Preferans deals
2,753,294,408,504,640
distinct deals, and every one of them is equally likely.
Preferans deals: the probabilities worked out
32 cards, three hands of 10 and a talon of two: the number of distinct deals is 2,753,294,408,504,640. With an honest shuffle every one of them is as likely as any other. From that single fact and an exact reading of hand strength you can work out how rare a clean nine, a clean misère and the other things people describe as "roughly once in a thousand games" actually are.
How many deals there are
A preferans deal is 32 cards split at random into three hands of 10 and a talon of 2. The number of ways to do that is 32! / (10! · 10! · 10! · 2!) = 2,753,294,408,504,640. With an honest shuffle each one is equally likely. Starting from that fact, we can calculate the probability of any particular distribution exactly instead of estimating it by eye.
Which cards go to whom
Before you pick up your cards, any particular card, say the ace of spades, has exactly 10 chances out of 32 of landing with you: 10/32 = 31.25%. Its chance of being in the talon is 2/32 = 6.25%. All three of you and the talon are completely equal in this respect. The deal does not favor anyone.
Probability of a clean game and a clean misère
Let us measure the probability of a clean game, one where the hand by itself guarantees the required number of tricks, without the talon. The values below come from 1,000,000 randomly generated deals (3,000,000 hands), with the weight of each suit evaluated by alpha-beta pruning (more on that below):
| Contract | Probability |
|---|---|
| 10 (the whole game on your own hand) | 0.0074% |
| 9 | 0.0890% |
| 8 | 0.5440% |
| 7 | 2.2200% |
| 6 | 6.1840% |
The drop from a clean six to a clean ten is a factor of about 1200 (9.04% against 0.0074%). Each additional guaranteed trick demands an ever more specific concentration of high cards in one or two suits, and among 2.75 quadrillion arrangements there are fewer and fewer of those. The fall is not linear but sharply accelerating.
A clean misère, a hand already safe for misère before the talon, comes up even less often than a clean nine: 402 times in 3,000,000 hands, or 0.0134%, roughly once in 7,463 deals. Where misère sits in this order is not obvious: it is rarer than a clean nine but more common than a perfect clean ten.
What if you add the talon?
All the numbers above are what the hand guarantees on its own, before the two talon cards. The talon can at worst leave the position unchanged, and it often improves it noticeably. Which talon card actually helps, and why it is not necessarily an ace, is covered in the article on what to expect from the talon.
Special deals
Besides the ordinary honest deal, SmartPref can deliberately skew the distribution described above. This is used in seasonal events, in tutorial scenarios, and simply for practice or fun. Ordinary rated games always use standard distributions:
| Type | What it does |
|---|---|
| Oblique | every player is guaranteed at least 5 cards of one suit, giving long, concentrated suits instead of the usual spread. The old Marriage program dealt the same way, if you remember it. |
| Skewed | each player is missing a suit that the other two hold. In other words, three of the four suits go to only two players. This was in Marriage as well. |
| Equalized | every player is guaranteed at least 2 cards of each suit, so there are no void suits and no long ones. Good for practicing ordinary contracts without extremes. |
| Average quality | players get hands of middling strength, with each hand weighted at no more than 5 tricks. Ideal for practicing hand evaluation and bidding. |
| Low quality | luck has abandoned everyone, with each hand weighted at no more than 2.5 tricks. Well suited to practicing all-pass. |
The contents of the talon can be fixed in the same way: two aces, two sevens, or a mariage (king and queen of one suit, chosen at random in advance) instead of the usual two random cards.
Generate your own deal
Pick a deal type, a talon type and a random number generator. The button below will generate a sample deal and recalculate the table of clean six through ten and misère right in your browser, for the settings you chose.
Deals and probabilities are recalculated right in your browser, by the same algorithms the SmartPref Pro app uses.
Summing up
- A deal is one arrangement out of 2,753,294,408,504,640 equally likely ones, not a vague piece of chance. Every probability in this article can be calculated exactly rather than guessed at.
- A clean six turns up in about 1 deal out of 11 (9.04%), but a clean nine or better in fewer than 1 in 1000 (0.096%), and a clean ten roughly once in 13,500 deals.
- A clean misère before the talon is rarer still than a clean nine: roughly once in 7,463 deals (0.0134%).
- All of these numbers are before the talon. What happens next is covered in the talon article.
- SmartPref can deliberately skew the deal in six ways, for seasons and tutorials, and offers two different random number generators under the hood. Try them in the generator above.
