the way; however after the first 10 cards are dealt, the percentage difference between in and out of position is very slight.

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