3 card flush draws vs 3 card straight draws

[Eric]

Suppose we have 2 9 J in back and the remaining 10 clubs are live. The math for completing our flush is pretty simple - we need 10 outs twice. We can just use combinations to compute our flush chances - we don't care about the order in which our last 2 clubs come.

The math isn't as simple if we have something like T J Q in back. Let's say all 4 kings, all 4 aces, all 4 nines and all 4 eights are live. Permutations (the order in which are "outs" come) are important. If we get A and then 9 we are unhappy.

Here is what happens when 1 of our "outs" hits:
9 ==> 8's and K's are still outs.
K ==> 9's and A's are still outs.
8 ==> only 9's are still outs.
A ==> only K's are still outs.

In other words, we can use a combination based outs calculator for the 3 card flush but not for the 3 card stratight.

Suppose we just set 6th street heads up as the button.

The flush draw is easily calculated. 12 cards are out and 40 are still in the deck. We still have 7 cards coming. There are 10 outs available and we need 2. A combination based outs calculator tells us we have about a 57% chance.

We can't do this type of analysis for the straight draw. Combinations like 8, A don't get us there at all. Other combinations may or may not get us there depeding on their permutations.

Let's look at 8, 9, A as an example:
8 followed by 9 followed by A ==> yes
8 followed by A followed by 9 ==> yes
9 followed by 8 followed by A ==> yes
9 followed by A followed by 8 ==> yes
A followed by 8 followed by 9 ==> no
A followed by 9 followed by 8 ==> no

[Eric]

We can break the straight draw up into 4 permutations.

8 is first out permutation
If we get an 8 before 9, K or A then we must get a 9 later.

9 is first out permutation
If we get a 9 before 8, K or A then we must get an 8 or a K later.

K is first out permutation
If we get a K before 8, 9 or A then we must get a 9 or an A later.

A is first out permutation
If we get an A before 8, 9 or K then we must get a K later.

[sixpeppers]

It's even a little simpler than that. If you are assuming you always place any straight card, you have 8 outs to 4 outs and 8 outs to 8 outs. So you just need to add up those equities. I am working on a little article about direct outs vs indirect right now, similar question, but not quite the same. Basically I am asking the question, should we always place the gut-shot card? And my results are rather inconclusive but leave room for discussion.

In any event, a 3-straight is clearly much weaker than a 3-flush because of the reasons you mentioned.

[Eric]

It's even a little simpler than that. If you are assuming you always place any straight card, you have 8 outs to 4 outs and 8 outs to 8 outs. So you just need to add up those equities. I am working on a little article about direct outs vs indirect right now, similar question, but not quite the same. Basically I am asking the question, should we always place the gut-shot card? And my results are rather inconclusive but leave room for discussion.

In any event, a 3-straight is clearly much weaker than a 3-flush because of the reasons you mentioned.

You're right, it is easier to think of it as 8 outs to 4 outs and 8 outs to 8 outs. However, unlike flushes, permutations matter because sometime we get 3 outs but in the wrong order such that we don't complete the straight: A followed by 9 followed by 8

In other words, we got the 8 outs to 8 outs but it was prefaced by an 8 outs to 4 outs so we lose.

[MadMojoMonkey]

I have a preliminary result for setting TJQ on 5th street in HU OFC when Hero is OOP.

31.7% of the time Hero improves to OESD before completing.
19.7% of the time Hero improves to GSSD before completing.
for a total of 51.4%.

This assumes that if Hero draws an 8,9,K, or A, that Hero always plays it to the straight.

Does anyone have a result that confirms or refutes this figure?

[Eric]

hauturi is talking about something similar on page 91 of the ofc strategy thread on 2p2.

Let's take a concrete example of 3 card open ended straight draw mathematics. Assuming 2 players, UTG, let's say you start with 678xx and put 678 in bottom hand. Now, at some later point you draw a four. Now you can either put the four in the bottom hand and hope for a five to make a straight, or place it elsewhere, and keep the 3c OESD. The probabilities for each option are (best oesd meaning that the remaining three 4's and all 9's and T's are alive):

6th street:
4 5's:
- gutshot 54.2%
- best oesd 54.1%
3 5's:
- gutshot 43.9%
- best oesd 49%
2 5's:
- gutshot 31.6%
- best oesd 42.9%
1 5:
- gutshot 17.1%
- best oesd 35.8%

7th street
4 5's:
- gutshot 50.2%
- best oesd 47.7%
3 5's:
- gutshot 40.3%
- best oesd 42.8%
2 5's:
- gutshot 28.7%
- best oesd 37.1%
1 5:
- gutshot 15.4%
- best oesd 30.5%

8th street:
4 5's:
- gutshot 45.6%
- best oesd 40.1%
3 5's:
- gutshot 36.2%
- best used 35.5%
2 5's:
- gutshot 25.5%
- best oesd 30.4%
1 5:
- gutshot 13.5%
- best oesd 24.7%

9th street:
4 5's:
- gutshot 39.9%
- best oesd 31.1%
3 5's:
- gutshot 31.3%
- best oesd 27.1%
2 5's:
- gutshot 21.8%
- best oesd 22.9%
1 5:
- gutshot 11.4%
- best oesd 18.3%

10th street:
4 5's:
- gutshot 33%
- best oesd 20.6%
3 5's:
- gutshot 25.6%
- best oesd 17.7%
2 5's:
- gutshot 17.6%
- best oesd 14.7%
1 5:
- gutshot 9.1%
- best oesd 11.5%

11th street:
4 5's:
- gutshot 24.5%
- best oesd 9.5%
3 5's:
- gutshot 18.7%
- best oesd 8%
2 5's:
- gutshot 12.7%
- best oesd 6.5%
1 5:
- gutshot 6.5%
- best oesd 4.9%

As can be seen, the gutshot is always better, if all four outs are alive. If not, keeping the oesd is better in earlier streets, but it's appeal diminishes in later streets. NOTE: Only the probablity of making the straight is considered here.

(with my program, OFCCalc, such analysis are easy to make)

[Eric]

I've been thinking about this in terms of branches. On the upper side we could have the following:
4 mid, 4 end
4 mid, 3 end
4 mid, 2 end
4 mid, 1 end
4 mid, 0 end

same with 3 mid, 2 mid and 1 mid for a total of 20

[Eric]

Taking the 20 up branches plus a 0 branch, I think we should make a chart with 21 column headings for the up branches and 21 row headings for the down branches. We can then fill in the cells. Of course the 0-0 cell will be 0 but everything else will be between 0 and 1. I think we should make 8 charts this way:
5th
6th
7th
8th
9th
10th
11th
12th

[MadMojoMonkey]

Let's gets some symbols to describe all 3-card OESD's.

3-card OESD's always look like this:
_ _ X Y Z _ _
where X,Y, and Z are 3 running values. We do not count A23, 234, JQK, QKA as 3-card OESD's because they do not have "full stretch".
E.g.
A23 looks like this: A 2 3 _ _ and acts like a 2-gap straight draw, same for QKA.
234 looks like this: _ 2 3 4 _ _ and acts like a 1-gap straight draw, same for JQK.


The complete list of 3-card straight draws is these 8 combos:
345, 456, 567, 678, 789, 89T, 9TJ, TJQ


Now let's get some symbols to describe the number of outs in each spot.

You've used the terms "upper and lower" as well as "mid and end"
Letting U,L,M, and E, we can name the outs as follows

_ _ X Y Z _ _ this spot is called lower end, or LE
^

So then the picture we're using is:
LE - LM - X - Y - Z - UM - UE
where XYZ represent a 3-card OESD, and LE,LM,UM,UE represent the number of live outs in each appropriate value.

So if you have 456, then
LE = # of 2's in the deck
LM = # of 3's in the deck
UM = # of 7's in the deck
UE = # of 8's in the deck

whew.


So in the 3-card OESD example
You want to make 20 charts
with LM = 4 and LE = 4 in all 20 charts
Each of the 20 charts will have the values UM = {0,1,2,3,4} and UE = {0,1,2,3,4}
So that's really 25 charts.

Note that the "0-0 cell" you mention is NOT 0%. There is still the chance to catch runner-runner for the lower end of the draw, which is fully live in all 25 charts.
The 0% cells you speak of would have 1 of 2 forms:
1) LM and UM are both 0
2) LE and UE are both 0 and either LM or UM is also 0

since these first 25 tables you want created have LM = 4 and LE = 4, there will always be some equity to make a straight.

The upper draws are not independent of the lower draws. 25 tables is not enough to capture the data.

[Eric]

I think we only need one chart for each street (or maybe 2 for each street (1 oop and 1 dealer)). Each chart has 21 columns, 21 rows and 441 cells with percentages.

[MadMojoMonkey]

I don't really think we're on the same page. I'll re-read your suggestions and see if I misinterpreted them.

Here's my best guess for the way to present the data:



This uses the above terminology as pertains to LE,LM,UM, and UE.

I arranged them in the order of greatest equity. by throwing numbers into the equity engine and grabbing likely candidates for top equity combos.

I do believe that the equities for each combo of outs are correct, but I don't have too much certainty that there aren't combos missing from that list.

[Eric]

Here are the 21 column headings I'm picturing:
UM=4, UE=4
UM=4, UE=3
UM=4, UE=2
UM=4, UE=1
UM=4, UE=0
UM=3, UE=4
UM=3, UE=3
UM=3, UE=2
UM=3, UE=1
UM=3, UE=0
UM=2, UE=4
UM=2, UE=3
UM=2, UE=2
UM=2, UE=1
UM=2, UE=0
UM=1, UE=4
UM=1, UE=3
UM=1, UE=2
UM=1, UE=1
UM=1, UE=0
UM=0, UE=Anything

[MadMojoMonkey]

It's so pretty!

Here's the equities for 3-card straight draws played in HU OFC on 5th street when Hero is IP:


The symmetry means it doesn't matter if you consider the top the uppers and the side the lowers or vise versa.

Play with this for a bit and see how amazing it is that you just count big boxes, then small boxes.

[Eric]

This is great stuff!

It shows that we can still be above 50% losing up to 3 ends but if we lose 2 or more mids then we're always under 50%.

There is something that confuses me. The 4,4 row heading and 1,1 column heading shows 38%.
The 4,4 row heading and the 0,0 column heading shows 33%. Based on our discussion last night,
I was expecting the 4,4 / 1,1 percentage to be lower because it allows for
the incorrect set with the end of the weak side such that we're down to one out.

[MadMojoMonkey]

This is great stuff!

It shows that we can still be above 50% losing up to 3 ends but if we lose 2 or more mids then we're always under 50%.

There is something that confuses me. The 4,4 row heading and 1,1 column heading shows 38%.
The 4,4 row heading and the 0,0 column heading shows 33%. Based on our discussion last night,
I was expecting the 4,4 / 1,1 percentage to be lower because it allows for
the incorrect set with the end of the weak side such that we're down to one out.

With the 4,4/1,X you still have an extra out to OESD if you catch LM first.

With 4,4/0,0, you're really looking at a fully live 2-gap 3-card SD. Same as 0,4/4,0 or 0,0/4,4... those are identical situations to a 2-gap SD.

Anytime it goes 0,X/Y,Z or X,Y,Z,0 it's equivalent to a 1-gap SD.

Anytime it goes 0,0/X,Y or 0,X/Y,0 or X,Y/0,0 it's equivalent to a 2-gap SD.

So really, the equity for every possible 3-card SD is represented in this table. For 2-players, w/ Hero in seat 2, on 5th street.

[MadMojoMonkey]

If you look at 4,4/1,4 (37%) and compare it to 4,4/1,0 (38%), you see the effect you mentioned.

You have higher equity when you have fewer outs, because setting the end which leaves you drawing to only 1 out is generally worse than NOT setting that card and keeping the 3-card OESD.

However, that affect is offset by the occasions that you draw the 1 out first, in which case you're drawing to a 4-card OESD.

So these competing effects keep the equities rather stable as you remove the outs beyond the 1 out.
I.e. 4,4/1,4 -> 4,4/1,3 -> 4,4/1,2 -> 4,4/1,1 -> 4,4/1,0
all have very similar equities.

[Eric]

A picture is worth a thousand words. This 5th street dealer picture/chart shows why a simple outs calculator isn't practical for 3 card straight draws.

For example, using the (M,E) notation, the 3,1 - 2,2 cell is a total of 8 "outs" and it shows 27%.
Using the (M,E) notation again, the 1,3 - 2,2 cell is also a total of 8 "outs" but its percentage is 19%.
We have 2 situations with the same number of "outs" but the percentages are totally different.

[givememyleg]

Yeah, that is really interesting.