August 15, 2010

Hoofbeats of Doom—Playing Zebra Hands

"When you hear hoofbeats, think horses, not zebras."

Medical aphorism attributed to Nobel Prize winner Dr. Theodore Woodward 

Medical students are taught to "think horses, not zebras" to remind them that in most patients, a common diagnosis is more likely than a rare condition.  This concept occurred to me recently when I saw a couple of hands posted in the strategy forum over at All Vegas Poker (AVP), and a similar hand on Vegas Poker Now (VPN).  In one hand, a player flopped a baby flush and ran into a bigger flopped flush, and in the other two hands, a player flopped the idiot end of a straight.  Of course, in each thread, the poster wanted to know how to avoid these potential coolers (well, other than by folding preflop, natch).

If you play poker regularly, you will run into the occasional monster cooler, something tougher to lay down than two pair versus a better two pair.  Sometimes you win, sometimes you lose, and sometimes you get to enjoy the slow motion carnage as a bystander to the train wreck.  The thing is, coolers tend to be memorable:

In Confessions of a Winning Poker Player, Jack King said, "Few players recall big pots they have won, strange as it seems, but every player can remember with remarkable accuracy the outstanding tough beats of his career."  It seems true to me, 'cause walking in here, I can hardly remember how I built my bankroll, but I can't stop thinking of how I lost it.

—Mike McDermott (Matt Damon), in "Rounders

Although set over set is bad enough, I have seen flopped set-over-set-over-set on two occasions, once with me involved; in both cases, bottom set turned quads for the win.  In a memorable stretch of runbad last fall, I got stacked three times in a single session of 2/5 NLHE, with a flopped King-high flush versus a flopped Ace-high flush twice, and a flopped set over set on the third hand.   Or, on my Festivus trip to Vegas last winter, I was involved in a hilarious hand with a couple of total yahoos, flopping a Yak-high flush against BWoP's Le Dawn-ed flush, and Poker Grump's Set O' Presto.  My hand did finish in a strong second place; unfortunately, I had bet the win, rather than the exacta box.

In any event, although weird coolers seem to happen frequently in poker, I decided to figure out the actual odds for the most common coolers:  set over set, flush over flush, and straight over straight.  Here's the executive summary (the "show your work" segment is below the jump).  Now, there are actually a couple of different ways of looking at the probability problem, with different relevance to our decision-making at the table:

The "Perfect Storm" calculation—Determining the odds that, as a deal begins, the right confluence of events will occur to create a cooler hand that will sink us or our opponent (or any two players at the table).

The "Doomswitch / Boomswitch" calculation—Determining the odds that, upon hitting a monster flop, an opponent has the necessary hole cards to complete the cooler hand.

Note that the odds below are "pure" odds, calculated without regard to player hand selection.  So, assuming players tend to muck certain hands preflop (e.g., J2s, 74, 43) as "junk", or fold small pairs to preflop raises, the realistic odds of flopped coolers are much longer.  Also, the odds below are merely a calculation of how often we will find ourselves in these flopped cooler situations, not how often we will be on the top or bottom side of these cooler flops.  The relative tightness/looseness of our and our opponent's starting hand ranges will dramatically affect how often we are winning or losing in these situations.

Set over set: 

Perfect storm odds—Thankfully, Brian Alspach has already calculated these probabilities in great detail for a Poker Digest article.  The perfect storm probabilities for a set over set situation vary somewhat based on the number of players in a hand, as more players mean a higher probability of players being dealt pocket pairs.  However, for a ten-handed game where all players take pocket pairs to the flop, flopped set over set should occur in roughly 1 out of every 167 hands.  Since we will only be dealt a pocket pair once every 17 hands, the odds we will be involved in a flopped set over set situation (assuming all pocket pairs see the flop) are 1 in 2,839.

Doomswitch / Boomswitch odds—We flopped a set, so what are the odds our opponent's random hand also flopped a set?—are 1 in 90

NOTE:  Not all flopped set over set hands will wind up as coolers. There will be cases where a draw heavy board will slow down the action.  I haven’t factored these situations out of the calculations above, because those hands typically will still end up as coolers if both players play their sets aggressively versus draws.  Also, there will be rare cases where the flop will be full house versus quads, which again are not factored out, as they are also coolers (just much colder).

Flush over flush:

Perfect storm odds—Unlike with sets, where more players in a hand mean more possible pocket pairs, which increases the odds of a flopped set over set situation, the odds of flush over flush situations decrease with more players, as more cards of the suit in question will be distributed to players, rather than being available for the flop.  To put it another way, having more than one opponent with suited hole cards decreases the odds of a flush flopping, while having more than one opponent with distinct pocket pairs increases the odds of sets flopping.  So, we need to look at the perfect storm odds of two players with random hands flopping flushes—1 in 19,491.

Doomswitch / Boomswitch odds—We flopped a flush, so what are the odds our opponent's random hand also flopped a flush?—1 in 39.

NOTE:  Not all flopped flush over flush hands will wind up as coolers.  There will be a few rare cases where the lower flush will in fact flop a straight flush or open-ended straight flush draw.  I haven’t factored these situations out of the calculations above, because those hands typically will still end up as coolers.

Straight over straight:

Perfect storm odds—To keep consistent with our flush odds calculation, what are the odds of two players with random hands flopping straight over straight?  This takes a little more thought, as there are three ways two players can get straight over straight—"bookend" straights with connectors (e.g., 34 vs. 89 on a 567 flop); "gapper" straights with a shared middle card (e.g., 35 vs. 58 on a 4-6-7 flop); and "bookend-gapper" straights with a shared middle card (e.g., 34 vs. 48 on a 5-6-7 flop, or 37 vs. 78 on a 4-5-6 flop).  The final odds—1 in 24,038.

Doomswitch / Boomswitch odds—We flopped a straight, so what are the odds our opponent's random hand also flopped a straight?—1 in 82.

NOTE:  Not all flopped straight over straight hands will wind up as coolers. A monochrome flop may easily slow down one or both players. Also, in the shared middle card straights, there will be a few rare cases where the lower straight will in fact flop a straight flush or open-ended straight flush draw. I haven’t factored these situations out of the calculations above, but it does mitigate somewhat the effect of straight over straight coolers.

Conclusions:

Realistically, the Doomswitch / Boomswitch odds are the ones we care about the most, since we rarely think about cooler hands if we aren't already in a potential cooler situation.  So, what can we conclude from these odds?
  • Flopped cooler hands are quite rare.  Flopping bottom/middle set, a small flush, or the idiot straight should be a happy occasion, not a time to start seeing monsters under the bed.
  • It is much more likely to see a flopped set over set situation, both because it is easier to hit the requisite perfect storm situation, and because player self-selection makes it more likely players will play pocket pairs to the flop, while many suited or connected/gapped cards get folded preflop as "junk" (e.g., J2s, 74o, 43).
  • Because flopped cooler hands are rare, if we encounter aggression after flopping bottom/middle set, a small flush, or the idiot straight, we are most likely up against a hand we can beat.  If we flopped a set, we are usually looking at two pair or a draw.  If we flopped a straight or flush, we are usually up against two pair, a set, or a draw (including pair-plus and combo draws). 
  • Because we are usually ahead on the flop with these hands, it might make sense, absent truly deep stacks, to play fast and aggressive with smaller sets, straights, and flushes. 
  • Conversely, with top sets, nut/big flushes, and nut/big straights, it may pay to raise smaller for value, or to slowplay, hoping our opponent will commit more chips on a safe turn card.  If our opponent in fact is on the wrong side of a cooler, he will likely let us know by getting his chips in on the flop without our help.
  • Although flopped coolers are rare, the odds of a cooler materializing greatly increase as the turn and river change the board texture.  Although we may flop the best hand, if the chips don't go in on the flop, we may well get run down on later streets.
Now, although the numbers tell us that flopped coolers are rare, sometimes those hoofbeats you hear are in fact from a herd of zebras.  Or gazelles.  Maybe wildebeests.  Anyway, good poker intuition can still play a valuable role in helping sniff out the trap hands where you seem destined to go broke.  For example, about a year ago I was playing $2/5 NLHE at the Meadows ATM.  There were a few limpers, and one of the regular maniacs raised to $25.  There were a couple of callers, all standard.  I found Yaks in the BB, so I popped it to $150 straight.  To my surprise, a young guy UTG smooth called, as did the original raiser.  The flop came out Qh-Jh-7d.  Yahtzee!  But, it was a busy board, so I decided to play aggressively, and bet $350.  The young guy UTG thought, then pushed all-in, and the maniac snap-folded.  Now, the young guy is a regular, and a solid player.  He isn't necessarily rock-tight, but his play smelled a lot like QQ.  I just couldn't see him playing 77 for $150 preflop, and I would have expected him to reraise with AA/KK.  AhKh made some sense, while AQ and QJs were longshots.  On the other hand, he is capable of a big move if he smelled weakness, though I couldn't think of a hand he might think I had that I would lay down, other than AK or an underpair.  We were each pretty deep, around $1,500 at the start of the hand.  So, calling would cost me ~$1,000.  My heart told me I was beat, but in the end, I just couldn't lay down the Yaks.  Sure enough, he rolled over QQ, and it was "good hand, good night" time for me.

A similar hand occurred at the 2009 WSOP, the infamous Billy Kopp blowup hand.  Essentially, Kopp went from being one of the dominating top three stacks with the final table bubble in sight, to busting out short of the final table, thanks mostly to gacking off a huge stack to Darvin Moon when both players flopped flushes.  Although there are many better poker players who have dissected and analyzed that pivotal hand, it seems to me that the idea the Moon had flopped a higher flush never even occurred to Kopp.  Although flush over flush was improbable, once Moon showed serious interest in the hand on the flop and the turn, alarm bells should have been going off.  I'm not saying Kopp should have laid his hand down, but at some point, a baby flush turns into a bluff catcher, and a deep stacked player needs to think about protecting his stack.




So, even though we should expect horses, not zebras, it pays to remember that sometimes:

Zebras can have you swimming with the fishes!

(Image from The Cute Report)

Detailed, boring math below the jump (feel free to point out math errors in the comments or via email!):

Set over set:

The basic calculations for the "perfect storm" odds are in the Alspach article.  Simply multiply the odds of any two players flopping sets (1/167) by the odds we will be dealt a pocket pair (1/17) to get the odds we will find ourselves in a flopped set over set situation—1 in 2,839.

Now, for the "doomswitch / boomswitch" odds, we need to:  a) assume we flopped a set, and b) calculate the odds our opponent has a pocket pair that also made a set on the same board.  Since we made a set with one of the board cards, our opponent must have a pocket pair that matched the rank of either of the remaining board cards (assuming they are distinct ranks).  Let's say the flop is J-9-5, and we have a set of 5s.  Our opponent can have JJ or 99 to be ahead.  Once this board flops, there are 6 ways for our opponent to hold a pocket pair of Js, and 6 ways for him to hold a pocket pair of 9s.  There are C(47,2) possible starting hands (after we know our hole cards and the flop), so the odds of our opponent holding a pocket pair that also flopped a set is:

(6 + 6) / C(47,2) = 12 / 1,225 = 0.0111 = 1.11% = 1 in 90

Flush over flush:

We can start with this calculation for flopping a flush for one player:
Total preflop combinations = C(52,2) = 1326
Suited combinations = C(13,2) * 4 = 78 * 4 = 312

Chances of being dealt 2 suited cards = 312/1326 = 23.53%

Once you have 2 suited cards chances of seeing a flop that gives you a made flush:
Number of possible flop combinations: = C(50,3) = 19,600
Flop combinations containing all of your suit = C(11,3) = 165

165/19600 = 0.008418367347 or about .84%

So, assuming you play any 2 suited cards, chances you'll get a flopped flush would be 312/1326 * 165/19600 =0.00198079232 or, as you figured, about .2%

The more important number here is the .84% chance that you'll get a flopped flush.

To adapt this methodology to two flopped flushes, just work in the odds for a second hand having been dealt two suited cards of your same suit out of the remaining 50 cards:

C(11,2) / C(50,2) = 55 / 1,225 = 4.49% (roughly 1 in 22)

Also, adjust the flop calculation to take out the two suited cards in villain's hand:

C(9,3) / C(48,3) = 84 / 17,296 = 0.49% (roughly 1 in 206)

So, the odds of two players having suited cards in the same suit, and flopping a flush, would be:

(312 / 1,326) * (55 / 1,225) * (84 / 17,296) = 0.005% (roughly 1 in 19,491)

However, once you flopped a flush, what are the odds your opponent with a random hand also flopped a flush?  This is a little easier.  There are 8 cards of our suit remaining after we flop our flush.  To get two hole cards dealt of our suit, we calculate:

(8/47) * (7/46) = 0.0259 = 2.59% = 1 in 39

Straight over straight:

First off, there are only 8 ways to make a “bookend” straight vs. straight (ignoring suits for the moment):

Low Hand / Flop / High Hand

A2 / 3-4-5 / 67
23 / 4-5-6 / 78


89 / T-J-Q / KA

Let’s start by finding the odds of being dealt the low side of one of these straights—“low bookend connectors”.  Now, the ranks 2-8 each work in two starting hand combos, while the Ace and 9 only work in one combo each.  There are 9 ranks for the first card, with four suits for each rank, giving us 9*4 = 36/52 odds of being dealt a qualifying first card.  Then, for each qualifying first card 2-7, there are 8 corresponding second cards that will give us connectors able to make a low bookend straight, while for the Ace and 9 there are 4 corresponding second cards that will give us connectors able to make a low bookend straight.  Thus, the odds of getting a connecting card for the second card is 7/9(8/51) + 2/9(4/51) = [(7*8) + (2*4)] / (9 * 51) = 64/459.  This gives us odds for getting low bookend connectors of (36/52 * 64/459) = 0.0965 = 9.65% = ~1 in 10.4.

Next we need our opponent to get the corresponding high bookend connectors, which fortunately are exactly one hand (disregarding suits).  So, for their first card, they can be dealt any of the eight available cards that will make the corresponding high bookend connectors (8/50), and the second card dealt to complete the high bookend set must be one of the four cards of the other rank (4/49).  So, the odds of our opponent getting dealt the high bookend connectors that match our low end are (8/50 * 4/49) = 0.0131 = 1.31% = ~1 in 77.

Finally, we need the gin flop, with cards of exactly the three ranks needed to complete the straight for both sets of bookend connectors. Needing one card out four from each of three ranks on the flop, means there are 64 (4*4*4) flop combinations (order doesn’t matter) that can complete the bookend straight. There are C(48,3) total flops = 17,296, giving us odds for hitting a gin flop of 64/17,296 = 0.0037 = 0.37% = ~1 in 270 (note: you could also calculate the flop odds as (12/48)*(8/47)*(4/46) = 0.0037).

Thus, the final odds for flopping the idiot end of bookend connector straights:  0.0965 * 0.0131 * 0.0037 = 0.0000047 = 0.00047% = ~1 in 213,796.

But wait, there’s more! (actually, a LOT more).  There are also straights where the two starting hands share a middle card and flop straight over straight.  These straights occur in the following pattern:

Low Hand / Flop / High Hand

A2 / 3-4-5 / 26
A3 / 2-4-5 / 36
A4 / 2-3-5 / 46
A5 / 2-3-4 / 56

23 / 4-5-6 / 37
24 / 3-5-6 / 47
25 / 3-4-6 / 57
26 / 3-4-5 / 67


9T / J-Q-K / TA
9J / T-Q-K / JA
9Q / T-J-K / QA
9K / T-J-Q / KA

So, ignoring suits, for each rank Ace (one) through 9, there are 4 starting hands that can flop the idiot end of a straight over straight (note that half are gapper vs. gapper, while the other half are gapper vs. connector, though each version has a shared middle card, so the distinction is not important).  So, we have 36 starting hands in total, without reference to suit.  Adding in suits, there are 16 (4*4) ways to be dealt each starting hand, and there are a total of C(52,2) = 1,326 starting hands, so the odds of being dealt an eligible idiot straight hand is (16*36)/1,326 = 0.4344 = 43.44% = ~1 in 2.3.

Now, for the high hand.  Once the low hand is set, our opponent needs the exact matching high hand.  These odds are slightly lower than with the bookend connectors, as there is one shared middle card between the low and high hands.  So, the odds of our opponent getting dealt the matching high hand is (7/50 * 3.5/49) = 0.0100 = 1.00% = 1 in 10.

Finally, we again need the gin flop, with cards of exactly the three ranks needed to complete the straight for both hands.  The flop odds calculation is identical to the flop odds we calculated for the bookend connectors: there are 64 (4*4*4) flop combinations (order doesn’t matter) that can complete the straight, and there are C(48,3) total flops = 17,296, giving us 64/17,296 = 0.0037 = 0.37% = ~1 in 270.

So, the odds for a flopped straight over straight where both hands share a middle card is: 0.4344 * 0.0100 * 0.0037 = 0.0000161 = 0.00161% = ~1 in 62,217.

Finally, to get the total odds of a flopped straight over straight, we add the bookend straights to the shared-middle card straights, and get 0.0000047 + 0.0000161 = 0.0000208 = 0.00208% = ~1 in 48,077.  But, this was calculated as merely being on the idiot end of straight over straight.  We could just as easily be on the high end of the equation, so we need to double these figures to get the final odds for being in any hand with a flopped straight over straight: 0.0000208 * 2 = 0.0000416 = 0.00416% = 1 in 24,038.

Onto the doomswitch/boomswitch odds—if we flop a straight, what are the odds our opponent has also flopped a higher or lower straight (identical straights are not included, only the coolers)?  Depending on whether it is a bookend vs. bookend situation, or a shared middle card situation, our opponent can have either 8 or 7 cards for the first hole card, and either 4 or 3 cards for the second hole card.  So, the odds become (7.5/47) * (3.5/46) = 0.0121 = 1.21% = 1 in 82.  For a bookend vs. bookend situation, the odds are slightly better: (8/47) * (4/46) = 0.0148 = 1.48% = 1 in 68.  If a gapper straight is involved, the odds are: (7/47) * (3.5/46) = 0.0113 = 1.13% = 1 in 88.

Now, why are the odds of a flopped straight over straight so much lower than the odds of a flopped flush over flush?  There are several factors in play.  First, it is easier to get an eligible straight starting hand than flush starting hand.  Conversely, it is easier for our opponent to get an eligible flush hand than it is to get an eligible higher straight hand.  But the real kicker is that the gin flop is easier (nearly 24% easier) to hit in a flush over flush hand than in a straight over straight hand:  84/17,296 flush flops rather than 64/17,296 straight flops.  Considering it is slightly easier to hit a flush draw than a straight draw certainly makes it “feel” correct that it should be tougher to flop straight over straight than flush over flush, even if a straight is an easier hand to make than a flush, considered ab initio.

4 comments:

  1. I don't know the odds - too lazy to figure them out - but in 5 years in the box, I've dealt AA vs KK vs QQ and put AKQ on the flop THREE times. I remember every one of them quite well ;)

    ReplyDelete
  2. @ --S: Since I'm watching the end of the PGA Championship, I decided to give your odds a whirl. This is for three-handed, which is easier to calculate than a full ring game, but you can see the Alspach article above to get a sense of how a full ring game changes the computation.

    Player A's odds of getting AA: 4/52 * 3/49
    Player B's odds of getting KK: 4/51 * 3/48
    Player C's odds of getting QQ: 4/50 * 3/47

    So the odds of just getting the starting hands dealt is already 1 in 8,482,717!

    Next, the flop needs exactly 1 of 2 Aces, 1 of 2 Kings, and 1 of 2 Queens:

    (2*2*2) / C(46,3) = 1 in 1898

    Odds of it all coming together:

    1 in 16,095,954,875!

    Now, making the game a full ring game likely drops these odds, but it seems like it's truly a one in a billion shot. For three times in five years, clearly you also deal for PokerStars on the side. :-p

    NOTE: Just getting set over set over set is ridiculously unlikely as it is, which I've seen twice (and both times bottom set turned quads). Numbers below again are for three-handed game:

    Player A: 52/52 * 3/51
    Player B: 48/50 * 3/49
    Player C: 44/48 * 3/47

    So the odds of three distinct pocket pairs are 1 in 4943. The odds of the gin flop are:

    (2*2*2) / C(46,3) = 1 in 1898

    The odds of the perfect storm occurring are 1 in 9,379,927. Yes, live poker is rigged!

    ReplyDelete
  3. My most memorable hand in a low limit online game was when we had AA vs KK vs QQ vs JJ. All baby cards came through the river and the betting went from capped pre-flop to a bet and everyone calling on the river. We all had over pairs to the board but you could just see everyone asking themselves 'what do these other guys have?'. I had the KK and was not surprised to see AA, but the QQ and JJ hands were a bit shocking.

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