Source:  Mr. Bridge  Length Attracts Shortage: The Principle of Vacant Spaces Other things being equal, there is a 36% chance that a suit will break 3-3, a 28% chance of a 4-1 break (although it feels Like more), a 50% chance that a simple finesse will succeed (although it sometimes feels like less) and so on. The trick is to know which ‘other things’ may not be ‘equal’. Truth be told, they very rarely are.
A 9 8 2 A 6 5 9 6 3 A 4 2
K J 10 7 5 K 8 A 7 4 K 8 5
You are in 4, after an uncontested auction, on a club lead from West. You are booked to lose two diamonds and a club and so to succeed, you will need to play the spade suit without loss. Plonking down the ace-king is clearly best. You will succeed whenever the spades are 2-2 and any time the queen is singleton. (You will also succeed if the spades are 4-0, so long as you guess which hand to start from.) Of course, circumstances alter cases. Suppose that you hold the same cards but East has opened with a pre-emptive bid of 4. Does this change how you should play the spade suit? You bet it does! Now the best play is to cash the king of spades, intending to finesse West for the queen. The reason for doing so is this: once you know East has a very long heart suit, he rates to be shorter than West in the remaining suits. Think of it like this – East probably has eight hearts, and thus five ‘nonhearts’.
Justin Corfield
Justin Corfield
West probably has no hearts, and so he has thirteen ‘non-hearts’. The queen of spades, being a spade, is a ‘non-heart’. This makes West (who has thirteen such cards) more likely to have it than East (who has only five). For the first half of the maxim – length attracts shortage – this is as far as we need to go. For the second half – the principle of vacant places – we need to look more closely, counting the cards each defender has played. Suppose you take the king of clubs, cash the king of spades (all follow), and continue with the jack, on which West plays the remaining low spade. What are the odds if you finesse? Well … we have seen two of East’s five non-hearts (a club and a spade) leaving him with three ‘vacant spaces’, or unknown cards. We have seen three of  West’s thirteen non-hearts (a club and two spades), leaving him with ten. So, at the key moment, the finesse is a ten to three favourite – a 77% chance. That, in a nutshell, is the principle of vacant spaces. (Some would ignore the clubs in the calculation as West had to lead something and East had to follow suit, but this is a small point.) Sooner or later, you will have to tackle a hand like this next one. Partner puts you into 7, and this is what you see:
A 2 K 10 K 10 9 3 A Q 6 5 4
K 9 A Q J A J 6 K J 10 3 2
The duplication in the majors is a trifle unlucky. Had you known, you would have stopped in 6NT. Meanwhile, West leads a heart and it is up to you to find the queen of diamonds. At this point, you have a simple 50-50 guess as to where she is. Can we improve on this? Maybe we can. Suppose you draw trumps, East turning up with all three of them. The odds on the diamond finesse have now changed. East has ten ‘ non-clubs’ to West’s thirteen, making it a thirteen to ten favourite, or 57%, to play West for the diamond queen. Most finesses start life as 50-50 propositions, but very few of them stay that way. To pass the time, you decide to cash your remaining spade and heart winners. East surprises you at this point by discarding a spade on the third round of hearts. Now we know that East began with three clubs, two hearts and two or more spades, leaving him with six vacant spaces, whilst West began with no clubs, six hearts and two or more spades, leaving him with five vacant spaces. The odds in the diamond suit have changed again and East is now a six to five favourite, or 55%, to hold the queen of diamonds. Crossing to the king of diamonds and running the ten is the best play. Conclusion The odds that a given thing will happen are not set in stone – on the contrary, during the play of most hands, declarer must confront a changing picture. The percentages I gave at the start of this article are perfectly correct. They are the chances that a given thing will happen assuming we know nothing about the concealed hands. Once infonnation starts coming in, those odds can change and often do.