Best possible drawing hands

[Poker Clif]

I've been thinking about pot odds and drawing hands (yes, I most definitely need to get a life). I don't remember ever reading what the best possible drawing hand in hold 'em would be, or put another way, what the largest possible number of outs would be at the turn or river.

I decided it would be useful to try to figure it out myself, rather than look for a chart or table. I would appreciate comments on either the correctness of either my answer, or the logic I used to get there.

The logical part: I came up with the following assumptions:

1. I should flop a straight flush draw with two overcards as a starting point.

2. I don't want to have an ace, since I could then only make the straight in one direction.

3. Since I don't have an ace, I don't want a scenario where an ace on the turn or river hurts me.

This is what I came up with:

My hole cards: Ks Qs
The flop: Jh Js Ts

So, my outs are:

Any spade, which makes a flush (9 outs)
And non-spade A or 9 makes a straight (6)
Any remaining K or Q makes top pair (6)

With 21 outs, I am less than 50% to hit on either street.

The moral of this story seems to be that a made hand is ALWAYS better than a drawing hand, since the best possible chance to hit a drawing hand is never 50% on any street, and in fact, in my scenario some of my outs could lead to a split pot.

Also a jack could make trips and a split pot, but since I'm only holding K-high at this point, a split pot would be worth some fraction of an out.

[MoonOrb]

Two thngs.

First, 6 of your outs aren't outs to someone holding a J. So you might want to make the flop JsTs2h or something. Also, with the board paired, there will be as many as ten cards that make a full house by the river, so 7 of your flush outs and all of your straight outs will dry up as well. All I'm saying is that if your goal is to think of the combination of a hand and flop with the most possible outs that is not yet a made hand, don't pair the board. It discounts too many of your outs because of all of the redraws and the possibility a K or Q won't win.

Second thing is that your hand is a favorite here to win by the river even if you're not a favorite on each street. You get 21 outs twice, so on the turn you're (21/47) to improve and (21/46) on the river. There will be a million people here who understand this better than I do, but I think you end up being better than a 2 to 1 favorite against, say, a lower pocket pair like 88 or something.

And I realize my post isn't too precise, but the moral of mine is that a made hand on the flop isn't always better than a drawing hand.

Oh, one other thing--this is even more true when the pot is multiway. I'm assuming for the sake of simplicity that this is heads up. And by "better," I think you mean which hand will win the most pots when you might want to consider which hand will win the most bets.

[MMagicM]

As Phil Gordon says: "A straight flush draw is a hand to bring to war!" If you hold Js Ts and the flop comes Qs Ks 6h you have 15 outs to make a straight or a flush which will happen 56.1% of the time with 2 cards to come.

[raze]

Take K [img]/images/graemlins/heart.gif[/img]Q [img]/images/graemlins/heart.gif[/img] on a J [img]/images/graemlins/heart.gif[/img] T [img]/images/graemlins/heart.gif[/img] 2 [img]/images/graemlins/club.gif[/img] board, and assume your opponent has J [img]/images/graemlins/diamond.gif[/img]5 [img]/images/graemlins/diamond.gif[/img].

Count your outs; you should be happy to get it all-in on the flop with two cards to come.

[Gonso]

An open-ended straight flush draw better, with overs better still.

[Poker Clif]

[ QUOTE ]
Two thngs.

First, 6 of your outs aren't outs to someone holding a J. So you might want to make the flop JsTs2h or something. Also, with the board paired, there will be as many as ten cards that make a full house by the river, so 7 of your flush outs and all of your straight outs will dry up as well. All I'm saying is that if your goal is to think of the combination of a hand and flop with the most possible outs that is not yet a made hand, don't pair the board. It discounts too many of your outs because of all of the redraws and the possibility a K or Q won't win.

Second thing is that your hand is a favorite here to win by the river even if you're not a favorite on each street. You get 21 outs twice, so on the turn you're (21/47) to improve and (21/46) on the river. There will be a million people here who understand this better than I do, but I think you end up being better than a 2 to 1 favorite against, say, a lower pocket pair like 88 or something.

And I realize my post isn't too precise, but the moral of mine is that a made hand on the flop isn't always better than a drawing hand.

Oh, one other thing--this is even more true when the pot is multiway. I'm assuming for the sake of simplicity that this is heads up. And by "better," I think you mean which hand will win the most pots when you might want to consider which hand will win the most bets.

[/ QUOTE ]

Thanks for the comments, they were just what I was looking for. As an intellectual and math exercise, I thought it was better for me to try to figure this out by myself, then have someone explain to me where I went wrong.

I understand the dangers of a paired board, and I'm surprised that in this context I didn't think about that.

I know you're right that I'm a favorite here. I have read that that's true. If anyone can help me with this, go for it, because I can't figure out why it's true.

If I have a whole bunch of outs, it's true that I have them twice, both on the turn and the river.

However one I thing I learned in my statistics classes is that "numbers don't have a memory".

There is no law of averges, that is, if my numerous outs give me a 48% chance of hitting, and I miss, that doesn't make it any more or less likely that I will hit on the river, which will be a 49% chance with one more "bad" card out of the way.

So, statistically speaking, we have fourth and fifth streets, two almost independent events (the extra known card on the river will make about a one percent difference in the likelihood of hitting one of my outs).

With each event, I have a less then 50% chance of hitting one of my outs. So how am I the favorite? What am I missing here?

Can any mathematicians out there help me with this?

[Peleus]

Yes you're correct in saying in both cases you have less then 50%, but because you're drawing to that twice, and you need to only hit it once.

Simplest example would be a coin flip. I get two flips of a coin, and I have to hit Tails once to win.

Each flip is independent of the other, each is a 50% chance.

Looking at the possibly out comes though, it can either be,

Heads + Heads (I loose)
Heads + Tails (I win)
Tails + Heads (I win)
Tails + Tails (I win)

We see its a 50% chance on each flip, but because I have multiple shots, and I only need to hit it once to win, I actually have a 75% chance of winning.

[MMagicM]

This might help you understand it:

http://en.wikipedia.org/wiki/Poker_probability_(Texas_hold_'em)

I haven't read everything so far, but the "After the flop" section should answer your question.

[Poker Clif]

[ QUOTE ]
Yes you're correct in saying in both cases you have less then 50%, but because you're drawing to that twice, and you need to only hit it once.

Simplest example would be a coin flip. I get two flips of a coin, and I have to hit Tails once to win.

Each flip is independent of the other, each is a 50% chance.

Looking at the possibly out comes though, it can either be,

Heads + Heads (I loose)
Heads + Tails (I win)
Tails + Heads (I win)
Tails + Tails (I win)

We see its a 50% chance on each flip, but because I have multiple shots, and I only need to hit it once to win, I actually have a 75% chance of winning.

[/ QUOTE ]

Well, just when I thought we had the subject settled, I ran across a poker odds chart online:

web page

It gives the odds for outs with one card to come after the flop, one card to come after the turn, and two cards to come after the flop.

Then, at the bottom is the following note:

"Remember! You should work out the odds only for the next card to come each time. Only use the third column [two cards to come] when your opponent is ALL IN."

[MoonOrb]

That comment doesn't change anything. There are two issues. First, whether or not you're a favorite to win the hand. Clearly, with a straight flush draw and two overcards, you are. Second, whether the pot odds on each street justify a call if you're bet into.

As a roughly 75% favorite to win the hand after the flop, you should have been raising for value. But let's suppose you just called (or suppose that you did raise for value and your bet was called). In any event, the turn comes and it's unhelpful for you. Your opponent bets out. Now you have a pot odds problem. With one street left, you have to count your outs again. It's possible now that the turn card helped your opponent, giving him either 2 pair or 3 of a kind, meaning your K-Q overcard outs are no good. You're now down to 15 outs. That gives you a slightly less than one-third probability of making your hand on the river. So just compare the size of the bet you have to call to the amount you expect to win if you make your hand and you'll know whether the call is appropriate.

This could put you in the situation where it would be correct to call an all-in bet (or make an all-in raise if you thought your opponent would call it) on the flop but incorrect on the turn.

But it still does not change the fact that after the flop, you're more than a 2:1 favorite to win.