Poker: Hold' Em Exercise

[primelord]

If you have seen this before please let the others work through it before you respond.

You are at a friendly home game and late in the night it is folded to your buddy who is in the SB. He completes the blind and flips over 7c 6d. Being that the game is friendly you go along and flip up your Jh Th and check your big blind.

The flop comes 9h 5s 8h.

Assuming both players are trying to play their hand optimally from this point forward. What should your friend do in the SB and what should be your response to his two options?

[rkmsuf]

18 outs for the big blind?

[BrianD]

I came up with 14 outs.

[Maple Leafs]

There's not enough information here! What type of player is he? What's your read? What are the blinds? What are the stacks? What is the game structure? What's your table image? How do you expect us to answer a simple question without this information? Do you realize we will just keep asking questions rather than say we don't know?

(Sorry, just about every poker question I've ever seen posed gets answered this way by the 'experts'. I just wanted to fit in.)

[rkmsuf]

9 hearts
4 queens
3 sevens

ok, 16

[BrianD]

Originally Posted by rkmsuf

9 hearts
4 queens
3 sevens

ok, 16

One of the queens and one of the sevens is also a heart. I think you counted them twice.

[rkmsuf]

Originally Posted by BrianD

One of the queens and one of the sevens is also a heart. I think you counted them twice.

ah, good point.

we never had this conversation.

[Maple Leafs]

So is BB the slight favorite here?

[BrianD]

My calculations have the BB at 51% chance of hitting at least one of the outs.

[Maple Leafs]

Is it close enough that the money already in the pot would make it worthwhile for either player to call any raise or reraise?

[rkmsuf]

Originally Posted by Maple Leafs

Is it close enough that the money already in the pot would make it worthwhile for either player to call any raise or reraise?

It comes back to chip count and whether a coin flip is worth it. Way ahead, maybe not. Way behind, probably is.

[Radii]

radii@wangband:~$ pokenum -h Jh Th - 7c 6d -- 9h 8h 5s
Holdem Hi: 990 enumerated boards containing 5s 9h 8h
cards     win   %win      lose  %lose       tie   %tie        EV
Jh Th     525  53.03       465  46.97         0   0.00     0.530
7c 6d     465  46.97       525  53.03         0   0.00     0.470

Just to clear that up.

[primelord]

Just in case it impacts your decisions at all this is a limit and not a no-limit game.

[BrianD]

Originally Posted by Radii

radii@wangband:~$ pokenum -h Jh Th - 7c 6d -- 9h 8h 5s
Holdem Hi: 990 enumerated boards containing 5s 9h 8h
cards     win   %win      lose  %lose       tie   %tie        EV
Jh Th     525  53.03       465  46.97         0   0.00     0.530
7c 6d     465  46.97       525  53.03         0   0.00     0.470

Just to clear that up.

That is a bit off from my calculations. Did I do something wrong, or is this just statistical variation from one sample?

[rkmsuf]

this question is useless without game info and chip counts

any fool can say yes or no to a coin flip

[primelord]

Originally Posted by rkmsuf

this question is useless without game info and chip counts

any fool can say yes or no to a coin flip

Well I cleared up that it is a limit game. That is the only other piece of information you need. There is a correct play for both players here.

[rkmsuf]

Originally Posted by primelord

Well I cleared up that it is a limit game. That is the only other piece of information you need. There is a correct play for both players here.

I really don't get the purpose fo this then unless you want to expound on the minor percentage advantage.

[BrianD]

The BB should probably bet each round to take advantage of the slightly better than 50% odds, and the SB should probably call each round since winning only 49% of the time is better than folding away the whole pot.

[rkmsuf]

Originally Posted by BrianD

The BB should probably bet each round to take advantage of the slightly better than 50% odds, and the SB should probably call each round since winning only 49% of the time is better than folding away the whole pot.

wow, that's a great puzzle.

[primelord]

Originally Posted by rkmsuf

I really don't get the purpose fo this then unless you want to expound on the minor percentage advantage.

I guess the purpose is to analyze the hand to help develop your ability to determine hand equity. On the surface when I first looked at this hand I thought the strategy was pretty clear for both players, but it's a more complicated problem than it first appears.

Once the hole cards are known there is a certain correct play for both players. So I guess it is a math exercise more than anything. It's not as simple as saying yes or no to a coin flip.

[BrianD]

If it matters, 30% of the time, the BB will have the hand won at the turn and the SB will have to fold.

[primelord]

Originally Posted by rkmsuf

wow, that's a great puzzle.

I am not certain why you are so bitter about this problem. FWIW Brian's answer is not correct.

[rkmsuf]

Originally Posted by primelord

I guess the purpose is to analyze the hand to help develop your ability to determine hand equity. On the surface when I first looked at this hand I thought the strategy was pretty clear for both players, but it's a more complicated problem than it first appears.

Once the hole cards are known there is a certain correct play for both players. So I guess it is a math exercise more than anything. It's not as simple as saying yes or no to a coin flip.

uh, ok.

perhaps if I play 365 days a year. not a friendly home game. a 3% edge is useless.

[rkmsuf]

Originally Posted by primelord

I am not certain why you are so bitter about this problem. FWIW Brian's answer is not correct.

I guess my point is that in reality in what you describe, both players with call out.

If they play hundreds of times then I guess maybe one of them makes out.

[BrianD]

I think it is safe to assume that the SB won't want to bet or raise since he has less than 50% odds. The BB probably should bet since he has better odds. The question is whether the SB should call or fold. I guess on how the math would come out and I guess that was wrong. I'll see how the math does come out unless someone beats me to it.

[primelord]

Originally Posted by rkmsuf

I guess my point is that in reality in what you describe, both players with call out.

If they play hundreds of times then I guess maybe one of them makes out.

And I guess my point is that is not the correct play. The point of the exercise is to determine what the correct play is. Just because something is not a significant edge does not mean it doesn't matter which option you choose.

Assuming you are not counting cards you should always hit a 16 in blackjack against a dealers T or face card. Statistically the difference between hitting 16 and standing on 16 is actually quite small. And over the course of one night the natural variance will likely not match up at all with the statistical probablity. It doesn't change the fact that it is still the correct play to hit 16.

[BrianD]

Turns out I was wrong. The SB should be betting, not the BB. The BB should fold.

Considering everything in the pot as a sunk cost, an assuming a bet and a call each round, there is a 30% chance of the BB hitting on the turn and winning 1 BB when the SB folds. There is a 21% chance of the BB hitting on the river and winning 3 BB. There is a 49% chance of not hitting at all and losing 3 BB.

.3*1 + .21*3 + .49*-3 = -.54 The average result for the BB is losing .54 BB by calling.
The average result for the SB is winning .54 BB by betting and getting called.

[Radii]

I assume that simple "bet when you're ahead, don't when you're behind" option is just not right here.

But I'll throw it out there just in case:

The SB(made hand) should check/call the flop. The BB(big draw) should bet/raise the flop. The BB is getting more equity out of each bet and should get as much money in as possible from a purely mathematical "correct play" perspective.

If the BB hits the turn, he wins the pot there. If not he no longer is a 50% favorite. But assuming he bet the flop I believe he has the pot odds to call a bet on the turn. The SB should bet the turn, and the BB should call.

The river plays itself, no money goes in.


Let me see if there's a good mathematical analysis of this to show the correct answer though, I'm not sure if that's right or not.

[Bonegavel]

As the SB here, I would check/call to try keeping the BB in the game. I know I have flopped a straight and if the BB flopped a straight, it is the very same one I have.

If the BB raises, I am going to assume that he has flopped a nut flush and will have to use my experience with him to know if he is bluffing.

[BrianD]

Originally Posted by Bonegavel

As the SB here, I would check/call to try keeping the BB in the game. I know I have flopped a straight and if the BB flopped a straight, it is the very same one I have.

If the BB raises, I am going to assume that he has flopped a nut flush and will have to use my experience with him to know if he is bluffing.

In this particular exercise, both hands were flipped up. No bluffing possible.

[digamma]

Originally Posted by Radii

I assume that simple "bet when you're ahead, don't when you're behind" option is just not right here.

But I'll throw it out there just in case:

The SB(made hand) should check/call the flop. The BB(big draw) should bet/raise the flop. The BB is getting more equity out of each bet and should get as much money in as possible from a purely mathematical "correct play" perspective.

If the BB hits the turn, he wins the pot there. If not he no longer is a 50% favorite. But assuming he bet the flop I believe he has the pot odds to call a bet on the turn. The SB should bet the turn, and the BB should call.

The river plays itself, no money goes in.


Let me see if there's a good mathematical analysis of this to show the correct answer though, I'm not sure if that's right or not.

This is right along the lines of how I was working out the problem.

[Bonegavel]

Originally Posted by BrianD

In this particular exercise, both hands were flipped up. No bluffing possible.

Why the hell are they flipped?

[sabotai]

Originally Posted by Bonegavel

Why the hell are they flipped?

Because this is not a real life situation, it's an exercise in poker math.

[albionmoonlight]

Remember one of the keys to problems like this is realizing that both players have perfect knowlege--and know that the other player does too.

So, when it comes to the small blind, he will say to himself, I can do X or Y. If I do X, then my opponent will be faced with a choice. He can do A, B, or C. And he will make that choice knowing that I know how best to respond to A, B, or C. Etc., etc., etc..

In other words, decisions made with perfect knowledge of perfect knowlege require you to take the logic tree all the way to the end before you can know which branch to take at the beginning.

[primelord]

I still have not seen the correct answer.

[sabotai]

Originally Posted by BrianD

Turns out I was wrong. The SB should be betting, not the BB. The BB should fold.

Considering everything in the pot as a sunk cost, an assuming a bet and a call each round, there is a 30% chance of the BB hitting on the turn and winning 1 BB when the SB folds. There is a 21% chance of the BB hitting on the river and winning 3 BB. There is a 49% chance of not hitting at all and losing 3 BB.

.3*1 + .21*3 + .49*-3 = -.54 The average result for the BB is losing .54 BB by calling.
The average result for the SB is winning .54 BB by betting and getting called.

Your math is a bit off I think. I have not redone it, just making a few obseravtions that may or may not influence the outcome.

For one, you don't seem to be taking into consideration the 1 big bet (2 bets) made prior to the flop (the completed SB and the BB). After the flop, there is 1 big bet in the pot to start with.

Secondly, if the BB makes his hand on the river, the SB will obviously not call. Therefore, wouldn't he win 2 BB post flop and not 3? Since the SB would call the flop and the turn, but not the river?

I may be missing something else, or be flat out wrong about what I said. (haven't thought about poker stuff in awhile and it's been a long day ).

[BrianD]

Originally Posted by sabotai

Your math is a bit off I think. I have not redone it, just making a few obseravtions that may or may not influence the outcome.

For one, you don't seem to be taking into consideration the 1 big bet (2 bets) made prior to the flop (the completed SB and the BB). After the flop, there is 1 big bet in the pot to start with.

Secondly, if the BB makes his hand on the river, the SB will obviously not call. Therefore, wouldn't he win 2 BB post flop and not 3? Since the SB would call the flop and the turn, but not the river?

I may be missing something else, or be flat out wrong about what I said. (haven't thought about poker stuff in awhile and it's been a long day ).

I was counting the money in the pot as a sunk cost, but if that is the case, I probably should add that to the winnings.

I had the BB winning 3 BBs on a river win counting 1 BB at the flop (not counting his own bet), and 2 BBs on the turn. Isn't the minimum bet on the turn in a limit game 2 BBs? I haven't played limit in a while, but I thought that was the case.

[sabotai]

Originally Posted by BrianD

I had the BB winning 3 BBs on a river win counting 1 BB at the flop (not counting his own bet), and 2 BBs on the turn. Isn't the minimum bet on the turn in a limit game 2 BBs? I haven't played limit in a while, but I thought that was the case.

Ahhh, there's the confusion.

No, the amount to bet on the turn and the river is considered 1 big bet (the amount to bet pre and post flop would be half a big bet).

[primelord]

Originally Posted by BrianD

I was counting the money in the pot as a sunk cost, but if that is the case, I probably should add that to the winnings.

I had the BB winning 3 BBs on a river win counting 1 BB at the flop (not counting his own bet), and 2 BBs on the turn. Isn't the minimum bet on the turn in a limit game 2 BBs? I haven't played limit in a while, but I thought that was the case.

The minimum bet in limit on the turn is 1 BB. The two small bets that are in the pot now make up 1 BB.

[BrianD]

Originally Posted by sabotai

Ahhh, there's the confusion.

No, the amount to bet on the turn and the river is considered 1 big bet (the amount to bet pre and post flop would be half a big bet).

Crap, I was counting the number of big blinds, not big bets. I think the math still works out, just cut in half to get big bets.

This gets more complicated though. If we count the 1 big bet (2 big blinds) in the pot as something to win but already a sunk cost (nothing to lose), at one bet each round, both players would average coming out ahead. The math is going to work out to multiple bets at some point before one player can expect to average out behind. I'll have to rework this a bit to figure out what this means.

[sooner333]

let me try to think about this...I believe that the small blind should check, then the big blind bet, and small blind call (because he's getting 2:1 on his money)...that means there will be 2 Big Bets in the pot for the turn. If a non-winning card comes out...the made hand is winning 68.18% of the time. At that point, he should bet, and the big blind should fold, because he is not going to win the 33% of the time to get proper pot odds. If a winning card comes out, the betting ends.

The small blind wouldn't start raising more before knowing that he'll fold 33% of the time, because any more money in the pot and the pot odds are good.

On second thought, because he only has a 30% chance of getting a card on the turn, the small blind should bet, and the big blind should fold because pot odds aren't good enough and there are no implied pot odds here because they both know eachother's cards.

[primelord]

You guys are all certainly thinking along the right lines, but almost all of you are veering off the correct path at various points.

As of sooner's post we still don't have a correct answer.

[primelord]

Perhaps the best way to attack this is to determine first specifically what the SB should do and then move on from there. I ahve seen some conflicting opinions on what the SB should do.

[Huckleberry]

The SB should bet/raise.

1.) At 47% he would bet 4 small bets if he could. He would put 4 small bets in to the 6 it would pay off (4 from the big blind plus the two pre-flop).

2.) Nobody else has said this.

[BrianD]

I created a spreadsheet (which may or may not have been accurate), and the results seem logical. After the flop, BB wasnts as much as possible in the pot. At this point, he still has a 51% chance of winning. Correspondingly, the SB wants as little as possible in the pot.

If the BB doesn't win at the turn, he now wants as little as possible in the pot while the SB wants as much as possible.

So....

After the flop, SB checks, BB bets, SB calls.
After the turn (assuming no winner), the SB bets and the BB calls due to favorable pot odds.

[primelord]

Getting closer...

[Brillig]

This is a close one. Since you have 7 cards exposed at the flop (not the usual five), the odds of you hitting one of your 14 outs on the turn are:

14/45 = 0.311

if you miss the turn, then hitting on the river is:

14/44 = 0.318

So, overall:

0.311 you hit on the turn
0.219 you miss on the turn, but hit on the river
0.470 you miss entirely


For simplicity, let's assume that the big blind is $1.

Let's look first at the "intuitively obvious option" - SB checks on the flop as a slight underdog, BB bets, SB calls. This puts $4 in the pot on the turn. If you missed the turn, you now are forced to call a turn bet from the SB by the pot odds.

The outcome would be:

.311 you win $2
.219 you win $4
.470 you lose $4

Net EV -$0.382


Now, you don't really like being forced by the pot odds to call a big bet on the turn as an underdog. Suppose you check it down on the flop in order to be able to fold on the turn? This would mean:

.311 you win $1
.689 you lose $1

Net EV -$0.378

This is actually fractionally better for you, so this is the course you should prefer. However, your opponent knows this. Suppose the SB bets on the flop in order to take this strategy away from you? If you just call, then the turn plays out as it did in the first scenario, and you end up with the same EV. However, you also have the option to raise on the flop, when you actually have a small edge. Assuming you do, and miss the turn, again each of you will put in one big bet on the turn.

Here's the EV for that:

.311 you win $3
.219 you win $5
.470 you lose $5

Net EV -$0.322

This is the most favorable of all for you. So, the SB should not bet out on the flop in order to prevent you from checking it down.

Therefore, the optimal play for both players would seem to be to check it down on the flop, and to fold on the turn depending on whether you hit or miss.

[primelord]

Brillig is my hero. I would have bet a large amount of money that he would be the one to solve it. This is of course correct.

Personally I found the problem interesting (rkmsuf clearly disagrees). On the surface this problem seems very simple. The SB does not have an equity edge on the flop so he/she can not bet the flop profitably even though he/she has the best hand right now. The BB can certainly bet the flop for a profit, but actually loses more when he misses the turn and has to call a bet than he made on the flop. So betting the flop is also not his best option despite the current edge.

[Draft Dodger]

i need aspirin

[BrianD]

Originally Posted by primelord

Brillig is my hero. I would have bet a large amount of money that he would be the one to solve it. This is of course correct.

Personally I found the problem interesting (rkmsuf clearly disagrees). On the surface this problem seems very simple. The SB does not have an equity edge on the flop so he/she can not bet the flop profitably even though he/she has the best hand right now. The BB can certainly bet the flop for a profit, but actually loses more when he misses the turn and has to call a bet than he made on the flop. So betting the flop is also not his best option despite the current edge.

If both players check with the intention of folding on the turn, why are the options winning $1 or losing $1? Since that money is alreay out of the players hands and in the pot, aren't they looking at winning $2 or losing $0? I thought you weren't supposed to make decisions based on previous bets.