Tuesday, August 28, 2007

The Odds and Ends of Bridge (Part 1)

I. Chance and Probability Theory

The likelihood of the occurrence of any event in Bridge is called a probability. For example, probabilities can refer to the location of specific cards, high card points, or distribution of the suits among the four hands. Probabilities are not guarantees. On one deal or a single play of the cards, calculated probabilities may lead you to an aberrant result, but over the long run probabilities will prove out within a very small variance. The moral of this story is that if you want the probabilities and odds to even out, play more bridge. The usefulness of probabilities is not to determine whether an isolated line of play will work, since if you have only one line of play, it makes no difference what the probabilities are, it either works or it doesn’t! On the other hand, if you have two or more chances to make your contract, and you are able to choose among them, then it is helpful to know which chance is more likely to succeed.

II. Using Probabilities to Help You Find Specific Cards

The underlying math of bridge sees 52 individual cards and each of 4 players having 13 pockets where cards can be located. Each time you identify that “x” pockets of a defender are occupied by certain cards, that defender has fewer empty pockets. If you also know the number of cards in that suit held by the other defender, you can fill up some of his empty pockets. We are only concerned with the opponents empty pockets!

For example, if LHO opens 2 spades and you end up playing a contract in 4 hearts, you can reasonably assume that LHO has 6 of his pockets occupied by spades, so he has now has 7 empty pockets. If Declarer looks at her hand and sees 2 spades and also sees 3 spades in the dummy, she knows that RHO started with 2 spades, so now RHO has only 11 empty pockets. This empty pockets stuff is the backbone of all probability calculations in bridge, called the Law of Vacant Places.

How can we make this useful at the card table? Let’s look at the above example. Assume that the success of the contract depends on locating a specific card in a suit other than spades. Since RHO has 11 empty pockets and LHO has only 7 empty pockets, the odds are 11 to 7 (64%) that RHO will have the card you are looking for. This little calculation can be applied any time you have the distribution in any single suit completely identified between the two opponents.

An adjunct to this principle is called the Law of Attraction. Length in one suit attracts shortness in another suit. Conversely, shortness in one suit attracts length in another suit. In this context, the terms “length” and “shortness” speak only to the relative holdings of a defender in two different suits, predicting that when the defender has length one suit he will have shortness in another suit. This is better explained by example.

Again, looking at our first example, since LHO has length in spades (very likely 6), the probabilities are that RHO has greater length in any other suit. Assume declarer has AJx in diamonds and dummy has K10x in diamonds, a two way finesse. Is it simply a random guess as to who has the Queen of diamonds? On this deal, declarer should always finesse RHO for the diamond queen. Two reasons:

1. Remember the “Law of Attraction?” We reason that LHO with 6 spades is likely to have the least number of cards in the diamond suit. At this point, it doesn’t matter how many more diamonds RHO has, as long as we recognize that the odds are that he will have more. Since RHO is most likely to have more diamonds, he is the most likely to hold the missing Queen of Diamonds (or any other specific diamond for that matter). What is important is not how many diamonds RHO has, as long he has more than LHO. If the seven diamonds break LHO 3 and RHO 4, the odds are 57.1% (4/7) that RHO has the Queen I.

If you remember the Law of Attraction, you don't even have to do the arithmetic or memorize card distribution tables. All you are trying to do is figure out which way to take a 50% finesse and you know the person with more diamonds is the favorite to hold the Queen! We don’t really care what the likelihood of success is, as long it is more than 50%. Alternatively, if your contract requires one of two finesses to work, in either diamonds or hearts, and one goes through RHO and the other through LHO, take the one through RHO for the same reasons. These conclusions come with the usual caveat about "all other things being equal." If you have seen the Q of diamonds fall out of LHO’s hand, don't ignore it!

2. I said two reasons. Here is the second! LHO opened a weak 2 bid. He likely has 6-10 hcps. Let’s say on average 8 hpcs. It is also likely that he has at least 5-6 of those points in spades. We could put a fine point on this, but this is not an arithmetic lesson. With 2 or 3 points outside of the spade suit, the odds are simply better that RHO has partner has the Queen of diamonds. This simply reinforces our decision to take our 2 way diamond finesse through RHO.

I have gone to great lengths (with endless repetition) to make this understandable for those who like to know “WHY?” It is better if you understand the analysis, but even if you forget, if LHO makes a weak 2 bid, just take all your finesses through RHO if you have a choice, and remember to save an entry to the dummy!

This example teaches another good lesson that even seems to escape good players. There is always a cost to entering a competitive auction. That cost is the information (strength, location and distribution) that you furnish by bidding, particularly when there is no substantial likelihood that your side will end up declaring a contract. One of the “Hallmarks” of good players is that they listen to the opponents during the auction and use the information they obtain against them.

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