Five card draw card removal effects

[LearnedfromTV]

I was playing around with some five card draw math, and I was wondering how your kickers predraw affect what hands your opponents can have, and if there is any consequent effect on proper opening ranges.

I looked at JJxxx. I calculated the exact number of ways your opponents can have QQ+ based on the number of x's that are Q, K, A. For two pair+, I just prorated the number of combinations to adjust from 52C5 to 47C5. This isn't quite correct, but I actually think it understates the effect I found, since holding one pair slightly decreases your opponents' chance of having two pair, trips, and full house relative to if you held no pair (I think). Either way, it shouldn't matter much.

Anyway, here are the estimated probabilities of a particular opponent holding QQ+ (including all two pair+), based on how many of your kickers are > J

JJ234, etc - 24.1%
JJ23A, etc - 21.3%
JJ2KA, etc - 18.6%
JJQKA - 15.9%

This translates to the likelihood of holding the best hand predraw in each position as:

Position - JJ234/JJ23A/JJ2KA/JJQKA
UTG - 25.1/30.0/35.6/42.1
MP - 33.1/38.2/43.8/50.1
CO - 43.6/48.6/53.9/59.5
BN - 57.5/61.8/66.2/70.8
SB - 75.9/78.6/81.4/84.1

The key factor is that any card you hold cuts in half the number of ways your opponent can have a pair of that card.

Additionally, holding x's > J reduces the odds an opponent with two pair has queens up or better, giving us extra outs, and reduces the chance an opponent who is holding QQ will catch trips to beat us if we catch two pair or trips.

So I'm wondering how often people who play a lot of draw use this type of reasoning. Also, I'm curious if anyone has done a full study of card removal effects (even something as small as holding a three straight will reduce very slightly your opponents' chances of holding a straight, etc.)

[bigpooch]

Interesting coincidence! One of my cats had erased some
notes for a book for limit draw (which I used to teach some
friends how to play) and some probabilities for JJ, JJX,
JJXY recorded in the book were inadvertently erased!

(Actually, the estimation in the OP is surprisingly quite a
bit off, although the idea of higher kickers is extremely
important.)

One can be more precise and look at the distribution of
hands once a hand of five cards are taken out. Since I've
worked out these calculations before, here is a summary if
you hold a pair for the combinations of better hands other
than straights and flushes:


quads...........387
full house.....2271
trips..........33075
two pairs...73683

total above: 109416

pair (u)......60198
pair (s)......32274
pair (i)......11559

u = unseen rank; s = seen rank; i = identical

For a pair of jacks, depending on the number of kickers
that are higher than a jack, it's easy to determine the
number of better hands EXCLUDING straights and flushes
from the above. With no higher kickers, there are
109416 + 3x60198 = 290010 combinations.


The only hands left now are flushes and straights, but we
lump straight flushes with flushes. This depends on the
"suit distribution" of the five cards we hold:

flushes:

4-1:......3492
3-2:......3288
3-1-1:....3123
2-2-1:....3003
2-1-1-1:..2838

(5-0 is excluded obviously)

The difference between the extremes is about 654
combinations. We'll assume the hand we have has 3-2
distribution, for consistency or argument's sake. Also,
it turns out that by doing so, we'll be conservative by
the order of about 0.1% or so in estimating the chances
that the hand is best.


straights:

These have to be worked out depending on the exact
distribution of ranks, and it's easier to compute the
number of straights including straight flushes and then
subtract the combinations for straight flushes.

Let's assume the 3-2 distribution (so we don't hold a
flush draw) and that the highest three ranks are of one
suit. Then for the hands JJ432, JJA32, JJAK2 and JJAKQ,
there are 26, 27, 29 and 30 possible combinations of
straight flushes. Now, as an example, compute the
combinations of straights including straight flushes:

JJ432: 16(4x16x2+2x64+1x16x3+1x4x9+2x27) = 6304
so subtracting 26, there are 6278 straight combinations.

Similarly,

JJA32: 16(1x4x6+3x16x2+3x64+1x16x3+1x4x9+1x27) = 6768
so there are 6768-27 = 6741

JJAK2: 16(1x18+1x4x6+2x16x2+4x64+1x16x3+1x4x9) = 7136
so there are 7136-29 = 7107

JJAKQ: 16(1x27/2+1x18+1x4x6+1x16x2+5x64+1x16x3) = 7288
so there are 7288-30 = 7258


Hopefully, all the above calculations are correct.


BETTER HANDS
------------

Then, these are the number of stronger combinations:
(e.g., for JJ432: 109416+3x60198+3288+6278 )

JJ432: 299576
JJA32: 272115
JJAK2: 244557
JJAKQ: 216784

Then, you should get these estimates (actually, the
approximation should give a "lower bound" since once there
is one hand beats a pair of jacks, it is more likely than
"average" that another hand also does; this is due to
pairs, trips, boats being more likely once more than one
player also holds a similar type of hand) for a pair of
jacks (an underestimate):


Position(6max) - JJ/JJA/JJAK/JJAKQ

utg - 33.7/37.8/42.0/46.7
hij - 41.9/45.8/49.9/54.4
co - 52.1/55.7/59.4/63.3
but - 64.8/67.7/70.7/73.7
sb - 80.5/82.2/84.1/85.9

The above percentages are also not quite correct because of
the "bunching effect": e.g., if the player utg will play
only KK or better, after he folds, the likelihood of getting
a king or ace is slightly higher; i.e., it is more likely
that someone will get a pair of kings in a six-handed game
AFTER the player utg folds (and he only plays KK or better)
than in a five-handed game. This effect will translate to
the percentages after the utg to be off by no more than
about 0.3% and could be lower than 0.2% once you get to the
button or small blind.


PRACTICAL CONSIDERATIONS
======================

For any marginal pair, not just jacks, the key kickers are
kings and aces. In practice, players (even experienced
ones) will cold call with AA (and even KKA) an opening raise
when it's incorrect to do so (especially if the raise is
from utg or if the hand is specifically AAKxy).

The LEAST important rank is the one just above the rank of
the pair because of the gap principle: even though you may
have the pair you are holding (a minimal legitimate opening
hand), you could have held much better; thus, cold calling
with a pair of exactly one rank higher than the minimum
opening pair is usually incorrect for the blind structure
that is commonplace.

This means that if you are in the hijack and hold a pair of
jacks, you normally can raise with JJAKx because it is very
unlikely that the cutoff, button or small blind will call a
raise cold with QQ. Also, it turns out that JJxyz (no
higher kicker than a jack) is a marginal open raising hand
from the cutoff (and submarginal in say, $5-10 with $2 and
$5 blinds), so holding even a queen kicker is just enough
to swing it into +EV at most tables. The raw percentages
don't reflect this, but just thinking about how the hand is
likely to play out and estimating the EV is sufficient.

For one pair hands WITHOUT an ace or king kicker, usually a
hand that has at least about 94% of the "money odds" for
opening for a raise is sufficient: e.g., in $5-10, since you
are risking $10 to win the middle of $7, you normally need
a hand that is about a 0.94(10/17) = 55.3% favorite of
being best before the draw. When a one pair hand has one
or more higher kickers, especially an ace, it may only need
to be about even money to be best before the draw to open
raise with. Of course, a lot depends on the opposition.

From the button, things change a lot, especially for one
pair hands such as 99 or 88. One reason is that some
players in the blinds will reraise with AA when facing an
opener from the button (and even with KK!). Thus, you
PREFER to have an ace with these one pair hands, but a hand
as good as 99K should still be opened because it's the best
hand over 55% of the time. With a pair of eights, you may
opt NOT to open raise with anything less than 88AJX since
it's not unusual to see many players in the big blind
defend with a pair of nines or a small blind cold calling
with TT.

[LLCoolDave]

Interesting concept you are analyzing there, havn't had the time to read it properly yet. What stuck out was this, though:

"since holding one pair slightly decreases your opponents' chance of having two pair, trips, and full house relative to if you held no pair (I think)"

Actually, I somehow think it's the other way around. If you hold a pair, there are 3 ranks he's less likely to have a pair and one where he is very unlikely to have a pair, where as holding 5 unpaired cards gives 5 ranks that are unlikely to pair up for an opponent. This is just an instinctive feeling, I havn't done the math to back this up, so I could very well be wrong. In any case, I think the difference is pretty minor.

Some of this math could very well help me play QQ and JJ better than I currently do, I still feel very unsure when playing these hands. I will take a closer look at this later.

[LearnedfromTV]

[ QUOTE ]
pair (u)......60198
pair (s)......32274
pair (i)......11559

u = unseen rank; s = seen rank; i = identical

[/ QUOTE ]

I see what I did wrong. I had these numbers at 84480/42240/14080. 84480 is of course the number of combinations of any pair for a randomly dealt hand (out of 52C5 possibilities). 6 ways to make a pair with 4 cards available, 3 if 3, 1 if 2, and I just divided the 84480 by the appropriate factor; but the other cards elminate some of the combinations also, i.e. JJ742, the number of combinations of 77xxx (in 47C5 given JJ742 dead)isn't just changed by the missing 7, but also the missing J, J, 4, 2.

I still don't understand why the ratios aren't 6:3:1. I guess if I have JJ742, an opponent with 77 and three random cards will have two pair relatively less often than one with JJ and three random cards? Because given that he has 77, JJ77 is tougher to make than if we give him JJ and try to make JJ77. Etc.

Thanks for the post, this is great stuff.

[cgull]

Bigpooch - Please write the book. I'll buy it. You consistently make wonderful posts on draw.

Since you know the game so well, do you still have fun playing ? Or is it a little mechanical ?

[Rampage_Jackson]

Wow this looks like a really good thread but I don't have time to check it now... gotta print it out later.

[AceOfClubs]

[ QUOTE ]
Bigpooch - Please write the book. I'll buy it. You consistently make wonderful posts on draw.


[/ QUOTE ]

[BBQbowser]

I am very big on this concept.

[mykey1961]

Here are some exact counts rather than estimates for what one opponent could have based on your 5 cards.

We have : JsJh4d3c2c

Opponent will have:
0.00001630 Straight Flush
0.00025229 Quads
0.00148050 Boats
0.00183384 Flushes
0.00409338 Straights
0.02156214 Trips
0.04803516 Two Pair
0.42385193 One Pair
0.49887447 High Card
0.19500580 QQ+

0.29550436 One Pair+
0.00004153 One Pair=
0.70445411 One Pair-

0.79745870 We win
0.00001760 We tie
0.20252370 We lose


We have : AdJsJh3c2c

Opponent will have:
0.00001760 Straight Flush
0.00025229 Quads
0.00148050 Boats
0.00183254 Flushes
0.00439457 Straights
0.02156214 Trips
0.04803516 Two Pair
0.42385193 One Pair
0.49857328 High Card
0.17710287 QQ+

0.23847251 One Pair+
0.00004153 One Pair=
0.76148597 One Pair-

0.82133057 We win
0.00001760 We tie
0.17865182 We lose

We have : AdKcJsJh2c

Opponent will have:
0.00001695 Straight Flush
0.00025229 Quads
0.00148050 Boats
0.00183319 Flushes
0.00463513 Straights
0.02156214 Trips
0.04803516 Two Pair
0.42385193 One Pair
0.49833272 High Card
0.15913931 QQ+

0.19236714 One Pair+
0.00004153 One Pair=
0.80759133 One Pair-

0.84063186 We win
0.00001760 We tie
0.15935053 We lose


We have : AdKcQcJsJh

Opponent will have:
0.00001760 Straight Flush
0.00025229 Quads
0.00148050 Boats
0.00183254 Flushes
0.00473357 Straights
0.02156214 Trips
0.04803516 Two Pair
0.42385193 One Pair
0.49823428 High Card
0.14103364 QQ+

0.14891958 One Pair+
0.00004153 One Pair=
0.85103889 One Pair-

0.85894876 We win
0.00001760 We tie
0.14103364 We lose

[bigpooch]

Actually, these are estimates; exact numbers would be ratios
over C(47,5) = 1533939 and the relevant combinations are in
my previous post from which exact numbers can be obtained
when heads up.

These approximations seem correct except that it is
misleading for the ties to be exactly the same hand with the
same kickers because in practice, JJT42 and JJ943 have the
same pot equity when these hands are in (with the intention
of drawing three to a pair). Hence, the ratio of
11559/C(47,5) or about 0.007535501738 should be thought of
and not 27/C(47,5) or about 0.000017601743.

Also, once we have numbers for heads up, an approximation
should be used when considering more than two opponents
because the inclusion-exclusion calculation is horrendously
difficult for an exact probability that the hand is best
predraw; besides, because of the factor in the next
paragraph, it is sufficient to have a good approximation.

In addition, if there has already been some action, such as
one or two players have folded ahead of you, the ranks are
no longer equally distributed but rather very slightly
skewed towards kings and aces.