## Saturday, February 26, 2011

### Cribbage

Cribbage is a fun card game. In it, you're dealt six cards, and you have to discard two. Afterwards, a starter card is drawn. The four cards you're left with and the starter card determine how many points your hand has. (There's of course more to the game than just that, but that's what's relevant.)

The scoring works like this:
His nobs - If you have a jack of the same suit as the starter card, that's 1 point. (If that starter card is a jack, that's something different and gets counted at a different time.)
Flush - If your hand has all cards the same suit, that's 4 points. If the starter card is the same suit too, that's another 1. After this, there's no difference between the starter card and the ones in your hand.
Pairs - For each pair you have, you get two points. Note that cards can be counted multiple times, so if you have a 4H, 4D, 4S, that's three pairs (4H, 4D), (4H, 4S) and (4D, 4S) for 6 points.
Straights - If you have three or more cards in a row (like 3, 4, 5) you get as many points as the straight is long. Personally, this seems somewhat inconsistent, since a straight of four is also two straights of three, which would get you 6 points instead of 4.
Fifteens - For each sum of fifteen you can make, you get 2 points. All face cards count as ten. Cards can again be counted multiple times, so if you have two tens, and two fives, that's four fifteens for 8 points.

Anyway, I was curious as to what the average hand value would be if you always discarded so as to give your hand as many points as possible (which you don't actually always want to do, because of other parts of the game, but oh well). So I made a program which goes through all 20,358,520 possible hands of six you can be dealt, and finds the best two to discard, taking into account the different possible starters cards.

All told, it ended up running for about 18 days. But it did finish. And here are the results.

The average hand value is 8.29 points.

The most frequently discarded rank is king at 11.56% of all discards.
After that is queen at 9.85%
8 at 9.30%
7 at 8.85%
Ace at 8.78%
9 at 8.37%
2 at 8.27%
10 at 8.02%
3 at 6.72%
6 at 6.55%
Jack at 6.14%
4 at 5.39%
5 at 2.17%