Discarding

In our last post we said that scoring an extra point per game will add an extra 1.7% on top of your win rate. Today we’re going to talk about how to discard to score that extra point. Discarding is such a ridiculously exciting topic that we’re not even going to give an overview. Let’s just jump right into the data!

Dealer discard table

This table shows the average number of points you can expect in your crib, based on the two cards you put in your crib.

	A	2	3	4	5	6	7	8	9	10	J	Q	K
A	5.2	4.4	4.6	5.2	5.2	3.7	3.7	3.7	3.3	3.3	3.5	3.3	3.3
2	4.4	5.8	6.9	4.6	5.2	3.9	3.9	3.7	3.7	3.6	3.8	3.6	3.6
3	4.6	6.9	5.9	5.0	5.9	3.8	3.8	3.9	3.7	3.6	3.9	3.7	3.7
4	5.2	4.6	5.0	5.5	6.3	3.9	3.7	3.9	3.6	3.4	3.7	3.5	3.5
5	5.2	5.2	5.9	6.3	8.5	6.4	5.8	5.3	5.1	6.3	6.7	6.4	6.3
6	3.7	3.9	3.8	3.9	6.4	5.6	4.9	4.6	4.9	3.0	3.2	3.0	2.9
7	3.7	3.9	3.8	3.7	5.8	4.9	5.8	6.4	4.0	3.1	3.3	3.1	3.1
8	3.7	3.7	3.9	3.9	5.3	4.6	6.4	5.3	4.5	3.7	3.3	3.1	3.0
9	3.3	3.7	3.7	3.6	5.1	4.9	4.0	4.5	4.9	4.1	3.7	2.8	2.8
10	3.3	3.6	3.6	3.4	6.3	3.0	3.1	3.7	4.1	4.6	4.3	3.3	2.7
J	3.5	3.8	3.9	3.7	6.7	3.2	3.3	3.3	3.7	4.3	5.1	4.5	3.8
Q	3.3	3.6	3.7	3.5	6.4	3.0	3.1	3.1	2.8	3.3	4.5	4.5	3.4
K	3.3	3.6	3.7	3.5	6.3	2.9	3.1	3.0	2.8	2.7	3.8	3.4	4.4

You can use this table when you’re having a tough time figuring out what to toss in your crib. For example, you’re holding 2-3-5-6-J-Q and it’s your deal. You probably want to give yourself either the 2-3 or the 5-6, but it’s not obvious which. You’ll end up with 4 points in your hand either way, so what should you put in your crib? Discard table to the rescue: if you toss the 2-3 in you’ll average 6.9 points in your crib; if you toss the 5-6 you’ll average 6.4 points. Keeping the 5-6 in your hand and tossing the 2-3 gives you a 0.5 point advantage in your crib. (5-6-J-Q is also a better pegging hand, but that’s a topic for a future article).

Michael Schell put together some good articles describing how to choose your discards using a discard table. I recommend using his technique, but with our discard tables. (We believe ours is more accurate, since it’s based on a very large sample of hands from actual human play, rather than a smaller sample or computer simulations).

Here’s a visual display showing how those discards stack up with each other. Pairs show up in blue, and cards totalling 15 show in red.

As expected, 5-5 is far and away the best discard. Also unsurprising: 10-K, 9-K, and 9-Q show up at the bottom of the rankings.

You’re probably not going to split up a pair or a 15

Hypothetical scenario: you deal yourself 3-3-6-7-J-J, and you’re not sure what you should put in your crib. Looking at the discard table you see that 3-3 will get you 5.9 points (on average), 6-7 will get you 4.9, and J-J will get you 5.1. It looks like 3-3 is the best discard, but the discard table is a little misleading here. You’ll get the “pair for 2” for the 3-3 or the J-J whether or not they’re in your crib, as long as you don’t split them, so you probably don’t want to consider those points when evaluating the discard. The entry in the table for the Jacks has even more built in points -- between the two Jacks you’ve got about a 50% chance of scoring “1 for his nobs”, for an extra 0.5 points or so on average.

If we ignore those extra points, let’s call them the “intrinsic points,” we see that 3-3 and J-J are actually poor discards, averaging 3.9 and 2.6 points respectively. The 6-7 is probably the right discard, beating out 3-3 by a full point.

Here are updated versions of the charts, with pairs in blue and cards totalling 15 in red. The left chart includes the intrinsic points in each average, while the right chart exclude those points. Note that Jacks have an intrinsic value of around 0.25 points, and we’ve accounted for that on the right chart, too.

Removing the intrinsic points changes everything. 5-5 is no longer the top of the heap. 2-3 beats it by almost half a point. And K-10 is no longer at the bottom, since it does better than pairs of 10s, Jacks, Queens or Kings!

Except for 5-5, the pairs look like pretty bad discards. It’s more important to throw yourself cards that add to five or could turn into a run.

Pone discard table

Here’s the corresponding table for pone, showing average number of points in dealer’s crib based on pone’s discard:

	A	2	3	4	5	6	7	8	9	10	J	Q	K
A	5.4	4.5	4.7	5.3	5.5	4.4	4.3	4.4	4.1	3.9	4.2	3.9	3.9
2	4.5	5.7	6.7	4.8	5.5	4.6	4.5	4.4	4.3	4.1	4.4	4.1	4.1
3	4.7	6.7	6.0	5.4	6.0	4.4	4.5	4.5	4.3	4.2	4.5	4.2	4.1
4	5.3	4.8	5.4	5.7	6.5	4.7	4.3	4.4	4.3	4.0	4.3	4.0	4.0
5	5.5	5.5	6.0	6.5	7.4	6.6	6.1	5.6	5.5	6.4	6.7	6.4	6.4
6	4.4	4.6	4.4	4.7	6.6	6.2	5.8	5.4	5.5	3.9	4.1	3.8	3.8
7	4.3	4.5	4.5	4.3	6.1	5.8	6.2	6.7	4.8	3.9	4.2	3.9	3.9
8	4.4	4.4	4.5	4.4	5.6	5.4	6.7	5.8	5.3	4.5	4.1	3.9	3.8
9	4.1	4.3	4.3	4.3	5.5	5.5	4.8	5.3	5.5	4.8	4.4	3.7	3.7
10	3.9	4.1	4.2	4.0	6.4	3.9	3.9	4.5	4.8	5.1	5.0	4.1	3.5
J	4.2	4.4	4.5	4.3	6.7	4.1	4.2	4.1	4.4	5.0	5.5	5.0	4.4
Q	3.9	4.1	4.2	4.0	6.4	3.8	3.9	3.9	3.7	4.1	5.0	5.0	4.0
K	3.9	4.1	4.1	4.0	6.4	3.8	3.9	3.8	3.7	3.5	4.4	4.0	4.8

Of course you should subtract the values in this table from your hand value when trying to decide what to toss into your opponent’s crib. As before, Michael Schell’s articles do a good job explaining the logic.

Here’s the visual layout of the average crib scores based on pone discard:

That’s it for this week

Want more details on these discard tables? Let us know and we’ll get it for you. Have a question about any other cribbage related topic? We want to answer it! Just leave us a comment.

Editor's Note: This is a guest post by Aaron Harsh continuing the series on cribbage strategy and tips. Aaron lives in Portland, Oregon with his wife Stacy and daughter Audrey. He spends his evenings analyzing cribbage strategy for Fuller Systems, and his days analyzing television viewership for Rentrak Corporation's Advanced Media & Information group. You can play him on Cribbage Pro Online as user "aaronhars", or in person at American Cribbage Congress grassroots club #28 (Oregon's Finest).

How Should You Respond to Pone’s Lead?

Today we’re going to talk about how dealer should respond to pone’s initial lead. As usual, all our figures are based on our database of millions of hands of actual cribbage play. None of this is guesswork on our part -- it’s all based on real games played by real players.

All our numbers are based on average dealer points pegged minus pone points pegged. So if dealer pegs 5 and pone pegs 3 we’ll consider that a net advantage of 2 points for the dealer. When we say something like “playing the 6 gives you an extra 1.3 advantage over playing the 3”, we mean “the difference between your score and your opponents will be, on average, 1.3 points larger if you play the 6 than if you play the 3”.

All the situations we describe below are relatively common. You won’t run into them every game, but you’ll see them often enough that playing them right will make a difference in your win rate.

These statistics don’t take into account situations where you want to play offensively or defensively. We’ll talk about offensive and defensive play in a future article, but for now you can use our recommendations as general guidelines for your play.

A few points makes a big difference

Before we get to the meat of the article, let’s talk about how big a difference a few points can make. Our data shows that 3.4% of games end up being decided by a single point. An average player winning 50% of her games will find herself stuck in the stink hole 50% * 3.4% = 1.7% of the time. If our average player fine tunes her strategy to squeeze out one more point per game she’d win those 1.7%.

Another 3.7% are decided by exactly two points -- the loser is stuck in peg 119 when the winner pegs out. That means you’re probably losing 50% * 3.7% ˜ 1.8% of your games by exactly two points.  If you could manage to score an extra 2 points per game you’d be winning about 1.7% + 1.8% = 3.5% more often. An extra 3.5% win margin is the difference between being good and being a great player.

Here’s a chart breaking down the difference between the winner’s final score (121), and the loser’s final score. These numbers only take into account games between two “A” level players.

In this article we’ll show common mistakes players make that cost them a few points per game. If you use them to flesh out your strategy you should see yourself scoring an extra point or two, and making a noticeable difference in your win rate.

Dealer holds 7-8-9-9, pone leads a 3

The right play here is a 9, but for some reason dealer’s are only playing it 46% of the time. On average when the dealer plays the 9 she ends up pegging 0.8 more points than pone. When dealer players the 8 she ends up scoring 0.2 points less than pone. And when she plays the 7 it’s even worse -- she pegs 0.4 points less than pone.

Why is the 9 the right play? Couldn’t pone have another 3 and make 15 for 2 points? Well, she could but that only happens 35% of the time. It’s almost as likely that she has a 4 (33%), so you’re not saving much there.

Why are so many people playing the 8 here? One theory is that they like to take the count to 11.  Pone will be nervous about playing a ten/face card to make the count 21, and might make a mistake. It’s not a bad idea, but the data shows you can score more by playing differently.

It’s more likely that she’s got a 2 and two face cards. When that happens you’re likely to score 31 for 2:

Pone	Dealer	Points
3
	9
10/J/Q/K
	9	+2 for dealer (31 for 2)
—
2
	7
10/J/Q/K
	6	+1 for dealer (go)

If dealer leads the 8 she’ll end up with two "GOs" for a total of 2 points, rather than a 31 and a go for a total of 3 points. So play your 9 on pone’s 3, and on average you’ll pick up an extra point. That’s a 1.7% increase in your likelihood of winning.

2-3-3-4, pone leads the 7

The 3 is the right play here, but most players are playing the 4 or the 2. Are they hoping to save their 3s for a pair later on? Are they playing the 4 to take the count to 11? Are they trying to avoid taking the count to 10 in case pone has a 5?  No matter what the reason, playing the 2 or the 4 doesn’t work.  On average the dealer pegs 2.6 more points than pone when she plays the 3. She only scores 1.3 more than pone when she plays the 4, and only 0.9 more when she plays the 2.

Go for the run if dealer plays a 5 for 15-2 on top of your 3.  In 32% of the cases where dealer plays the run she scores big:

Pone	Dealer	Points
7
	3
5		+2 for pone (15 for 2)
	4	+3 for dealer (run of three)
6		+5 for pone (run of five)
	2	+6 for dealer (run of six)
	3	+7 for dealer (run of six, and a go)
—
?		+1 for pone (go)

Dealer scores 3+6+7=16 points, to pone’s 2+5+1=8, for a net gain of 8 points.

Executive summary, play the 3 instead of the 4 and you’ll gain 2.6 points on your opponent instead of just 1.3.

Some other popular hands

You’re holding 4-5-6-9 and pone leads the 7. A lot of players are playing the 4 here, but the 9 will, on average, give you an additional 1.6 points advantage over your opponent.

Dealer’s holding 5-10-J-K and opponent leads a king. We’ve seen that 48% of players will pair the king. The 5 is the better play, giving you an extra 2.2 point advantage over the king.

Dealer has 4-5-6-J and opponent leads a 6. Most players go for the pair, but they shouldn’t. Play your jack instead. The jack gives you a massive 2.4 points extra pegging advantage over the 6.  (And please don’t play the 4 or the 5 instead -- they do even worse than the 6).

Last one for today: dealer holds 6,6,7,8 and opponent leads a 10. Play the 6 instead of the 7 for a 1.6 point increase in your advantage over pone.

Let us know what you think

Want to see the results for another hand? Want more details on why these plays work so much better? Leave us a comment below and we'll get back to you.

Editor's Note: This is a guest post by Aaron Harsh continuing the series on cribbage strategy and tips. Aaron lives in Portland, Oregon with his wife Stacy and daughter Audrey. He spends his evenings analyzing cribbage strategy for Fuller Systems, and his days analyzing television viewership for Rentrak Corporation's Advanced Media & Information group. You can play him on Cribbage Pro Online as user "aaronhars", or in person at American Cribbage Congress grassroots club #28 (Oregon's Finest).