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Old 11-05-2005, 08:40 PM
Jim Morgan Jim Morgan is offline
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Default Hand probabilities.

Suppose I have Ad2d3hKh in a 10-handed game.
Somewhere I have seen a stat that says that there is a 36% chance that someone was dealt the nut heart draw.
However, I am not so sure this number is accurate (though in real play, I am sure it is close enough)
If the flush never shows, then I really don't care about that opposing AhXh. The only time I care is when I see the board. But now the deck composition is very different. The pool of unseen cards is no longer 48 cards with 11 hearts.
Assuming we are at the river, the pool of unseen cards is down to 43 and only 8 of them are hearts.

It seems this makes the K-high flush quite a bit more likely to be the winner. On the other hand, I don't know how the 36% figure was calculated or if it accounted for the board cards in this way.

I believe a similar situation exists with nut low (A2 for simplicity). But in this case I think it weakens our holding. Once the 5 cards on the board have no A or 2, it is MORE likely that we are facing another A2 than it was before we see those cards. What I don't know is how this impacts the probability or if that is already accounted for when people speak of the odds of having the only nut low.

Anybody know the gory details here?

Jim Morgan
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  #2  
Old 11-05-2005, 10:21 PM
sy_or_bust sy_or_bust is offline
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Default Re: Hand probabilities.

I'm not an odds guru, but be careful not to assume that Pr(nut draw dealt) equals Pr(nut draw in play). You can make this assumption for A2, but you need to seriously discount for many other hands.
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  #3  
Old 11-07-2005, 02:44 AM
Buzz Buzz is offline
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Default Part 1. Your flush questions addressed.

[ QUOTE ]
Suppose I have Ad2d3hKh in a 10-handed game.
Somewhere I have seen a stat that says that there is a 36% chance that someone was dealt the nut heart draw.

[/ QUOTE ]

Jim - Before you see the flop, you know the whereabouts of 4 cards and the whereabouts of the other 48 are unknown.

When you see the five card board on the river, you know the whereabouts of 9 cards and the whereabouts of the other 43 are unknown.

Before the flop, the probability one of your nine opponents was dealt the ace of hearts is 36/48. After all five board cards are known, and if the ace of hearts is not one of them, the probability one of you nine opponents was dealt the ace of hearts is 36/43. Those probabilities are simply the number of cards your opponents have collectively been dealt divided by the total number of cards possible.

Assuming one of your opponents was dealt the ace of heart, the probability that opponent was also dealt another heart has three cases. (1) one other heart, (2) two other hearts, and (3) three other hearts. All three must be considered and then combined (by adding).

Then the probability an opponent was dealt the ace of hearts is multiplied by the probability that opponent was dealt one, two, or three other hearts. Iím sure I waded through that at one time or another and posted the result both here on 2+2 and on r.g.p. I would have been more interested in nine handed than ten handed, but I imagine I computed it for both.

[ QUOTE ]
However, I am not so sure this number is accurate

[/ QUOTE ]

And you still wonít be sure if I wade through the calculations again.

[ QUOTE ]
The only time I care is when I see the board.

[/ QUOTE ]

Fair enough. In addition to the only time you care, I think I might have been interested in knowing how often a king high flush draw might run into an ace high flush draw, figuring from immediately after a flop that had two cards of the flush suit. There was some crap that kept getting posted about never drawing to anything but the nut flush. (Iím not advising anybody to draw to anything but the nut flush, but I do think non-nut flush draws, especially to the king, do add value).

[ QUOTE ]
But now the deck composition is very different. The pool of unseen cards is no longer 48 cards with 11 hearts.
Assuming we are at the river, the pool of unseen cards is down to 43 and only 8 of them are hearts.

[/ QUOTE ]

Good thinking. Which ever one youíre interested in, they clearly are two distinct problems - or three distinct problems if figured from immediately after the flop - or four distinct problems if you want to throw in the turn.

But if there are two distinct problems, then there are also two distinct answers.

[ QUOTE ]
It seems this makes the K-high flush quite a bit more likely to be the winner. On the other hand, I don't know how the 36% figure was calculated or if it accounted for the board cards in this way.

[/ QUOTE ]

Tell you what. Iíll run a ten handed simulation, giving Hero
K[img]/images/graemlins/heart.gif[/img], K[img]/images/graemlins/club.gif[/img], 2[img]/images/graemlins/heart.gif[/img], 3[img]/images/graemlins/diamond.gif[/img], and making the board on the river
<font color="white">_</font>
Q[img]/images/graemlins/heart.gif[/img], 8[img]/images/graemlins/heart.gif[/img], 7[img]/images/graemlins/heart.gif[/img], J[img]/images/graemlins/club.gif[/img], T[img]/images/graemlins/diamond.gif[/img]. The only way Heroís hand can be beaten is by an ace flush.

10,000 deals good enough? That should get us reasonably close to the truth. Iím predicting Heroís hand will win about 64% of the time.
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.
.
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6389 wins and 3611 losses. Well... that rounds off to 64%.

Next Iíll run another ten handed simulation, giving Hero
K[img]/images/graemlins/heart.gif[/img], K[img]/images/graemlins/club.gif[/img], 3[img]/images/graemlins/heart.gif[/img], 2[img]/images/graemlins/diamond.gif[/img], and making the flop Q[img]/images/graemlins/heart.gif[/img], 7[img]/images/graemlins/heart.gif[/img], 2[img]/images/graemlins/heart.gif[/img]. This time Hero will still probably end up with the second nut flush on the river, but this time the board may pair on the turn or river. In addition, 44/990, Hero figures to end up with the nut flush on the river. Also, thereís a very remote chance of an opponent ending up with a straight flush. Finally, note that Hero could win with a full house or quads. Will hero win more or less than in the last sim?
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.
.
Ready?
After this flop, Hero has 4291 wins for flush, 242 wins with full house, 15 wins with quad kings. 4548 total wins.
Hero has 181 losing full houses, 5271 losing flushes.
Hero loses with a flush more than he wins, but more to full houses than to ace-high flushes.
Opponents collectively have 5597 wins.
ē 12 winning straight flushes,
ē 301 winning quads,
ē winning full houses 2985 times, and
ē 2154 winning flushes.
The simulator doesnít provide a way to know how often an opponent had an ace-high flush that was pre-empted by a full house. (Or if there is a way, I donít know how).

But since there were two more non-hearts, Heroís opponents figure to have a heart flush slightly less often than on the river. kind of a moot point when you notice how often Hero gets beaten after the board pairs. Getting beaten by the board pairing and an opponent making a full house or quads is more prevalent than getting beaten by a better flush by about a three to two ratio.

At any rate, Hero wins more with the second nut flush on the river than with the second nut flush on the flop. That seems entirely reasonable.

Letís back it up to before the flop. Weíll give Hero K[img]/images/graemlins/heart.gif[/img], K[img]/images/graemlins/club.gif[/img], 3[img]/images/graemlins/heart.gif[/img], 2[img]/images/graemlins/diamond.gif[/img] again but leave the board blank. ten handed, 10,000 runs. Of course this time Hero can make a full house and a straight, but is not guaranteed a flush.
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This time Hero wins more with a full house or quads (607+113) than he wins with a flush (294/10000) and Hero loses more with the flush (369/10000) than he wins with it. But with 23KKs, the hand wins for low more than for high and the suited king just changes some of those low wins to scoops and some of those low quarters to three quarters. Youíre not really playing the hand exclusively for the suited king, but rather for a combination of different potential ways to win which include the suited king.

Bear in mind that these sims involve non-folding opponents. In a real game you have to adjust for folding opponents.

Buzz
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  #4  
Old 11-07-2005, 02:52 AM
Buzz Buzz is offline
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Join Date: Sep 2002
Location: L.A.
Posts: 598
Default Part II. Your low questions addressed.

[ QUOTE ]
I believe a similar situation exists with nut low (A2 for simplicity). But in this case I think it weakens our holding. Once the 5 cards on the board have no A or 2, it is MORE likely that we are facing another A2 than it was before we see those cards. What I don't know is how this impacts the probability or if that is already accounted for when people speak of the odds of having the only nut low.

[/ QUOTE ]

Jim - Your odds always depend on your perspective. The more information you have, the better able you are to pin point your chances. But I think you already know that and are just wondering from what perspective the odds are figured.

The answer is when you know the whereabouts of 9 cards, then the odds are figured from a 9 card perspective. When you only know the whereabouts of 4 cards, then the odds are figured from a four card perspective.

In the interest of clarity, the vantage point from which the odds are figured and how many hands are involved should always be stated.

Letís make Heroís hand A2KKn and the board 34589n. In that case, Hero scoops or shares 10,000/10,000 regardless of the number of players. As the number of players increases, so do the chances one of Heroís opponents also has the nut low (obviously). In a ten player game, Hero, as simulated, scoops 5760 and wins a total of 2035 for the 4240 non-scoops.
1017.58 = Q/4 + S/6 + E/8
4240 = Q + S + E
Two equations and three unknowns, hmm...
Fiddle faddle ...
Letís try:
3736 = Q
494 = S
10 = E
check:
934=Q/4, 82.33=S/6, 1.25 =E/8
934+82.33+1.25 = 1017.58
check:
3736+494+10 = 4240.

O.K. looks like Q = 3736, S = 494, E = 10 works. Lucky guess. (Got it on the first approximation).

A different simulation would be slightly different. This time, it looks like A2KKn, with a final board of 34589n,
ē wins outright for low 5760/10000,
ē gets quartered for low 3736/10000,
ē gets sixthed for low 494/10000
ē and gets eighthed for low 10/10000.
About 57.6% sole posession of the low, about 37.4% getting quartered, about 4.9% getting sixthed, and about 0.1% getting eighthed. Something like that.

I calculated that a few years back. I got slightly different results for a ten player game:
56.88% sole possession
37.69% getting quartered
5.31% getting sixthed
0.1% getting eighthed.
Well... I guess thatís reasonably close to tonightís sim values.

Thatís in a ten player game with the board known and with no aces or deuces on the board. With aces and deuces on the board, as you have reasoned, things would be different.

Before you know there are no aces or deuces on the board, from your perspective youíre less likely to encounter an ace or deuce in the hands of an opponent (because thereís a chance that ace or deuce could end up on the board).

But Iím with you - what does it matter if that happens, because if it does, youíre screwed (counterfeited) anyway?

[ QUOTE ]
I believe a similar situation exists with nut low (A2 for simplicity). But in this case I think it weakens our holding. Once the 5 cards on the board have no A or 2, it is MORE likely that we are facing another A2 than it was before we see those cards. What I don't know is how this impacts the probability or if that is already accounted for when people speak of the odds of having the only nut low.

[/ QUOTE ]

Well, thatís true except I simulated (and previously calculated) how it impacts the probability with no ace or deuce on the river. Going backwards to before the flop, it would be LESS likely. I neither figured not simulated the probability of another opponent with an ace-deuce before the flop, because as you have implied, itís kind of a moot point. But anyhow, I think itís approximately somewhere in the neighborhood of one third.

More importantly, if you do make it to the river with a bare ace-deuce nut low, then in a ten player game youíll have to share the low about 43% (turned out to be ~42% in the simulation I ran tonight and for which I gave the low results above).

I mostly play in nine player games, and there I figure Iíll get quartered or sixthed (or, ugh, eighthed) approximately two times out of five with the most commonly played nut lows (A2XY, A3XY, or 23XY). Itís close to that in a ten player game.

When will you get quartered or sixthed?

On the river.

[ QUOTE ]
Anybody know the gory details here?

[/ QUOTE ]

I think thatís it.

Buzz
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  #5  
Old 11-07-2005, 11:03 AM
hachkc hachkc is offline
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Join Date: Feb 2005
Location: Lake Orion, MI
Posts: 69
Default Re: Hand probabilities.

[ QUOTE ]
Suppose I have Ad2d3hKh in a 10-handed game.
Somewhere I have seen a stat that says that there is a 36% chance that someone was dealt the nut heart draw.
However, I am not so sure this number is accurate (though in real play, I am sure it is close enough)
If the flush never shows, then I really don't care about that opposing AhXh. The only time I care is when I see the board. But now the deck composition is very different. The pool of unseen cards is no longer 48 cards with 11 hearts.
Assuming we are at the river, the pool of unseen cards is down to 43 and only 8 of them are hearts.

It seems this makes the K-high flush quite a bit more likely to be the winner. On the other hand, I don't know how the 36% figure was calculated or if it accounted for the board cards in this way.


[/ QUOTE ]

Another question to ask yourself, how willing our you to defend your 2nd nut flush if you are bet into or raised?

In PLO8, I've often taken pots down on the flop or by simply holding the A and betting like I have the flush or flush draw. Obviously this requires the right board or your hand to have some strong redraws for the high and a low shot. For limit, the 2nd nut flush is probably much more playable and profitable.
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