Odds of being dealt particular melds in pinochle

The odds of receiving particular hands in poker are well known; a quick search reveals many pages explaining them (such as Wikipedia's page on the topic). For Pinochle, I was never able to find a similar listing of probabilities for the various melds. There are, of course, incredibly many questions we might be interested in, and we probably can't answer them all. Some of them, though, are natural and obvious to ask. What are the odds that I will be dealt a straight on this hand? What are my odds of being dealt four aces? What are my odds of having no meld at all? The purpose of this page is to answer a few questions like this.

The following probabilities were obtained using a program I wrote. The code and instructions for using it are given at the bottom of this page. A lot of the numbers can be justified with brief math as well; if there is any interest, I will add the explanations when I get the time.

This chart assumes the standard 48-card deck and 3 or 4 players, with all the cards being dealt at the beginning of the hand.

meld odds with 16 cards
(3 players)
odds with 12 cards
(4 players)
straight (A-10-K-Q-J in a suit) 15.91% 3.83%
two straights in same suit 0.00049% 0.0000040%
marriage (K-Q in a suit) 79.94% 58.91%
pinochle (J-♠Q) 30.54% 18.57%
double pinochle 0.94% 0.25%
4 aces (of different suits) 8.25% 2.76%
all 8 aces 0.0034% 0.00013%
exactly 7 aces 0.097% 0.0076%
no aces 2.79% 8.02%
roundhouse (marriage in each suit) 0.36% 0.022%
no meld (except possibly 9's) 11.19% 31.34%

WARNING: Don't try to combine rows of this chart for combined melds (like having 4 aces and a straight), because the probabilities here are not enough information to find the answers. If you are fearless, you can try to edit my program to give you the answer. Or, you can just ask me; I'll try my best. It can't handle questions about more than one hand, such as the probability of two people getting a straight in the same round. The only questions it can answer are questions of the form "What are the odds that a randomly dealt hand of X cards will contain a meld of type Y?"

Note that the odds for aces also apply to kings/queens/jacks — for example, the odds of being dealt 4 kings in a 16-card hand are 8.25%.

A roundhouse is a holding of a marriage in each suit. It scores meld for 4 kings, 4 queens, 1 trump marriage and 3 non-trump marriages.

The last line of the chart excludes nines from consideration because the value of a nine depends on the trump suit.

This C program was used to generate the above numbers. Compile it (under Unix, you can do gcc -O3 pin.c -o pin) and then run it with the commands ./pin 16 or ./pin 12 (the parameter is the hand size). It uses some interesting techniques, but for now this brief blurb will have to suffice. On my three-year-old machine, ./pin 16 runs in about 12 minutes; ./pin 12 finishes in a minute. It should work for any hand size from 0 to 48, but the running time may be too long for your patience. If you don't feel like running the program yourself, you can view the results for 16 cards and 12 cards here.

I release the program under the terms of the GNU General Public License version 3. I welcome comments, questions, or corrections: you can find my email address here.

Copyright Tom Plick 2011