Calypso in Numbers

One of the unusual features about the card game Calypso is that it is played with a 208-card quadruple deck of cards — a very unusual feature, especially for a trick-taking game. Relatedly, it is also somewhat unusual to play the game out in four deals, where each subsequent deal uses the remainder of the deck. Consequently, any considerations about probability will be rather different to similar questions in other games (such as Contract Bridge).

This page contains various bits-and-pieces related to probabilities, statistics, and any other numerical aspect of the game.

A note on probabilities

When looking at values of probabilities, it can be useful to to have some rough rules to use as a guideline, particularly when it comes to rare probabilities. Let's be specific to probabilities about hands of cards, i.e. 'what is the probability of getting a hand of cards of such-and-such a type?'. Then we can make a fairly conservative estimate about the number of hands you will be dealt in your lifetime. Let's say you are dealt 250 hands a week (over 35 a day!), 50 weeks a year, for 80 years. It's possible the hungriest card-players amongst you may exceed this, but such exceptions will be very rare. This gives a total of a million (\(250\times50\times80 = 10^6\)) lifetime hands. Therefore we can reasonably take the reciprocal of this (\(10^{-6} = 0.000001\)) as an upper bound for hands which are practically impossible. In other words, any events more rare than this, you can pretty safely assume you will never see personally.

Of course this doesn't apply when it comes to looking at the greater scale of many people playing. For instance looking over hands played online/in tournaments may lead to counts well in excess of a million (not in Calypso of course, but say for Contract Bridge), in which case one shouldn't be particularly surprised to see rarer events occurring, in which case these lower-probability values become relevant again.

For Calypso, when talking about probabilities for 13-card hands, the discussion really only applies for the first hand of the game. After that we have some information about the cards that have already been out, which changes the possible distributions of future deals. Using this information in any useful way would also require players remembering relevant details of the cards that have been used up already. For now, I am going to keep that can of worms closed, and focus on the initial hand probabilities, which can still serve as a rough guide for the further hands.

Headline numbers

There is a naïve count of 149,389,005,978,091,284,720 hands of size 13 from a deck of 208 (about 149 quintillion). This is just \(208 \choose 13\). However, this does not account for the fact that we have many duplicate cards, which are indistinguishable in our hand, and thus overcounts the true number.

The actual count of distinct hands is much lower, at 13,021,583,189,580 (around 20 times the 635,013,559,600 one has for single-deck hands) — just a shade over thirteen trillion. However each of these hands is not equally likely - roughly speaking hands with more duplicates are less likely than those with fewer. To each hand we can associate a weight, which is proportional to the relative probability of each hand being dealt from a perfectly shuffled deck. The sum of these weights is simply our naïve count above.

Calypso suit distribution probabilities

In Calypso, as in many other trick-taking games, one of the key considerations that governs your fortune in a hand is the distribution of suits in your hand. Do you have a similar number of cards in each suit, or lots of cards in a single suit, or in two suits? As with all questions of probabilities, it is important to make the question we are asking precise, or else we can easily get into a complete muddle. So as a start, we can simply ask what the probability we have a given distribution of suits, such as '5 in one suit, 4 in another, 3 in another, and 1 in the remaining suit'. Initially we will just look at these probabilities ignoring the status of each suit (so counting all hands of that distribution the same, regardless of whether the 5-card suit is your trump suit, or your left-hand-opponent's, etc.)

The following table gives the approximate probabilities of being dealt a given suit distribution from a quadruple deck, arranged in descending probability, along with cumulative values. For comparison, the corresponding single-deck values are also given. Distributions with less than half a percent probability are not shown. See below for exact values, other deck values, and more detail.

The rough pattern that one can observe is that 'more extreme' distributions become slightly more likely, with flatter distributions being consequently a bit less likely. The ranking of distributions is similar, but one can see that the ordering is, in a few cases, slightly different than the single-deck case (i.e. when single-deck probabilities increase when reading down the column).

We can see this a bit more directly by looking at just the longest suit. In this table we see the probability of different lengths of longest suit (only those with probability more than \(0.001\)):

Here we specifically see that the quad-deck means we are less likely (than with a single deck) to have a four- or five-card longest suit (by nearly 10%), and more likely to have six or more cards of the suit.