Combinatorics Meets Luck

poker hand combinatorics

Just returned from a poker weekend at MGM National Harbor, just outside Washington DC. It was a good and profitable trip.

One hand from a $2/5 no limit Texas Holdem game on Saturday night deserves extra attention.

We were 6-handed. Under the gun wIth AT, I raised to $20. Three players called.

Flop ($80): T8♣3♣

At first glance, this looked good for me. With top pair / top kicker, I should have the best hand. At second glance, a majority of the deck could make this very difficult on the turn –> any club, king, queen, jack, nine, seven, or six could complete a draw or make a higher pair. I could play this conservatively or aggressively, but trying to milk value out of a slightly weaker hand with small bets is out of the question.

My plan was to bet aggressively on the flop, and if called, re-evaluate based on the turn card and possibly just try to check it down from there. The big blind checked, I bet $55, and the next two players folded. Then the big blind check-raised all-in for $330. Yikes! Calling would risk nearly half of my stack, and I hadn’t included this possibility in my plan upon seeing the flop.

At $2/5, his bet is 66 big blinds. That’s a relatively short stack for that game. With that stack size, his check-raise shove looks an awful lot like a semi-bluff with a draw, on a very drawy board. Is that my read? Is it correct? Do I trust it? In the spirit of writing a blog titled They Always Have It, the answer is emphatically yes. In this spot, with this stack size, he [almost] always has a draw when he check-raises all-in. He doesn’t want a call and is exerting maximum pressure to get me to fold. I know, on some weird days, he’ll have JJ or T8 or 33 and I’ll be wrong and pay him off. That’s poker. On that day, however, I’m was going to trust my read.

If I call and my read was wrong – say he flopped a set of 8’s or 3’s – will losing this pot cause me to tilt? If I call and my read was correct and he hits one of his outs, will losing this pot cause me to tilt? If either of these answers is yes, I should fold and quit for the day. Now that I’m safely back at home, let’s do some math and combinatorics.

There is now $465 in the pot and it will cost me $275 to call. From a strictly math perspective, I need to have equity of 37.7% or higher to justify calling (call amount of $275 divided by final pot amount of $730 including my call and subtracting $10 for rake and tip).

Let’s start with the flush draws (excluding 8♣ and 3♣ that are now on the board), and assume his pre-flop calling range with suited cards is fairly wide and includes all suited aces, kings, queens, connectors and 1- and 2-gappers:

A♣Q♣ A♣2♣ = 9 combinations (left out A♣K♣ as he should re-raise pre-flop with that)

K♣Q♣ K♣2♣ = 9 combos

Q♣J♣ Q♣2♣ = 8 combos

J♣T♣, J♣9♣ J♣7♣, J♣6♣ = 4 combos

T♣9♣, T♣7♣, 9♣7♣, 9♣6♣ = 4 combos

7♣6♣, 7♣5♣, 7♣4♣, 6♣5♣, 6♣4♣, 5♣4♣, 5♣2♣, 4♣2♣ = 8 combos

That’s 42 possible flush draw combinations.

Against that range, my equity is 53.8%. I should call, especially considering the amount of dead money in the pot.

A majority of this range includes some non-flush outs, with the flush draw supplemented by a straight draw or at least one live king, queen or jack. In the worst case, J♣9♣ would have a total of 18 outs – any club, queen, jack or seven would turn this into a winning hand. Q♣J♣ also would have 18 outs – any club, queen, jack or nine. My equity against these exact hands is right at my break-even percentage.

There are also 30 more combinations of open-ended straight draws. There are 16 combinations each of J9 (in which case the jack also is a live card) and 97, less one each where both are clubs and I’ve already included in the flush draw combinations. Against only the open-ended straight draws, my equity is 60.6% and I should call.

Combining the ranges of both flush and straight draws, my equity is 56.6%. This assumes I’m correct in my read that I’m up against a drawing hand. Let’s take this a step further and see how many combinations he could have that are already ahead. That might include JJ (6 combos), T8 (6 combos), TT (1 combo), 88 (3 combos), and 33 (also 3 combos). Adding these 19 combinations to his range would lower my equity against the entire range to 47.6%, still plenty high enough to justify calling.

I took a slow deep breath and pushed out calling chips.

The dealer delivers both the turn and river cards quickly, the K♣ followed by 7♣. Now the board is T8♣3♣ K♣ 7♣ As I wait for him to show the winning hand, I can almost feel my AT shriveling up beneath my fingertips. All of the flush draws got there. All of the J9 hands got there. The 97 hands with 9♣ got there. Hell, even KT got there, even though I didn’t include this in the range. And of course any 2-pair+ hands on the flop are still winners. 63 of the 72 possible drawing hands are now winners. I only beat exactly 97 without a club.

I look at him and his head droops slightly. “I missed,” he says very weakly. “A pair of sevens.” The words are barely audible as he rolls over a red seven, keeping his other card face down. As I started stacking my new chips, several players looked at him in disbelief. He just shrugged and said he had a straight draw, but no clubs.

When someone wishes you “Good Luck!” for a poker game, this is what they’re talking about.

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