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How the probabilities change with new information |
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Probability theory is a closed book for many players but to succeed at bridge a basic understanding is required. Most players know that with four cards outstanding including the queen it is right to play for the drop. In fact the percentage advantage is not much – less than one success in every twenty hands. Any good player would prefer to back their judgement based on lots of other clues. There is the opening lead for a start. If it was clearly a safe lead maybe the defender has no safe lead outside that suit. If it was a risky lead a similar deduction can be made. If your judgement is right once in 20 hands and neutral on the remaining 19 (i.e. you gain as much as you lose) then finessing is OK. But I know few bridge players who will admit to being that poor in judgement. So any indication is better than the mantra “eight ever, nine never”, surely one of the stupidest ever coined. But this paper is
about the other side of the coin – how much does additional
information change the odds?
To understand what follows it is important to know about the
concept of “vacant spaces”.
Nothing to do with the When a hand is dealt the opponents have 13 unknown cards each. The first information comes with the bidding. If either opponent bids that indicates something about their distribution and point count. Even a failure to bid can give information. As an example let’s assume that West overcalled in spades and East supported her. West’s spade suit can be assumed to be five and East’s three. So West has eight vacant spaces outside spades (13-5) and East has ten (13-3). If West opened 1NT you can assume that she had no singletons or voids. Also the chance of a six card suit is very low. But you need to be cautious about the negative inferences you draw. The key factor I introduce in this article is the difference between the vacant spaces in West’s hand and those in East’s. If you have no information the result is zero. In the example above it is -2, i.e. West has two fewer than East. Mathematicians like to use Latin to show how erudite they are. The term we use is “a priori”. This identifies the odds before anything is known. So the a priori odds of dropping the queen are 52.17% and the odds of finessing are 47.83%. Actually that is rubbish; the true odds are 20.34% and 18.65%. There are lots of other possibilities besides. Someone can show out or the queen could drop on the first round. Since in these cases there is no guess we eliminate the relevant probabilities and then consider only the cases where there is a guess. There are three positions of interest: Missing Kx in a suit The relative “a priori” odds of dropping the king (after West has followed with a small card he cannot be void nor can he hold the king alone) are 52% and finessing is 48%. But if the difference between West’s and East’s vacant spaces is not zero these odds change. If West has fewer than East (i.e. more cards are known in his hand) the odds improve for playing for the drop. But if West has more vacant spaces than East the odds move the other way. Not surprising. But how much? In fact if West has one more vacant space than East the odds for the finesse and the drop are equal. And if the difference is greater it will pay to finesse. Missing Qxxx in
a suit Here the odds are 52.17% for dropping the queen and 47.83% for finessing. If West has one more vacant space than East there odds move to 50% each. And any further difference will make the finesse a good bet. Missing Jxxxxx
in a suit. The typical layout here is something like: ªA 10 3 2 ªK Q 4 After ªK and ªQ have been played and nothing untoward has happened ª4 is played and West plays the last small spade. The finesse is 47.62% while the drop is 52.38%. Once again, if West has one more vacant space the odds become 50% for both. So a rule can be adduced: If there is a possible finesse or drop guess in a suit in one of the three positions (i.e. missing Kx, Qxxx or Jxxxxx) and West (the player through whom the finesse could be taken) has one more vacant space than East you should back your judgement based on any extraneous information. And if he has more than one you should finesse unless there is extraneous information. But please don’t blame me when it goes wrong. You will be playing “against the room” if the other players follow rules blindly so will get a “top or zero”. And it does not take into account any factors such as if you have been gifted a trick you will not want to go down by taking a losing finesse. Happy finessing. |