Michael Schell  Tip #1
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Card frequency in cribbage
Cribbage is played with a standard 52card deck, divided into
four suits and thirteen ranks. Although suit enters the picture when
a player gets a flush or His Nobs, it's rank that's of
primary importance. Not all ranks are created equal. Because face
cards have a pip value of ten, and because we award two points for a
fifteen combination, 5s are, a priori, the
most valuable cards in the deck (making cribbage perhaps the world's
only card game where this is the case). Points are also scored for
runs, and since these are not allowed to cycle around from a
K to an A, the usefulness of edge cards
(A, 2, Q and K)
is correspondingly reduced.
Since different card ranks tend to have different values, it
stands to reason that certain ranks are more likely to be retained
after discarding. We can calculate that any particular rank has a
39.7% chance of appearing among your opponent's initial six cards,
but do we know how likely is it to appear among her final four?
Well, thanks to some work by a few enterprising researchers, we do.
The measure in question is card frequency by rank, which
is expressed as the percentage likelihood that, on a random deal,
your opponent will be holding at least one card of the specified
rank after the toss. Obviously these numbers are linked to
discarding decisions, and thus will be different for pone and
dealer.
I know of three good sources for card frequency data. All are
encapsulated in the following table. The figures are for normal,
nonendgame situations.
Card frequency in fourcard hands
Rank 
Pone 
Hessel 
Schempp 
Ras 
A 
21.5% 
20.9% 
19.9% 
2 
25.4% 
25.0% 
25.2% 
3 
28.1% 
27.8% 
27.0% 
4 
28.7% 
28.7% 
29.5% 
5 
38.5% 
38.6% 
39.0% 
6 
28.3% 
28.7% 
28.2% 
7 
25.6% 
25.1% 
25.8% 
8 
25.5% 
25.2% 
26.0% 
9 
25.0% 
25.2% 
25.6% 
10 
23.5% 
23.2% 
22.8% 
J 
29.6% 
30.0% 
28.4% 
Q 
19.8% 
19.0% 
20.3% 
K 
13.4% 
12.3% 
14.7% 
Rank 
Dealer 
Hessel 
Schempp 
Ras 
A 
25.6% 
25.6% 
25.0% 
2 
24.0% 
23.8% 
23.9% 
3 
24.2% 
24.1% 
25.1% 
4 
27.9% 
28.0% 
28.4% 
5 
28.2% 
28.5% 
28.7% 
6 
26.1% 
25.6% 
25.6% 
7 
22.5% 
21.1% 
21.8% 
8 
22.8% 
21.7% 
22.7% 
9 
25.7% 
25.3% 
25.0% 
10 
29.1% 
29.0% 
29.1% 
J 
26.9% 
27.3% 
27.3% 
Q 
26.9% 
27.5% 
27.0% 
K 
27.0% 
26.9% 
26.7% 
Like discarding averages, these figures are standardized results
drawn from thousands of samples and calculations. While they are
useful, it is important to remember that they are an abstraction. In
a real game the likelihood of your opponent having a particular rank
will depend on the starter and the six cards you were dealt,
since she obviously cannot have any of these. And naturally, once
she starts playing her cards, you will get a more concrete idea of
her hand composition.
The three sets of statistics were generated using three distinct
methodologies. Craig Hessel's figures come from the same calculation
routine he developed to compile his table of discard averages (see
my articles
Discarding to your crib and
Discarding to opponent's crib). Hessel's original routine
disregarded suit entirely, and since the possibility of retaining a
flush was thereby removed, the incidence of pairs was somewhat
overstated. He has since revised his algorithm to correct this, and
it is the newer and more accurate results that I cite above.
Tim Schempp's percentages are derived from the hand frequency
data included with his
table of pegging averages. The source data were sampled from
computer simulations of nonendgame discarding situations. Since
Schempp's discarding algorithm ignores suit, the incidence of pairs
is slightly exaggerated. This explains why his totals for each rank
are, on average, a trifle lower than Hessel's.
George "Ras" Rasmussen's statistics come from the discard data he
gathered from his own overtheboard games between 1990 and 1998. It
is the same data set used to generate his
discard averages. Although Ras
did not specifically track card frequency in fourcard hands, he did
note the exact incidence of each of the 91 possible discards in his
sample set, making it possible to extrapolate card frequency numbers
by assuming a uniform distribution of ranks among the original
sixcard hands. Like the other numbers, Rasmussen's pertain to
nonendgame situations. His methodology was to throw out data for
cribs that were not actually counted in the game (i.e., because one
player had already won).
Although they were gathered using very different techniques —
empirical calculation, computer simulation, sampling from expert
games — these three sets of numbers agree very closely. Just how
closely can be seen by graphing the results, which also makes them
easier to visualize.
Card frequency in fourcard hands (pone)
Card frequency in fourcard hands (dealer)
Legend
None of the results for any rank vary by more than 2.4% between
sources. The majority are within 1%. As a further check on their
reliability, I compared these statistics to those generated from a
log file of 500 games I played against
Cribbage for Windows 97. I found that my results correlated
extremely closely with those of Hessel, Schempp and Rasmussen. We
can be confident that these two graphs are an accurate
representation of card frequency in cribbage hands.
Pone's cards
Let's take a closer look at the numbers, starting with pone. It's
no surprise that the card pone is most likely to hold is a 5.
This rank occurs in 38.5% to 39.0% of pone's fourcard hands.
Comparing that to the 39.7% likelihood of being dealt one or
more 5s should give you a idea of how seldom this
card gets passed to an opponent's crib. The second most frequently
held card is the J, which is obviously preferred over
other tencards because of the possibility of His Nobs.
Pone will hold onto midcards with average frequency. There is a
slight bump on 4s and 6s, probably
because these cards are often held in combination with a 5.
A more significant feature is the dropoff in edge cards. Whereas
Js appear in about 30% of pone hands, Qs
appear in only 20%, and Ks are present in only 13%.
There is a similar, though less dramatic, dropoff with 3s,
2s and As, all of which appear less
often than 4s, with progressively decreasing
frequency. Although they are valued as pegging cards, As
and 2s just don't factor into enough runs to maintain
parity with their higher neighbors.
What are the implications for practical play? Well, the
conventional wisdom that 5s and Js are
the easiest cards to trap when pegging against pone is clearly
upheld. If you're dealer, saving combinations like 34,
46 and 67 for last can be remarkably
effective — pone will often have a lone 5 lurking
behind a trio of low and/or midcards. Likewise the wellknown trick
of retaining a pair of Js for last in hopes of
tripling a lone J is a valid tactic against pone. But
trying the same ploy with a pair of 10s, Qs
or Ks is much less likely to succeed.
If you're trying to hold down pone's pegging, then the
percentages tell you he's most likely to be able to pair a 5,
and least likely to be able to pair a K. It's
important to keep this in perspective though. If you're defending
with 4510K and pone leads a 3, your
safest reply is ordinarily the K. But if the starter
is a 10, then your safest reply is the 10,
since the duplication of this rank by the starter is more
significant than pone's general preference for 10s
over Ks. In general, a matched card is statistically
safer than an unmatched card, but 5s and Js
are exceptions. A duplicated 5 is no safer against
pone than a lone A, 2, 3
or any other card besides a J. And a duplicated
J is more likely to be paired by pone than a lone
K, and about equally likely to be paired as a lone
Q. So if you're holding JJQK as
dealer, your safest reply to an A through 10
lead is your K, not one of your Js.
Take another look at pone's Q/K
dropoff. If your opponents aren't hanging onto these cards, then
where are they going? Into your crib, of course. Now you know why
certain wide card combinations return a bit more in your crib when
they contain a Q or K instead of a
10.
Toss: 

Average in your crib: 


Consensus 

Hessel 
Bowman 
Ras 
A10 

3.43 

3.37 
3.42 
3.51 
AQ 

3.43 

3.39 
3.40 
3.50 
AK 

3.41 

3.42 
3.44 
3.36 







210 

3.57 

3.51 
3.50 
3.71 
2Q 

3.62 

3.52 
3.49 
3.86 
2K 

3.54 

3.55 
3.49 
3.57 







310 

3.56 

3.61 
3.56 
3.51 
3Q 

3.61 

3.62 
3.55 
3.65 
3K 

3.70 

3.66 
3.55 
3.89 







410 

3.59 

3.62 
3.56 
3.60 
4Q 

3.60 

3.63 
3.55 
3.63 
4K 

3.61 

3.67 
3.55 
3.61 







510 

6.66 

6.68 
6.61 
6.70 
5Q 

6.63 

6.71 
6.59 
6.59 
5K 

6.67 

6.70 
6.59 
6.73 







610 

3.20 

3.15 
3.13 
3.31 
6Q 

3.31 

3.08 
3.12 
3.73 
6K 

3.15 

3.13 
3.12 
3.21 







710 

3.28 

3.10 
3.14 
3.59 
7Q 

3.25 

3.17 
3.19 
3.39 
7K 

3.29 

3.21 
3.19 
3.47 







8Q 

3.18 

3.16 
3.20 
3.19 
8K 

3.15 

3.20 
3.21 
3.04 







9Q 

3.00 

2.97 
3.03 
2.99 
9K 

3.07 

3.05 
3.08 
3.07 
Dealer's cards
With pone, the difference in frequency between the most and least
favorite rank is over 25%. With dealer, the corresponding spread is
less than 8%. Since dealer's most valuable cards can find themselves
in the hand or crib, the distribution of ranks in dealer's
fourcard hands is much more flat.
The edge card dropoff we saw for pone is completely absent for
dealer. An A gets held a trifle more often than a
2 or 3. And Js,
Qs and Ks are each held with equal frequency.
There are slight bumps on 4s and 6
(which are often held with a 5), and a notable dip on
7s and 8s. The latter feature results
from dealer's tendency to toss midcards from mixed hands like the
following:
AA88QQ
A367QK
237810J
Knowing about this midcard dip can influence how you play your
cards as pone. Suppose you're holding the following:
78JJ
You have a choice between leading a J and leading
one of the midcards. Knowing that dealer is statistically less
likely to be holding a 7 or 8 than a
5 or J, you decide to lead the
8. Good choice, but not just because of card frequency. It
also helps you stay out of trouble if dealer does score on
your lead. If he takes a 152 with a 7, you can
either pair him or else break with a J, leaving you
with the flexible 7J. But if he scores a 152 off a
J lead instead, you're then stuck playing your second
J, since you cannot safely play your 7
or 8. This gives up a 312 if dealer has a 6 —
not too unlikely if he has a 5 — and leaves you with
touching cards, which are more likely to get trapped into a run
later.
The 7/8 dip also helps explain why
midcards often seem more productive in your opponent's crib than in
your own. If you've studied
discarding averages for dealer and pone, you may have noticed
that while most twocard combinations are worth roughly one point
more to your opponent's crib than they are to your own crib, with
midcard combinations (such as 67 and 88)
the difference is closer to 1¼ points.
Since 7s and 8s are
disproportionately absent from dealer's hands, something else must
be taking up the slack. That something is 5s and
tencards, which are slightly more likely to be retained than
discarded. This is not surprising when you consider that although
there are sixteen low cards (A through 4)
and sixteen midcards (6 through 9) in
the deck, there are a total of twenty 5s and
tencards. If dealer gets a 5 among her original six
cards, it's likely she'll also have a couple of tencards, in which
case she'll usually keep them all together. If you're pone, it's
worth paying particular attention to possible 5
traps, since if dealer does have one of these cards, it'll often be
accompanied by tencards, leaving the door open for standard traps
like these:
^{6}^{ }_{K}_{ }^{6 }^{ }_{5}_{ }^{4}^{ (315)}
^{9}^{ }_{K}_{ }^{4 }^{ }_{5}_{ }^{3}^{ (315)}
^{8}^{ }_{K}_{ }^{J}^{ (281) }_{ }_{K}_{ }^{7}^{ }_{5}_{ }^{6}^{ (284) }_{ }_{5}_{ (51)}
Let's see if we can apply what we know about
dealer card frequency to an old conundrum: leading from tencards.
Say you're holding 5JQK as pone. For one reason or
another, you've ruled out leading the 5.
Which card should you lead? The current prevailing opinion is
articulated thusly by DeLynn Colvert in
Play Winning Cribbage:
"Lead the K. The least likely tencard held by
Jake will be the K. You do not want to
entice a pair here."
Colvert specifically advises against leading a lone J,
"the most likely tencard held by Jake". Dan Barlow, in
Cribbage for Experts, concurs with the K
lead. But a different view is offered by George Rasmussen:
"Although [leading the K] is commonly believed
[to be best], and is espoused in DeLynn Colvert's book, it is not
so in actual play. The reason is that the J is the
key connector among the tencards and will be discarded to
dealer's crib more than any other tencard: with the 5,
with another J, as a 10J or
JQ combination. The J will also be
discarded with a middle card or a small card. This will nearly
always be done in preference to putting the K in
the crib. Dealer will often pair your K lead
thinking you led it believing it was safest lead of the
tenpointers. Think of how often you discard the J
to your own crib, and your rationale for doing so. Once you've
done this, it will become readily apparent why the J
is the safest tencard lead you can offer the dealer."
Moreover, Ras states that by leading a single J
"...you're rid of it on the opening play. That danger
represented by the trap of a single J is
considerable. The J is considered by most to be the
second most available card to trap in cribbage. Only the 5
is more frequently trapped."
What to do then? Well, the numbers tell us that dealer is equally
likely to have a J, Q or K,
so it turns out the conventional wisdom, as rendered by Colvert, is
wrong. To be sure, Rasmussen also misstates the case somewhat when
he suggests that dealer is less likely to have a J
than another tencard, but his point is still sound. A J
is no more vulnerable to being paired than a Q or
K, but it is more vulnerable to being trapped
into a run, both because it has more neighboring cards, and because
dealer is likely to be specifically gunning for it. Therefore the
safest plan is to get it out of the way first by making it your
opening lead.
A more aggressive tencard lead would be the K,
which saves the JQ combo for later, thereby
maximizing your chances of trapping dealer into a pair or run on the
second play series. Leading the Q, advocated by
Jarvis and Wergin and formerly my own personal choice, turns out to
be inferior. It offers no more safety than leading the J,
and because it breaks up your run, it provides less offensive
potential than leading the K. It's true that a
Q lead prevents dealer from safely playing a 10,
something he could do if you led the K. But if dealer
is really trapped with four tencards, he'll probably just pair your
Q lead anyway instead of risking a run or pair
with a 10 reply.
Of course, if one of your tencards is duplicated by the starter
or by a card you tossed, then that card will be the safest lead
regardless of its rank.
One other bit of conventional wisdom dispelled by card frequency
stats concerns J traps. We've seen that these are
very effective against pone, but most authorities, including both
Colvert and Rasmussen, will encourage you to try them against dealer
as well, usually by saving a pair of Js for last in
an attempt to triple a lone J. (With JJQK,
for instance, you would lead the K, hoping to triple
dealer's J at the end.) This is certainly a
legitimate tactic if you need to peg six points, but there's really
no reason to reserve this treatment for Js as opposed
to, say, Qs and Ks. In fact, a card
you're more likely to be able to triple is a 10,
the one tencard that dealer holds in disproportion to the others (10s
are not particularly valuable in dealer's crib, and they'll often be
held in combination with midcards). If you're holding a pair of
10s as pone, consider leading away from them. You
should also think twice before leading a lone 10.
The last word
It's worth reiterating that this discussion of card frequency
pertains to normal play. In peggingcritical situations, you
can count on a very different distribution of opponent's cards,
including a preponderance of low cards and magic elevens.
Nevertheless, it's worth getting familiar with these numbers. They
can tell you things about your opponent's hand that he doesn't want
you to know.
 Republished from
Cribbage Forum by permission. Text copyright © 2002 by
Michael Schell. All rights reserved.
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