## Poll

13 votes (46.42%) | |||

3 votes (10.71%) | |||

7 votes (25%) | |||

9 votes (32.14%) | |||

1 vote (3.57%) | |||

2 votes (7.14%) | |||

4 votes (14.28%) | |||

1 vote (3.57%) | |||

3 votes (10.71%) | |||

5 votes (17.85%) |

**28 members have voted**

August 31st, 2017 at 12:38:44 PM
permalink

Error correction: Strategy for Pre-Flop Decision when you have three unsuited cards

I found an error in my model that slightly changes the strategy and calculated EVs that I posted yesterday. This post is a replacement for yesterday's post with the errors corrected.

I've modeled the Chase the Flush hands when you start with three different suits- what I call a a rainbow hand. The player will be dealt a 3-suit rainbow hand about 39.76% of the time. As you might imagine, this type of hand should almost always be checked. However there are some exceptions: when the three unsuited cards in your hand are high enough, it is better to Bet 3X than check. These exceptions are:

Bet 3X with these "3 unsuited card" hands:

AA + 3 or higher ( AA3 to AAA)

AK + 6 or higher (AK6 to AKK)

AQ + 8 or higher

AJ + 9 or higher

KK + J or higher

KQQ

Here are the calculated EVs for some hands with 3 unsuited cards

For the preflop check EV calculation, I used the following strategies for the Postflop and River decisions:

- Postflop: Bet 2x with 3-card flush, otherwise check

- River: Computer-perfect decisions as to whether to Bet 1x or Fold

This strategy will have you betting 3X on 5.8% of all 3-suit starting hands and preflop checking on the other 94.2% of those hands. All together, this strategy of Betting 3x on very high unsuited hands will add 0.27% to a player's EV while playing Chase the Flush.

I found an error in my model that slightly changes the strategy and calculated EVs that I posted yesterday. This post is a replacement for yesterday's post with the errors corrected.

I've modeled the Chase the Flush hands when you start with three different suits- what I call a a rainbow hand. The player will be dealt a 3-suit rainbow hand about 39.76% of the time. As you might imagine, this type of hand should almost always be checked. However there are some exceptions: when the three unsuited cards in your hand are high enough, it is better to Bet 3X than check. These exceptions are:

Bet 3X with these "3 unsuited card" hands:

AA + 3 or higher ( AA3 to AAA)

AK + 6 or higher (AK6 to AKK)

AQ + 8 or higher

AJ + 9 or higher

KK + J or higher

KQQ

Here are the calculated EVs for some hands with 3 unsuited cards

3 suit hand | EV, Bet 3X | EV, Check |
---|---|---|

AAA | +0.659 | +0.221 |

AAQ | +0.345 | +0.038 |

AA7 | -0.174 | -0.266 |

AKJ | +0.051 | -0.127 |

AQQ | +0.036 | -0.129 |

AJT | -0.276 | -0.316 |

KKJ | -0.215 | -0.250 |

KQQ | -0.243 | -0.258 |

KQT | -0.454 | -0.385 |

For the preflop check EV calculation, I used the following strategies for the Postflop and River decisions:

- Postflop: Bet 2x with 3-card flush, otherwise check

- River: Computer-perfect decisions as to whether to Bet 1x or Fold

This strategy will have you betting 3X on 5.8% of all 3-suit starting hands and preflop checking on the other 94.2% of those hands. All together, this strategy of Betting 3x on very high unsuited hands will add 0.27% to a player's EV while playing Chase the Flush.

So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.

September 1st, 2017 at 11:44:43 AM
permalink

Strategy for “BET 1X or FOLD” Decision on the River When Your Longest Flush is a 2-Card Flushsuited

Situation You have checked to the river in Chase the Flush.. You arrive at the river with a hand that has no flushes longer than a 3-Flush - a “2-Flush Hand.” (Actually, by definition, such a hand must contain three 2-card flushes in three different suits and a 7th card in the fourth suit.) When should you BET 1X? and when should you FOLD?

There is no published strategy for this. This post provides a strategy that is almost computer-perfect.

Factors to Consider

1. Does the Board have 2 suited cards, i.e., a 2-card flush on the board? If it does, then you must FOLD. This is an absolute rule, correct in 100% of situations. When the board has 2 cards in the same suit, the probability of the dealer having a 3-card (or longer) flush is simply too high to justify Betting 1X on the river.

Another way to say this is:

When your longest flush is 2 cards on the river, never consider betting 1X unless your starting hand has 3 suits and the board has all 4 suits. Both your starting hand and the board must be rainbow, otherwise FOLD.

2. How many dealer “OUTS” are there such that the dealer can make a 2-card flush that is higher than your highest 2-card flush? I will give many examples of this definition of dealer OUTS below and show you how to apply it. The short rule is:

- Dealer OUTS = 0-8: always Bet 1X

-Dealer OUTS = 9-11: it depends upon the third factor below.

- Dealer OUTS = 12 or higher: always FOLD.

3. Given a board with 4 different suits, how many cards on the board are in the range 2-8, i.e., are <9?

This surprising factor arises from the Ante payout rules in which the Ante Bet always pushes if the dealer does not qualify and the player has not folded - and the dealer must have a 9-high 3-card flush or higher to qualify. With a rainbow board, each card on the board lower than a 9 means that if the dealer does make a 3-card flush in that suit then the probability of that 3-card flush ‘not qualifying’ is about 20%. Conversely, for each hand on the board that is 9 or higher, there is zero chance that a dealer 3-card flush in that suit will not qualify. So, when you have no flush longer than a 2-card flush on the river and the board is rainbow, every card on the board that is lower than a 9 increases the EV of Bet 1X by about 5-6%. In plain English, low cards on the board increases the EV of going to showdown.

So, given a rainbow board and a player hand with nothing longer than a 2-card flush, there are two things to count: (1) Dealer “OUTS” to make a higher 2 card flush than the player and (2) number of cards on the board in the range 2-8. The table below gives the exact rule on how to combine them.

Criteria on When to BET 1X on the River with a rainbow board and a longest flush of only 2 cards.

*To make a higher 2-card flush than player.

Examples

I am providing 5 examples of how to count dealer “OUTS,” because it can be tricky and how to apply the rule I’ve presented above. Note that all graphics refer to suits simply as suit A, suit B, suit C or suit D

Example 1. Player: As 9h 4d, Board: Qs Kh Kd Ac

You have made a high 2-card flush, an AQ but the rest of the board is scary: A, K and K. The graphic below illustrates that there are 4 dealer “OUTS” to have a better 2-card flush than AQ and that you should Bet 1x on this river.

Example 2. Player: Ts Qh 4d, Board: As Kh 3d Ac

You highest 2-card flush is AT in spades. Counting dealer “OUTS” on this board is tricky. The graphic below shows that there are only 7 outs and that you should BET 1x.

Example 3. Player: Ts 6h 4d, Board: Ks Jh 3d Jc

Your highest 2-card flush is K-T of spades. The board is dangerous, because there are two other suits with cards higher than the Ten that you are contributing from your hand. The graphic below shows that there are 8 outs.

Example 4. Player: Qs Th 7d, Board: 7s 8h 8d 9c

Your highest 2-card flush is Q-7 of spades. This is not a very high 2-card flush and the graphic below shows that there are 11 dealer “OUTS” to make a 2-card flush that is higher than Q-7. However, the board also has three suits in which the only card is in the range 2-8. The correct decision is BET 1X. This example, hand is at the very outer limits of qualifying for a BET 1x. If there had only been 2 board cards in the range 2-8, then the correct decision would be FOLD.

Example 5. Player: 6s 5h 2d, Board: As 3h 3d 4c

Your starting hand is terrible: 6-5-2 rainbow. The board has given you an A-6 as your highest 2-card flush –not a hand that you can often BET 1X with. However, the other 3 cards on the board are all lower than than the 6 that your starting hand contributes to the A-6 flush. The graphic below shows that there are only 7 dealer “OUTS”, namely the 7-K of spades. The rules say to BET 1X.

Summary

When you are on the river with a 2-card flush hand, you should not base your BET vs. FOLD decision strategy simply on how high your highest 2-card flush is. I propose a BET vs. FOLD strategy in which you count (1) the Dealer OUTS to make a higher 2-card flush than yours and (2) the number of board cards in the range 2-8. Except for a small number of rare situations in which your hand has 2 or more cards in the range of 2-8 and the board has cards<9 in the same suits, this proposed strategy appears to always provide the computer-perfect optimal decisions.

Situation You have checked to the river in Chase the Flush.. You arrive at the river with a hand that has no flushes longer than a 3-Flush - a “2-Flush Hand.” (Actually, by definition, such a hand must contain three 2-card flushes in three different suits and a 7th card in the fourth suit.) When should you BET 1X? and when should you FOLD?

There is no published strategy for this. This post provides a strategy that is almost computer-perfect.

Factors to Consider

1. Does the Board have 2 suited cards, i.e., a 2-card flush on the board? If it does, then you must FOLD. This is an absolute rule, correct in 100% of situations. When the board has 2 cards in the same suit, the probability of the dealer having a 3-card (or longer) flush is simply too high to justify Betting 1X on the river.

Another way to say this is:

When your longest flush is 2 cards on the river, never consider betting 1X unless your starting hand has 3 suits and the board has all 4 suits. Both your starting hand and the board must be rainbow, otherwise FOLD.

2. How many dealer “OUTS” are there such that the dealer can make a 2-card flush that is higher than your highest 2-card flush? I will give many examples of this definition of dealer OUTS below and show you how to apply it. The short rule is:

- Dealer OUTS = 0-8: always Bet 1X

-Dealer OUTS = 9-11: it depends upon the third factor below.

- Dealer OUTS = 12 or higher: always FOLD.

3. Given a board with 4 different suits, how many cards on the board are in the range 2-8, i.e., are <9?

This surprising factor arises from the Ante payout rules in which the Ante Bet always pushes if the dealer does not qualify and the player has not folded - and the dealer must have a 9-high 3-card flush or higher to qualify. With a rainbow board, each card on the board lower than a 9 means that if the dealer does make a 3-card flush in that suit then the probability of that 3-card flush ‘not qualifying’ is about 20%. Conversely, for each hand on the board that is 9 or higher, there is zero chance that a dealer 3-card flush in that suit will not qualify. So, when you have no flush longer than a 2-card flush on the river and the board is rainbow, every card on the board that is lower than a 9 increases the EV of Bet 1X by about 5-6%. In plain English, low cards on the board increases the EV of going to showdown.

So, given a rainbow board and a player hand with nothing longer than a 2-card flush, there are two things to count: (1) Dealer “OUTS” to make a higher 2 card flush than the player and (2) number of cards on the board in the range 2-8. The table below gives the exact rule on how to combine them.

Criteria on When to BET 1X on the River with a rainbow board and a longest flush of only 2 cards.

Number of Board Cards <9 | Max Dealer ‘OUTS” * |
---|---|

0 | 8 |

1 | 9 |

2 | 10 |

3 | 11 |

*To make a higher 2-card flush than player.

Examples

I am providing 5 examples of how to count dealer “OUTS,” because it can be tricky and how to apply the rule I’ve presented above. Note that all graphics refer to suits simply as suit A, suit B, suit C or suit D

Example 1. Player: As 9h 4d, Board: Qs Kh Kd Ac

You have made a high 2-card flush, an AQ but the rest of the board is scary: A, K and K. The graphic below illustrates that there are 4 dealer “OUTS” to have a better 2-card flush than AQ and that you should Bet 1x on this river.

Example 2. Player: Ts Qh 4d, Board: As Kh 3d Ac

You highest 2-card flush is AT in spades. Counting dealer “OUTS” on this board is tricky. The graphic below shows that there are only 7 outs and that you should BET 1x.

Example 3. Player: Ts 6h 4d, Board: Ks Jh 3d Jc

Your highest 2-card flush is K-T of spades. The board is dangerous, because there are two other suits with cards higher than the Ten that you are contributing from your hand. The graphic below shows that there are 8 outs.

Example 4. Player: Qs Th 7d, Board: 7s 8h 8d 9c

Your highest 2-card flush is Q-7 of spades. This is not a very high 2-card flush and the graphic below shows that there are 11 dealer “OUTS” to make a 2-card flush that is higher than Q-7. However, the board also has three suits in which the only card is in the range 2-8. The correct decision is BET 1X. This example, hand is at the very outer limits of qualifying for a BET 1x. If there had only been 2 board cards in the range 2-8, then the correct decision would be FOLD.

Example 5. Player: 6s 5h 2d, Board: As 3h 3d 4c

Your starting hand is terrible: 6-5-2 rainbow. The board has given you an A-6 as your highest 2-card flush –not a hand that you can often BET 1X with. However, the other 3 cards on the board are all lower than than the 6 that your starting hand contributes to the A-6 flush. The graphic below shows that there are only 7 dealer “OUTS”, namely the 7-K of spades. The rules say to BET 1X.

Summary

When you are on the river with a 2-card flush hand, you should not base your BET vs. FOLD decision strategy simply on how high your highest 2-card flush is. I propose a BET vs. FOLD strategy in which you count (1) the Dealer OUTS to make a higher 2-card flush than yours and (2) the number of board cards in the range 2-8. Except for a small number of rare situations in which your hand has 2 or more cards in the range of 2-8 and the board has cards<9 in the same suits, this proposed strategy appears to always provide the computer-perfect optimal decisions.

So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.

September 1st, 2017 at 12:30:37 PM
permalink

There is a tremendous amount of great information in your posts, gordon. Thank you for sharing!

Playing it correctly means you've already won.

September 1st, 2017 at 3:30:49 PM
permalink

Ditto.

Many thanks.

Many thanks.

September 1st, 2017 at 4:31:10 PM
permalink

Wow! Excellent job.

“Man Babes” #AxelFabulous

September 2nd, 2017 at 8:48:21 AM
permalink

Thank you all. Your recognition means a lot to me.

Special thanks to MIPLET for his excellent Chase the Flush River calculator, which I have used in this work and to verify my own code.

Special thanks to MIPLET for his excellent Chase the Flush River calculator, which I have used in this work and to verify my own code.

So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.

December 21st, 2017 at 8:12:40 AM
permalink

This game has arrived to Motor City Casino recently. What would the house edge be on the Same Suit pay table if the payouts were as follows:

7-card straight flush: 500 to 1

6-card straight flush: 500 to 1

7-card flush: 500 to 1

5-card straight flush: 400 to 1

6-card flush: 50 to 1

4-card straight flush: 20 to 1

5-card flush: 5 to 1

4-card flush: 1 to 1

7-card straight flush: 500 to 1

6-card straight flush: 500 to 1

7-card flush: 500 to 1

5-card straight flush: 400 to 1

6-card flush: 50 to 1

4-card straight flush: 20 to 1

5-card flush: 5 to 1

4-card flush: 1 to 1

December 21st, 2017 at 9:03:03 AM
permalink

Quote:FourFiveFaceThis game has arrived to Motor City Casino recently. What would the house edge be on the Same Suit pay table if the payouts were as follows:

7-card straight flush: 500 to 1

6-card straight flush: 500 to 1

7-card flush: 500 to 1

5-card straight flush: 400 to 1

6-card flush: 50 to 1

4-card straight flush: 20 to 1

5-card flush: 5 to 1

4-card flush: 1 to 1

Using the Wizard's combinations for the Same Suit bet gives a house edge of 10.95% for the above pay table.

June 15th, 2018 at 4:37:21 AM
permalink

Discountgambling - https://discountgambling.net/2018/05/21/chase-the-flush/ - has recently posted a Basic Strategy Card and narrative analysis on "Chase the Flush". One interesting comment is that the Same Suit Bonus @ 5.67% HE was "reasonable"!!

June 15th, 2018 at 8:43:48 AM
permalink

Very interesting to read Stephen Howe's work and strategy. I have not looked at the mathematical strategy for Chase the Flush for many months now, (if you scroll up on this thread you will see some of my posts on this subject) but I always found that it was a rich and interesting subject. I think that Stephen has done an excellent job, as usual.

It is interesting that Howe's posted strategy is still 0.3% from optimum! I suspect that his strategy for betting on the river are probably where the inefficiencies lie. There are just so many different suit distribution and overcard/undercard combinations that can arise between the board and the player's cards on the river that its very hard to develop a simple river strategy that captures all of the EV.So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.

It is interesting that Howe's posted strategy is still 0.3% from optimum! I suspect that his strategy for betting on the river are probably where the inefficiencies lie. There are just so many different suit distribution and overcard/undercard combinations that can arise between the board and the player's cards on the river that its very hard to develop a simple river strategy that captures all of the EV.