Cribbage Hand Statistics - You Ask, We Answer

We’ve received a couple of statistics questions recently from Cribbage Pro players. We love getting these types of questions, and to show our appreciation we’re devoting this week’s entry to their answers.

First dealer’s advantage

Cribbage Pro player Zerrick wondered what the odds were that the player with first deal will win the game.  Across our entire sample we saw that the first dealer won the game 55.5% of the time.  Broken down by the players’ skill levels (using the skill levels from the first post in this series), we get:

Probability first dealer wins game

	First dealer’s skill level
First pone’s skill level	A B C D A 56.0% 53.3% 50.5% 44.8% B 58.0% 55.2% 51.2% 45.8% C 61.6% 59.4% 55.5% 53.7% D 66.8% 64.2% 57.7% 55.2%

It’s interesting to see that when the two players’ skill levels are equal the first dealer wins 55% or 56% of the time.  If you work through the math you see that no matter what skill levels are involved, a player is around 11% more likely to win if she deals first than if she deals second. [Editor's Note: This should be incentive as well for those "D" level players who are not seeing an advantage when facing the higher skilled "A" and "B" players and winning only 45% of the time even as first dealer. Start taking advantage of this and win more cribbage games by improving your cribbage strategy.]

Hand statistics

Last weekend a Cribbage Pro player named Kip beat me two out of three games, then stumped me with a cribbage statistics question: “How often do players hold double runs in their hands?” Good games, Kip. Here’s the answer to your question and a few more, with lots of colorful charts:

Type of hand	% Dealer	% Pone	% Dealer's Crib
Flush	3.0%	3.3%	0.9%
Single pair	50.6%	51.0%	46.7%
Two pair	9.7%	11.4%	6.6%
Pairs royale	5.7%	5.7%	3.7%
Full house	0.6%	0.7%	0.3%
Double pairs royale	0.2%	0.2%	0.1%
Single run	22.1%	21.4%	16.2%
Double run	16.7%	16.1%	6.6%
Triple run	0.7%	0.7%	0.2%
Double double run	1.7%	1.6%	0.4%

As usual, all the numbers in the blog post come from our sample of several million cribbage hands. There’s some overlap in these numbers – a hand could hold a flush and a run, or a pair and a run. Interesting to see that double double runs (for example, 4-4-5-5-6) are more common than triple runs (like 4-4-4-5-6).

What about “15”s, you say? Here’s a little chart for you:

# fifteens in hand

The bars for 5, 7, and 8 fifteens in a hand are too small to be visible, but it’s still possible to count 5, 7, or 8 fifteens in one hand. About one hand in 800 has 5 fifteens, one in 600 has 7 fifteens, and one in 15,000 has 8 fifteens.

Note that only 20% of dealer or pone hands have 0 fifteens, which means that most hands have at least one fifteen in them. Not only that, most hands actually have at least two 15s in them.

Points per hand

Put those all together, and how many points do you get? You get this!

# points in hand

Points per hand look fairly similar for dealer and pone.  The most common number of points in a hand is 8. Dealer’s hand holds 8 points 16.9% of the time, and pone’s holds 8 points 18.2% of the time.

Hands with an even number of points are more common than hands with an odd number of points. Looking at the first table explains why – most hands have pairs and 15s (which score even number of points), while only 23% of hands have single or triple runs (which can score an odd number of points).

It’s impossible to get a 19, 25, 26, or 27 point hand, so those entries don’t show up in the chart. 22, 23, 28, and 29 are possible, but show up so rarely that their bars aren’t visible. About one in 15,000 hands on Cribbage Pro scores 28 points, and about one in 220,000 scores 29 points.

The typical crib doesn’t have as many points as the typical hand, so this next chart looks a lot different. You can score more than 16 points in a crib, but we left those bars off because those scores happen so rarely that you wouldn’t be able to see them. Only about 1 in 1,000 cribs scores more than 16 points.

# points in crib

Pegging probabilities

The last chart in today’s post shows pegging probabilities:

# points pegged

Dealer always pegs at least one point, so dealer gets 0% in the “0 points pegged” column (we’ve excluded hands where pone pegs out before counting). The most points we’ve seen pegged in one hand is 29, and in the millions of hands analyzed for this blog, this has only happened once. In that hand dealer and pone each held two aces and two 2s. Ouch.

That’s all for now

Thanks for the questions, Kip and Zerrick. We know more of you have questions, and we’d love to answer them. Just leave us a comment below.

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).

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).