[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.