High Card Flush Advanced Strategy

Introduction

The casino game High Card Flush has been growing in popularity. The only published strategy for High Card Flush has been the Charles Mousseau Strategy, which is to make a maximum Raise bet on any four-card or higher flush, and to Raise any three-card flush of rank T-8-6 or greater, otherwise FOLD. That strategy for three card flushes was developed without regard to cards that are not part of the flush.

Below is a slightly more powerful strategy that does take into account the off-suit cards in a hand with a three-card flush.

Three-Card Flush Strategy

The strategy for a three-card flush depends first upon the ranks of the cards in the flush:

• J32 to AKQ — always raise
• T32 to T98 — it depends upon the off-suit cards
• 432 to 987 — always fold.

So, the off-suit cards need only be considered when you play a 10-high three-card flush, which will occur with a frequency of about 5.3% of all your hands.

Let’s look at the EV of 2 hands with the same T86 flush but with very different off-suit cards:

• Th-8h-6h + Js-Ad-Qd-Jc — EV = -0.86655 — CALL
• Th-8h-6h + 7d-4d-2d-9c — EV = -1.06993 — FOLD

The differences in the ranks of the off-suit cards, and, less importantly, their difference in suit distribution are the reason these T-8-6 flushes have such different expected values. In order to better decisions when playing a 10-high three-card flush, this strategy requires that you count the number of high cards in your hand and, sometimes, check your suit distribution.

The presence of high cards in your four off-suit cards increase the power of your 10-high three-card flush because they reduce the probability that the dealer will beat your hand with a higher ranking three- card flush. Similarly, cards lower than a 10, 2-9, in your off-suit cards, weaken your 10-high flush because they reduce the probability that the dealer will have a low-ranking three-card flush that you can beat.

The optimal CALL/FOLD decision on your 10-high three-card flush will also occasionally depend on your suit distribution. Here is the nomenclature for suit distributions that is used here.

• 3211 — a three card flush, with 2 cards in a second suit and one card in each of the other two suits
• 3220 — a three card flush, with 2 cards in each of two different suits and with no cards in the fourth suit
• 3310 — a hand with two three card flushes in different suits and one card in one of the other two suits. With this hand, of course, the higher-ranking flush is always played.

The 3211 distribution will occur the most frequently, and the 3310 distribution the least frequently.

Basic Strategy for a 10-high three-card flush

CALL with any 10-high three card flush (T32 to T98) in which all four offsuit cards are in the range T-A.

Otherwise, the table below provides the minimum three-card flush hand for which you should make a CALL bet as a function of the suit distribution and the number of off-suit high cards, J-A, in your hand. With lower-ranking flushes than those shown in the table, you should FOLD.

Here is how to implement the strategy when you have a 10-high 3-card flush:

1. First, check to see if all four of your off-suit cards are in the range 10-A. If they are then CALL, if not go to step 2.
2. Count the number of cards in your hand that are in the range J-A, if it is zero then FOLD. If the number of high cards, J-A, are in the range 1-3, go to step 3.
3. Check your suit distribution and check whether your flush is at least as high as the appropriate entry in the table above.

Examples

Example: Ts-3s-2s +Ac-Qc-Jh-Td

This is a relatively weak T32 flush but it has four off-suit cards in the range T-A, so the best action is CALL. The EV of the CALL bet is -0.98119, which is better than FOLD = -1.0.

Example: Th-9h-3h + 8c-8d-4c-3d

This hand, which has a T93 flush, does not have 4 off-suit cards that are 10 or higher, indeed it has no cards that are J or higher. The best action is FOLD. The CALL EV = -1.05224

Example: Th-9h-3h + Kc-8d-4c-3s

This hand does not have 4 off-suit cards that are 10 or higher, however it has one card (the king of clubs) that is J or higher. It is a T93 flush and has the suit distribution 3211, i.e., 3 hearts, 2 clubs, 1 spade and 1 diamond. The T93 flush is higher than T87 in the above table and the best action is to raise. The EV by raising is -0.99302.

Note that if the suit distribution had been 3310, then the table above indicates that the T93 flush with one card J-A should be folded. For a 3310 suit distribution, the Raise EV = -1.01846.

Example: Th-7h-4h + Ks-4s-Ac-Jc

This hand does not have 4 off-suit cards that are 10 or higher, however it has three cards, that are J or higher. Regardless of the suit distribution a T74 flush with 3 high (>T) offsuit cards is high enough to raise. With the 3220 suit distribution shown here, the raise EV = - 0.93468.

House Edge

A simulation of High Card Flush using 200 500 million trials produced these values of house edge:

• Mousseau strategy: House Edge = 2.74446%
• This strategy: House Edge = 2.6855%

So, this strategy produces an additional increment of expected value to the player of almost 0.059%. There are a few rare hands for which this strategy is not mathematically perfect, but overall it should be very close to an optimal strategy.

Acknowledgement

Thanks to James for performing these house edge calculations.

About the Author

Gordon Michaels is a research scientist who has been Chief Technology Officer of Oak Ridge National Laboratory and Senior Scientist in the U.S. Department of Energy's Office of Intelligence. He was a card counter at Blackjack in the 1980s, and has since developed novel mathematical tools for analyzing casino games. He is sought out by "advantage players" from around the world for his help with advanced topics in gaming mathematics. He goes by Gordonm888 on the Wizard of Vegas forums.

Written by: Gordon Michaels