Discarding to your crib

This article begins a discussion of discarding technique. The general principles of discarding are amply covered by the books recommended in A course of study, but to get beyond the basics you need to look at some hard numbers. Grinding statistics is not very exciting work, but it will eventually pay off as your discarding decisions are governed by sound mathematics rather than guesswork. So hang in there.

What we're going to examine are discarding tables. These indicate the average number of points a crib will be worth if you discard a particular two-card combination. Many such tables have been published over the years, but the majority of them have been compiled using statistically dubious means, and have little real value to cribbage players. I'm going to cite three tables that do have value. All are from reliable sources, and were compiled using different methodologies. They don't agree on every detail — the composition of any crib is the result of a decision by two independent players, so it is impossible to arrive at absolute numbers. But taken together, they paint a good picture of what you can actually expect in your games.

We'll start with the statistics compiled by Craig Hessel, author of Cribbage for Windows 97, a shareware cribbage program. Hessel wrote a special routine that goes through each of the 18,395 different six-card hands (independent of suit), calculating optimal discards and reckoning the point value of the resulting cribs based on the available starter cards. The process is iterated (repeated) until the average values for each of the 91 possible discard combinations have stabilized. Because of this rigorous methodology and the high number of iterations, these statistics are more reliable than your garden variety computer-generated averages.

Hessel ignored crib flushes in generating his statistics — an understandable simplification, since these happen approximately once every 675 deals, representing a difference in non-pair tosses of less than .01 points. Hessel found that the average value of all cribs was just under 4.8 points.

Discarding to your crib (Hessel)

  A 2 3 4 5 6 7 8 9 10 J Q K
A 5.26 4.18 4.47 5.45 5.48 3.80 3.73 3.70 3.33 3.37 3.65 3.39 3.42
2 4.18 5.67 6.97 4.51 5.44 3.87 3.81 3.58 3.63 3.51 3.79 3.52 3.55
3 4.47 6.97 5.90 4.88 6.01 3.72 3.67 3.84 3.66 3.61 3.88 3.62 3.66
4 5.45 4.51 4.88 5.65 6.54 3.87 3.74 3.84 3.69 3.62 3.89 3.63 3.67
5 5.48 5.44 6.01 6.54 8.95 6.65 6.04 5.49 5.47 6.68 7.04 6.71 6.70
6 3.80 3.87 3.72 3.87 6.65 5.74 4.94 4.70 5.11 3.15 3.40 3.08 3.13
7 3.73 3.81 3.67 3.74 6.04 4.94 5.98 6.58 4.06 3.10 3.43 3.17 3.21
8 3.70 3.58 3.84 3.84 5.49 4.70 6.58 5.42 4.74 3.86 3.39 3.16 3.20
9 3.33 3.63 3.66 3.69 5.47 5.11 4.06 4.74 5.09 4.27 3.98 2.97 3.05
10 3.37 3.51 3.61 3.62 6.68 3.15 3.10 3.86 4.27 4.73 4.64 3.36 2.86
J 3.65 3.79 3.88 3.89 7.04 3.40 3.43 3.39 3.98 4.64 5.37 4.90 4.07
Q 3.39 3.52 3.62 3.63 6.71 3.08 3.17 3.16 2.97 3.36 4.90 4.66 3.50
K 3.42 3.55 3.66 3.67 6.70 3.13 3.21 3.20 3.05 2.86 4.07 3.50 4.62

The next table is taken from the Third Edition of DeLynn Colvert's Play Winning Cribbage (p. 117). Colvert does not reveal the source of these statistics, but they appear to come from Bruce Bowman's Cribbage Master program. These numbers were generated from computer simulations of real game discarding situations.

Discarding to your crib (Colvert)

  A 2 3 4 5 6 7 8 9 10 J Q K
A 5.4 4.1 4.4 5.4 5.5 3.8 3.8 3.8 3.4 3.4 3.7 3.4 3.4
2 4.1 5.7 6.9 4.4 5.4 3.8 3.8 3.6 3.7 3.5 3.8 3.5 3.5
3 4.4 6.9 5.9 4.7 5.9 3.7 3.7 3.9 3.7 3.6 3.8 3.5 3.5
4 5.4 4.4 4.7 5.7 6.4 3.8 3.8 3.8 3.7 3.6 3.8 3.5 3.5
5 5.5 5.4 5.9 6.4 8.6 6.5 6.0 5.4 5.4 6.6 6.9 6.6 6.6
6 3.8 3.8 3.7 3.8 6.5 5.8 4.8 4.5 5.2 3.1 3.4 3.1 3.1
7 3.8 3.8 3.7 3.8 6.0 4.8 5.9 6.6 4.0 3.1 3.5 3.2 3.2
8 3.8 3.6 3.9 3.8 5.4 4.5 6.6 5.4 4.6 3.8 3.4 3.2 3.2
9 3.4 3.7 3.7 3.7 5.4 5.2 4.0 4.6 5.2 4.2 3.9 3.0 3.1
10 3.4 3.5 3.6 3.6 6.6 3.1 3.1 3.8 4.2 4.8 4.5 3.4 2.8
J 3.7 3.8 3.8 3.8 6.9 3.4 3.5 3.4 3.9 4.5 5.3 4.7 3.9
Q 3.4 3.5 3.5 3.5 6.6 3.1 3.2 3.2 3.0 3.4 4.7 4.8 3.4
K 3.4 3.5 3.5 3.5 6.6 3.1 3.2 3.2 3.1 2.8 3.9 3.4 4.8

Colvert's statistics match up closely with Hessel's. The average he gives for each toss is within 5% of its counterpart in Hessel's table. Perhaps the main difference is that Hessel gives higher figures for certain near card tosses (in which the two cards are adjacent or separated by one rank). Compare, for example, the 3-4, 6-7 and 6-8 discards in the two tables. Note that Colvert only resolves his figures to one decimal place (though Bowman's original numbers have two). Like Hessel, Colvert considers an average crib to be worth about 4.8 points.

The third table is perhaps the most interesting in that, unlike the others, it is based on real results from over-the-board play, rather than computer simulations. It is the result of years of meticulous note-taking by George "Ras" Rasmussen, an ACC Life Master and Pacific Northwest cribbage legend who tabulated the results of 86,755 discards to his own crib in games played between 1990 and 1998, most of them against expert players. Ras has released his statistics to the public, and I'm pleased to publish them here in their entirety for the first time on the Web.

Discarding to your crib (Rasmussen)

  A 2 3 4 5 6 7 8 9 10 J Q K
A 5.51 4.35 4.69 5.42 5.38 3.98 4.05 3.77 3.49 3.51 3.57 3.50 3.36
2 4.35 5.82 7.14 4.64 5.54 4.15 3.78 3.82 3.91 3.71 4.05 3.86 3.57
3 4.69 7.13 6.08 5.13 5.97 4.05 3.33 4.13 4.09 3.51 4.07 3.65 3.89
4 5.41 4.63 5.12 5.54 6.53 3.95 3.61 3.77 3.82 3.60 3.98 3.63 3.61
5 5.38 5.53 5.97 6.53 8.88 6.81 6.01 5.56 5.43 6.70 7.09 6.59 6.73
6 3.97 4.15 4.05 3.95 6.80 5.76 5.14 4.63 5.11 3.31 3.45 3.73 3.21
7 4.05 3.77 3.33 3.61 6.00 5.14 5.87 6.44 4.06 3.59 3.83 3.39 3.47
8 3.76 3.82 4.13 3.77 5.56 4.63 6.44 5.50 4.77 3.72 3.93 3.19 3.04
9 3.49 3.90 4.08 3.82 5.43 5.11 4.06 4.76 5.21 4.40 4.01 2.99 3.07
10 3.50 3.71 3.51 3.60 6.69 3.31 3.59 3.72 4.39 4.72 4.76 3.17 2.84
J 3.56 4.05 4.06 3.98 7.08 3.45 3.83 3.92 4.01 4.75 5.28 4.83 3.92
Q 3.50 3.85 3.64 3.63 6.59 3.73 3.38 3.19 2.99 3.16 4.82 4.93 3.48
K 3.36 3.56 3.89 3.61 6.72 3.20 3.46 3.04 3.07 2.83 3.92 3.48 4.30

Rasmussen found that the average value of his cribs was 4.84 points.

The great advantage of Rasmussen's statistics is that they reflect the bias of human, rather than computer, players. On the other hand, they are based on raw samples, and have a higher margin of error than the computer generated averages. In general I would consider values within 3-4% to be statistically equivalent. It's always wise to compare the values from all three tables in evaluating a particular discard.

There are a few disparities between Rasmussen's statistics and those from Hessel and Colvert. The most significant are given below:

Toss:

Average crib:

Rasmussen Colvert (Diff.) Hessel (Diff.)
6-7 5.14 4.8 +0.3  4.94 +0.2 
3-4 5.13 4.7 +0.4  4.88 +0.2 
A-3 4.69 4.3 +0.4  4.47 +0.2 
K-K 4.30 4.8 -0.5  4.62 -0.3 
2-6 4.15 3.8 +0.4  3.87 +0.3 
3-9 4.09 3.7 +0.4  3.66 +0.4 
A-7 4.05 3.8 +0.3  3.73 +0.3 
2-J 4.05 3.8 +0.3  3.79 +0.3 
3-6 4.05 3.7 +0.4  3.72 +0.3 
8-J 3.93 3.4 +0.5  3.39 +0.5 
2-9 3.91 3.7 +0.2  3.63 +0.3 
3-K 3.89 3.5 +0.4  3.66 +0.2 
2-Q 3.86 3.5 +0.4  3.52 +0.3 
7-J 3.83 3.5 +0.3  3.43 +0.4 
6-Q 3.73 3.1 +0.6  3.08 +0.7 
7-10 3.59 3.1 +0.5  3.10 +0.5 
7-K 3.47 3.2 +0.3  3.21 +0.3 
3-7 3.33 3.7 -0.4  3.67 -0.3 

I have distilled these three tables into a single set of figures for use in my own personal analysis and over-the-board decision-making. Click here to see my numbers.

These three discarding tables confirm many things you probably already know. The most productive tosses are those that are already worth points going in. This includes 5s, combinations totaling five (A-4 and 2-3), 15s (5-x, 6-9 and 7-8) and pairs. (Note that any five-card hand that includes a 5, A-4 or 2-3 must be worth at least two points. Therefore a toss of any of these cards is automatically worth two points). Next best are touching cards such as 3-4, 6-7 or J-Q. At the bottom of the list are wide cards, particularly combinations comprised of a mid-card and a high card (e.g., 9-K), since the two cards cannot possibly be combined to form a score, unless they're the same suit and form part of a five-card crib flush.

A less obvious point revealed by the tables is the relative futility of tossing pairs to your crib. With the significant exception of 5-5, pairs gain surprisingly little added value when they go in your own crib. The reason is clear if you think about it. A 6-7 tossed to your crib is worth zero points going in, but can improve to two points in combination with a 2, 6 or 7, three points in combination with a 5 and five points in combination with an 8. A 6-6 toss is worth two points going in, and can improve with a 3, 6 or 9, but otherwise will require two additional cards to increase in value.

Accordingly, the average for a 6-7 toss is 5.14 points (Rasmussen), all of which is added value. The average for 6-6 is 5.76, but this represents only 3.76 points of added value. High pairs get even less added value: a pair of kings, for example, is worth less in your crib than a 6-7. The upshot is this: if you have a choice between tossing a stray pair and tossing two stray touching cards, the latter is normally the more productive choice.

For example, suppose you deal yourself the following lamer of a hand:

A-3-8-9-Q-Q

Toss the 8-9, not the pair of queens. This gives you a two-point hand (before the cut) and a crib worth an average of about 4.7 points. If you tossed Q-Q instead, you would be left with a bust hand and a crib worth an average of about 4.8 points (depending on whose averages you consult). Clearly the former will get you the most points in the long run.

Of course, in making any discarding decision you need to consider your strategic objectives. In this case, your board position may well dictate keeping A-3-8-9, since it is a better pegging hand than A-3-Q-Q.

Speaking of near cards, "edge" combinations such as A-2 or Q-K are, expectedly, less productive than combinations such as 3-4 or 6-7, which can be extended into runs in either direction. A surprising exception is J-Q, which outscores 10-J and Q-K in all three tables. Perhaps this is due to the proclivity of most players to toss kings to your crib.

Another point worth keeping in mind is that since these discarding averages are derived from large sample sets, they are somewhat abstracted. The actual average crib you can expect from a particular discard is influenced by four cards that cannot possibly combine with your two, namely the four cards you didn't toss to your crib! If you're dealt 6-7-8-8-9-10 and discard 9-10, don't expect to get the full 4.3 points in your crib. Since the chances of pone tossing or cutting an 8 are significantly reduced by the two 8s remaining in your hand, your average crib will be degraded by to point, a point that is not reflected by the static values in the above tables. For more on this subject, see the article Discard averages in the real world.

Star power

The ACC Tip Library includes an article by Rasmussen based on some slightly earlier discarding figures. Rasmussen divides the possible discards into four groups:

Toss:

Average crib:

Rasmussen Colvert Hessel

 four stars

5-5 8.88 8.6 8.95

 two stars

2-3 7.14 6.9 6.97
5-J 7.09 6.9 7.04
5-6 6.81 6.5 6.65
5-K 6.73 6.6 6.70
5-10 6.70 6.6 6.68
5-Q 6.59 6.6 6.71
4-5 6.53 6.4 6.54
7-8 6.44 6.6 6.58
 one star
3-3

6.08

5.9 5.90
5-7 6.01 6.0 6.04
3-5 5.97 5.9 6.01
7-7 5.87 5.9 5.98
2-2 5.82 5.7 5.67
6-6 5.76 5.7 5.74
5-8 5.56 5.5 5.49
2-5 5.54 5.4 5.44
4-4 5.54 5.7 5.65
A-A 5.51 5.4 5.26
8-8 5.50 5.4 5.42
5-9 5.43 5.4 5.47
A-4 5.42 5.4 5.45
A-5 5.38 5.5 5.48

 also-rans

all others

Here is a simplified grouping, designed to be easier to remember. If you memorize nothing else from this article, you should memorize the top three groupings, since the 23 tosses they comprise are usually the basis of the most troublesome discarding decisions:

four stars (one toss)
averages roughly 9 points

  • 5-5

two stars (8 tosses)
averages roughly 6 to 7 points

  • 2-3
  • 4-5 and 5-6
  • 5-x
  • 7-8

one star (14 tosses)
averages roughly 5 to 6 points

  • all other combos with a 5
  • A-4
  • pairs up to 8

also-ran (69 tosses)
averages less than 5 points

  • 6-9
  • pairs higher than 8
  • everything else

Actually, Ras lumps the 7-8 toss with the one-stars, since it returns slightly less than 6.5 points (per his averages). But because it is much closer to the two-stars than it is to its nearest neighbors among the one-stars, I prefer to place it with the two-stars instead. This also keeps all the 15 combinations together (except for 6-9), making it a bit easier to remember.

What, you may ask, do the stars mean? Well, since an average crib is worth just under five points, you can see that the four star toss (5-5) is worth roughly four points more than that. Likewise, the two star tosses are worth two points more than an average crib, and the one star tosses are worth one point more than an average crib. If you have the opportunity to toss 5-5 to your crib, it is worth sacrificing up to four points from the value of your hand to do so. Likewise, it is worth sacrificing up to two points from your hand to make a two star toss, and one point to make a one star toss. Other tosses do not justify sacrificing even a single point.

For example, holding 5-5-J-Q-K-K, you have four possible tosses:

Keep:   Toss:   Hand value
(without starter):
  Crib value
(without starter
or pone's toss):
5-5-K-K J-Q 12 0
J-Q-K-K 5-5 8 2
5-J-Q-K 5-K 9 2
5-5-J-Q K-K 10 2

Tossing J-Q keeps the most points in your hand, but tossing 5-5 is a legitimate option, since 5-5 is a four star toss. Tossing 5-K is not advisable, however, since the three point reduction in the value of your hand is not justified by the two star toss. Tossing K-K is similarly not worth the two point sacrifice to your hand value.

Of the four tosses, the 5-5 will earn you the most points on average between your hand and crib, and in most cases, this will be the correct play. However, if your board position requires you to hold down pone's pegging, you'll have an easier time defending with 5-5-J-J than with four ten-cards.

Another example is 2-3-5-8-9-K. Tossing 8-9 keeps the most points in your hand. But a more productive play is to toss 2-3, a discard that justifies a two point sacrifice to the value of your hand.

The addition method

Knowing the relative value of these discards can be helpful if you have two or more ways to play a particular six-card hand. A quick and dirty way to evaluate a possible toss is to add the points remaining in your hand to the average value of the contemplated discard. For example, holding 2-3-5-8-K-K, you could toss 2-3 or 5-8, either way keeping six points in your hand. But the 2-3 toss (a two-star) returns returns about seven points in your crib, which added to the six points in the hand makes thirteen. The 5-8 toss (a one-star) only returns about 5 points in the crib, making 11 points total. Clearly the 2-3 is the more productive toss.

Or consider 5-6-7-8-8-8. Keeping the double run (6-7-8-8) is the most obvious play, but the resulting 5-8 toss is inferior to the 5-6 toss you make keeping 7-8-8-8. Either way you'll keep twelve points in your hand, so toss the 5-6 to maximize your offensive potential.

Likewise, holding A-A-5-6-6-8, you could keep A-A-6-8 or A-6-6-8, retaining six points either way. A-6-6-8 looks tempting, because of the fourteen points you'll get on a 7 cut. But the right play is to keep A-A-6-8 instead. This puts a 5-6 in your crib, worth a full point more than the A-5 you'd toss if you kept A-6-6-8. In the long run, you'll get the most total points this way.

Of course, in any discarding decision you must consider the pegging implications of your toss. But if your goal is to maximize the combined value of your hand and crib, this simple addition method will work with most hands, provided you have a reasonably good grasp of the averages. And you can obtain one by studying the above tables.

It's worth knowing that the occasions where you can't rely on this method usually involve 2-3 or 5-5 combos. The classic example is 2-2-3-3-6-6, where tossing 6-6 works out best, even though it keeps four points less than tossing 2-2. This is because the 2-2-3-3 hand is so powerful, getting sixteen points on an A or 4 cut, and twelve points on any ten-card cut. Another example is 5-5-6-7-8-K, where the addition method suggests tossing 5-5, but where the most productive move turns out to be tossing 5-K instead. This counterintuitive play — which breaks up both the pair and the double run — outscores all the alternatives, including tossing 5-5, 7-8 and 8-K.

If you find the addition method too challenging, you might try this "shorthand" variant. It's cruder, but requires less memorization, and is based on Rasmussen's star system. Take the value of the four cards remaining in your hand, then add four points if the toss is 5-5, two points if it's a two star toss, one point if it's a one star toss, and zero points if it's any other toss (even something like K-K that's worth points going in). Though simple, the shorthand method will still guide you to the correct toss most of the time.

In a future article, we'll examine a more precise method for evaluating discards. Next time, however, we'll turn our attention to discarding to your opponent's crib.

- Originally written February 2000. Last updated April 2002


 
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