Vacant Places

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In the card game bridge, the law or principle of vacant places is a simple method for estimating the probable location of any particular card in the two unseen hands. It can be used both to aid in a decision at the table and to derive the entire suit division probability table.

At the beginning of a deal, each of four hands comprises thirteen cards and one may say there are thirteen vacant places in each hand. The probability that a particular card lies in a particular hand is one-quarter, or 13/52, the proportion of vacant places in that hand. From the perspective of a player who sees one hand, the probable lie of a missing card in a particular one of the other hands is one-third. In Contract bridge, once the play commences, the dummy is exposed and so, for any player, there are only two unseen hands where a card may lie.

The principle of vacant places is a rule for updating those uniform probabilities as one learns about the deal during the auction and the play. Essentially, as the lie of some cards becomes known – especially as the entire distributions of some suits become known – the odds on location of any other particular card remain proportional to the dwindling numbers of unidentified cards in all hands, i.e. to the numbers of so-called vacant places.

The principle of vacant places follows from Conditional Probability theory, which is based on Bayes Theorem. For a good background to bridge probabilities, and vacant places in particular, see Kelsey;^[1] see also the Official Encyclopedia of Bridge.^[2]