Sunday, January 30, 2011

The High-Only Four-Flush

One of the most misunderstood hands in Seven-Card Stud High-Low is the high-only four-flush. These are hands such as 2, 9, Q, K suited that can never qualify for the low-pot, but need just one additional suited card to complete a flush for a powerful high-hand. These hands occur frequently on Fourth Street when players who automatically play any three suited cards catch their suit. These players almost always stay to the end no matter what action follows.

In high-only Seven-Card Stud, a four-flush is usually a positive expectation holding. With a live draw and three cards to come, the probability of completing the flush by the end is about 50%. That means as long as the pot is paying better than even money--which it always is because of the antes--you will win money over the long run. It is the equivalent of betting on coin-flips that pay back more than you wager.

But the math is very different if you are in a Stud High-Low game and completing the flush will only win half the pot. If you win half the pot, half the time, your pot equity is only 25%. The pot must be paying at 3 to 1 for you to just break-even on a wager. That means heads-up play against a player with a better high hand who is also drawing to the low-pot is usually not worth the risk, unless you have some other way of backing into the high-hand.

To get the proper pot odds to play a high-only four-flush usually requires a multi-way pot. With three or more players your bets are being multiplied, not just matched. But, even then it is difficult to get a positive expectation. For example, in a three-way pot your bets are essentially being matched 2 for 1, which is not the 3 for 1 payoff needed to break even. Imagine a betting game in which three players put up equal amounts of money and 50% of the time you win 50% of the amount wagered. If the bets were $10 it would cost you $100 to play this game 10 times. Your expected winning of 5 x $15, or $75, doesn't cover your cost.

Playing a high-only four-flush against two other players with possible low-hands is essentially this kind of a betting game. The difference is that in poker, there is usually some dead money in the pot from antes and bets on prior streets. But, the dead money has to be large and your cost small to profitably compete for it.

To see how this works lets consider a specific example. Consider a Fifth Street scenario with the following conditions.

Stakes: $1-2
Hands Dealt: Seven
Current pot size: $10

Remaining hands:

You (2-Diamonds, 3-Diamonds) K-Diamonds, 10-Diamonds, 9-Spades
Alice (X, X) A-Clubs, 6-Spades, 7-Clubs
Bob (X, X) 2-Spades, A-Hearts, A-Spades

The number of exposed cards is 11 plus the 4 door cards that were dealt and mucked, a total of 15. If none of the mucked door cards were diamonds, you have 9 outs with 2 cards to come and 37 unseen cards. Under these circumstances, the probability of completing the flush by the end is 43%. But, this board appears to give someone a low-hand and you are currently behind for high. You are attempting a draw for half of a $10 pot. The expected values for some possible betting scenarios can be calculated.

Scenario 1: You believe Bob will lead the betting for the next three streets and Alice will call, in which case you will call (likely to happen if Alice is drawing for a low-hand). If this happens the final pot will be $28 ($10 + $2 x 3 players x 3 streets). You will have to invest $6 for a 43% chance of winning $14. Because 43% of $14 is $6.02 this is essentially a break-even proposition.

Scenario 2: You believe Bob will lead the betting for the next three streets and Alice will raise, and if you call, Bob will re-raise and Alice will cap (likely if Alice already has a lock on the low-pot and is free-rolling. If this happens the final pot will be $82 ($10 + $2 x 3 players x 4 bets x 3 streets). You will have to invest $24 for a 43% chance to win $41. Because 43% of $41 is $17.63 this is a losing proposition.

These scenarios are all thinking ahead after seeing the fifth card, but prior to any Fifth Street action. At that moment your expected return from further investment in the hand is between break-even and negative. It will require more than $10 of dead money to shift Scenario 1 to a positive expectation and much more to shift Scenario 2 to your favor.

The lesson from this analysis is that you should not automatically play any three suited cards if one or more is high. If Alice and Bob are showing high door cards go ahead. A low-hand is unlikely to develop and if you catch your suit on Fourth Street, you have an advantage. Plus if the King is the high card on the board, you can represent it as a pair of Kings and put pressure on Bob and Alice to fold without completing the flush. But, if Alice and Bob have low door cards and/or Aces, even catching your suit might leave you with a negative expectation if Alice and Bob develop viable low-hands. You cannot pressure anyone with a lock on the low-pot, so your fold equity is gone.

All this re-enforces the fundamental concept for playing Stud High-Low that is: if you can't foresee a scenario in which you have a high probability of scooping, don't play the hand.

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