Recent discussions with friends have revealed that many club players do not know about the ‘Law of Vacant Places’. This is a useful method of assessing the probability of where a particular card lies when you have clues as to the suit lengths in the various hands. If someone pre-empts at the three level, for example, they probably have seven cards in the suit. If they overcall an unusual two no trumps then they probably have ten or more cards in the two lowest unbids suits and so on. This leaves much less space in their hand for the cards from the other suits or fewer ‘vacant places’ as this law calls them. Counting the vacant places in a hand is the key to determining probability of finding a particular card there.
Let’s look at a specific example: As North you hold: S AJ, H AQJT92, D K32, C J7 and you are playing in four hearts after opening one heart, East overcalling two no trumps and partner bidding three hearts which you raise to four. East leads the king of clubs, indicating that they hold the queen, and partner puts down: S 75, H 87543, D A654, C A4
At first sight your contract appears to depend upon the location of the king of hearts since you have a spade, a club and a diamond to lose. You win with the ace of clubs, lead a heart and West plays the six. What is the chance of the finess winning?
You may think that the probability of the finesse succeeding is 50% since the king could be either side. This takes no account of East’s two no trumps overcall, however. Lets count the vacant places assuming East is five – five in the minors. Here’s what we know with x being a single card and ? an unknown number of cards:
AJ AQJT92 K32 J7 ? ? 6? ? x xxxxx xxxx KQxxx 75 87543 A654 A4
With ten known cards, East has only 3 vacant places i.e. cards which as far as you know could be the king of hearts. Looking at our own hands we can deduce that West has four clubs and one diamond. We have also seen the six of hearts so we know six cards that cannot be the king of hearts and hence West has seven vacant places giving a probability of 7/(7+3) or 70% that the king of hearts is with West and the finesse will succeed. If East is 6:5 in the minors then he has one less vacant space and West has one more making it even more likely that the finesse will succeed (8/(8+2) or 80%).
This hand was quickly constructed to give a simple example of how to use the Law of Vacant Spaces to calculate the probability of a finesse succeeding. If you analyse the hand more carefully, however, you will see that you should not take the finesse even though it has a high probability of success! If West holds the king of hearts then you can make your contract by an elimination play. You lay down the ace of hearts, hopefully dropping the king, but if the king does not drop you eliminate clubs and spades from your hands, remove West’s presumed singleton diamond and can throw him in with the king of hearts when he has to return a black suit giving you a ruff and discard. By playing the Ace of hearts, you also win when East has the singleton king of hearts.
You must be careful to ensure that West does not get the lead until his diamond exit card has been removed and you have eliminated the black suits from your two hands. Luckily, leading the jack of clubs from dummy will force East to win with his presumed queen (he led the king from at least five clubs). You win a diamond return in the South hand, lead a spade towards the ace and return the jack. If West wins the trick he is thrown in. If East wins then he must return a diamond to avoid giving you a ruff and discard. West can ruff but he is thrown in and must still give you the all-important ruff and discard. If East returns a spade after taking his queen of clubs you win with the ace, take a diamond trick and play a spade.
The moral of this story is always look for a better line of play than taking every finesse in sight. Also note how much information a two suited overcall gives to your opponents!
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