Suit Breaks, Vacant Spaces, and the Numbers Behind Play Decisions

How the outstanding cards split

Before a card is played, the probability that the missing cards divide in a particular way depends only on how many are missing. These are the numbers every declarer should know cold:

Two rows deserve to be memorised as a pair, because they decide contracts. Missing five cards, the three-two break comes home just over two thirds of the time — this is the number behind almost every "draw trumps and claim" line. Missing four, the even two-two break is the minority case at 40.70%, less likely than three-one.

Why an even number of missing cards breaks badly

The rule of thumb is that an even number of missing cards tends to split unevenly, and an odd number splits as evenly as it can. It sounds like folklore, and it is straightforward counting.

Splits are decided by counting arrangements of individual cards, not by counting patterns. With four cards missing there are six ways to give two specific cards to West and two to East, but eight ways to produce a three-one split once both defenders are counted — four with West long and four with East long. The weighting by the space left in each hand nudges the balanced case up a little, but not enough to overturn the count: three-one finishes at 49.74% against 40.70% for two-two.

The practical version: missing four cards to the queen, do not plan on the drop. Missing five, plan on the three-two and have a fallback for the four-one.

All of these figures assume you know nothing about the defenders' hands. Once the bidding or the early play says otherwise, they stop being the right numbers — and the shift is often large enough to reverse a decision.

Vacant spaces: updating as evidence arrives

By the time you face the critical decision you usually know something: a preempt has shown seven cards in a suit, or three rounds of a side suit have been played out. The tool for updating is the vacant-spaces count.

Count the cards still unaccounted for in each defender's hand. If West has shown up with seven spades and East with two, then once those cards are accounted for West has six unknown cards left and East has eleven. A missing queen in another suit is therefore roughly eleven-to-six to sit with East — not even money, and a finesse that looked like a coin flip is a substantial favourite in one direction.

This is why good declarers cash side suits before taking a critical finesse. Every round played reveals more about the shape behind the decision, and the finesse is not going anywhere. The cost of the delay is the risk of a ruff; the benefit is a better-informed guess.

The same reasoning applies to the auction. A defender who opened a weak two has already told you about six cards, and applying the a priori table after that is simply using the wrong table.

Eight ever, nine never — and what the numbers say

The best-known maxim in card play says that with eight cards missing the queen you should always finesse, and with nine you should play for the drop. Like most maxims it compresses a real calculation, and it is worth seeing what the calculation gives. The reference tables on this site model the standard holding — A K J T and small opposite small cards — and report the all-win probability of the best line:

The maxim survives, but the margins are far narrower than its confident phrasing suggests. The recommended nine-card line is worth 57.91%, and the eight-card finesse 52.83% — real edges, but small enough that a single inference from the bidding can overturn them. "Nine never" is a default for when you know nothing, not a law.

If a defender has shown length elsewhere, the vacant-spaces count almost always outweighs the few percentage points separating these lines. Evidence beats maxims.

Restricted choice

One inference deserves singling out because it is genuinely counter-intuitive. Missing the queen and the jack, if a defender drops one of them on the first round, the odds favour finessing against the other rather than playing for the drop.

The reason is that a defender holding both honours could have played either one, while a defender holding only that card had no choice. The card you saw is therefore about twice as likely to have come from the singleton holding, and the finesse becomes the percentage play. The principle generalises: when a defender makes a play he could have made in more than one way, the fact that he chose that particular way is evidence about what he holds.

Combining chances

A single percentage is rarely the end of the analysis, because most contracts have more than one route home. The skill is ordering the chances so that a failure of the first still leaves the second available.

• Test the suit that can be abandoned. If a three-three break would solve everything and a finesse is the fallback, play the suit you can still recover from first.
• Count the tricks you need, not the tricks available. At teams, a line that brings home the contract 90% of the time and the overtrick 60% beats one that delivers both 75% of the time.
• Prefer the line that survives the bad break. Where two lines are close, the one that still delivers on a four-one is usually the better practical choice, because bad breaks are where contracts die.

From table to conditional probability

A printed table can only answer the unconditional question. Once you have information — a preempt, a shape shown by a raise, three rounds of a side suit — the number you need is conditional, and it is in no book, because it depends on exactly what you know.

That is the gap the Probability Solver fills: you state the constraints you have actually established, and it computes the split probabilities that follow from them rather than the ones that would apply to an unknown deal.

Frequently asked questions

Is the three-two break really 68%?

67.83%, before any evidence. That figure assumes you know nothing about the defenders' hands; after an opposing preempt or a revealing auction it can move a long way in either direction.

Why is two-two less likely than three-one when only four cards are out?

Because there are more ways for the cards to fall three-one than two-two once both defenders are counted. Splits are decided by counting arrangements of individual cards, not patterns, and the count favours the uneven case: 49.74% against 40.70%.

Does the vacant-spaces calculation apply to honours as well as shape?

It applies to the location of any specific unseen card, honours included. It is a good approximation rather than an exact posterior, because the bidding carries information about strength as well as shape, and vacant spaces only track the shape half.

When should I abandon "nine never"?

As soon as you have evidence. The line is worth a few percentage points over the alternative on an unknown deal, and a defender who has shown six cards in another suit shifts the odds by considerably more than that.