Total Trumps with a 4-4-4-1 Shape

In To Bid or Not To Bid, his book on the Law of Total Tricks, Larry Cohen often estimates the number of total trumps on the assumption that partner holds a singleton and a 4-4-4-1 shape. On the surface this is a rather a strange assumption, because a 5-4-3-1 shape is more likely on an a priori basis. One might think vaguely that 4-4-4-1 is reasonable as the average of 3, 4, and 5 is 4, but that is not a fruitful way of thinking. In this study we consider the exact consequences of Cohen’s assumption with regard to probability considerations to be taken into account during the auction.

Here are 6 such consequences of assuming partner holds a 4-4-4-1 shape:

1) your side’s best fit is in the longest suit in your own hand;

2) the total number of trumps is 16 + the difference between your longest suit and the opponents’ trump suit;

3) the estimate of total trumps thus obtained is optimistic on average;

4) given that partner has a singleton in the opponents’ trump suit, the resultant distribution of sides is the single most likely one on a random deal;

5) the 4-4-4-1 shape represents a condition of maximum uncertainty with regard to the number of cards held by partner in the potential 3 trump suits.

6)  The assumption conforms to Jaynes’ Principle, which states that in a condition of partial knowledge one should assume the distribution that is most probable, that is, the one associated with the greatest number of card combinations on the basis of a random deal and whatever else is otherwise known.

We shall now illustrate how the consequences apply to common competitive bidding situations in which the Law of Total Tricks plays a central role.

The Total Trump Calculation

When the dummy appears a declarer can count the total trumps by subtracting number of cards in the longest combined suit from the number of cards in the shortest and adding the result to 13. The total of trumps is an unambiguous characteristic of the division of sides.

The number of total trumps = 13+ (longest combined suit – shortest combined suit)

Of course, the defenders’ hands produce that same number of total trumps albeit generally with different independent hand distributions. Their division of sides may be different from that of the declarer. For example, if declarer has an 8-7-6-5 division, so will the defending side, but in the order 5-6-7-8. The total number of trumps is 16. If declarer has a 9-6-6-5 division, the defenders will have a 4-7-7-8 division, and the total number of trumps is 17, no matter which division of sides one uses to calculate it. A difference in the division of sides is a characteristic of deals that produce an odd number of total trumps.

The problem faced by a player in the middle of an auction is to estimate the total number of trumps knowing only for sure his own hand shape. Naturally, one would attempt to use the estimate that is most probable given what is assumed at the time from the auction. Suppose the opponents have preempted and partner has doubled for takeout. The calculation of total trumps is easiest for a 4-4-4-1 shape opposite. That shape contributes a difference of 3 between the long suit and the short suit regardless of which of 3 suits provides the best fit. This produces a minimum of 16 total trumps. To obtain an estimate of the total trumps with both hands taken into account, partner needs to add to 16 the difference between his longest suit and the presumed trump suit of the opponents. Thus, with a 5-3-3-2 shape the number of total trumps will be either 18 or 19 depending on whether the player holds 3 or 2 cards in the opponents’ trump suit.

The Most Probable Conditions

To illustrate how to estimate relative probabilities, we consider the following simple example from Cohen’s book where a player holds this hand: ♠ A43 ♥ QJT54 ♦ 963 ♣ 82, a 3=5=3=2 shape. The LHO opens the bidding with a preemptive 5♣. Partner doubles and the RHO passes. Cohen assumes the LHO holds 8 clubs for his bid and partner holds a singleton in that suit for his double. What is the number of total trumps one should use as a basis for the decision of whether or not to pass the double for penalty? Cohen assumes partner holds a 4-4-4-1 shape, so the division of sides should be 7=9=7=3. The total trumps add up to 19 (13+9-3). The division of sides for the opponents is 6=4=6=10. Cohen concludes that one should pass the double with such a low number of trumps available.

Let’s look at the relative probabilities of a 4-4-4-1 or a 5-4-3-1 shape opposite. Here are 6 possible distributions.

I                 II                III              IV               V             VI

♠ 3 – 4        ♠ 3 – 4        ♠ 3 – 5       ♠ 3 – 5      ♠ 3 – 3      ♠ 3 – 4

♥ 5 – 4       ♥ 5 – 3        ♥ 5 – 3       ♥ 5 – 4       ♥ 5 – 4     ♥ 5 – 5

♦ 3 – 4        ♦ 3 – 5        ♦ 3 – 4       ♦ 3 – 3       ♦ 3 – 5      ♦ 3 – 3

♣ 2 – 1        ♣ 2 – 1       ♣ 2 – 1      ♣ 2 – 1       ♣ 2 – 1     ♣ 2 – 1

Sides    7=9=7=3    7=8=8=3    8=8=7=3    8=9=6=3    6=9=8=3   7=10=6=3

Trumps     19                  18            18                  19                 19           20

Weights    100                 96            96                  69                 69          46

The probability weights are a reflection of the number of card combinations available for each pairing. Once a player sees his hand the distribution on the left is fixed (at 3=5=3=2) and it becomes a question of how many card combinations are available to his partner on the right-hand side. The more combinations available on a random deal basis, the greater the probability that the given condition exists. Condition I encompassing a 4-4-4-1 shape is the single most likely distribution, however, it is much more likely overall that partner holds a 5-4-3-1 shape as there are 6 such possibilities(only 5 are shown).

Calculation of Weights

There are available these many cards with which to fill out the hand: 10 spades, 8 hearts, and 10 diamonds. Conditions I and II has these many card combinations chosen at random from the pool of unknown cards:

Condition I    (10! 8! 10!) divided by (4! 6!)(4! 4!)(4! 6!)

Condition II    (10! 8! 10!) divided by (4! 6!)(3! 5!)(5! 5!)

Ratio II to I    24 divided by 25 which is the same as 96 to 100.

Similarly for Conditions III through VI. These calculations don’t take into account high-card content, but they provide reasonably accurate guidelines in this free-wheeling age where consideration of shape plays the dominant role in competitive bidding.

Note that the most even distribution in the 3 suits, 4-4-4, produces the maximum number of card combinations. Condition I is the condition of maximum likelihood, that is, it is the single most likely configuration given the player has counted his own hand. The fit in the heart suit is the key factor. There is a 9-card heart fit for Conditions I, IV, and V, hence 19 total trumps, and a total weight of 238 (46% of all cases). There is an 8-card heart fit for Conditions II and III, hence 18 total trumps and a sum of weights of 192(37%). It is most likely that the total number of trumps is 19. Least likely is that the total trumps are 20, 2 conditions with a total weight of 92(18%). There is a low probability that the longest suit in one hand matches the longest suit in the hand opposite. The average number of total trumps is 18.5, so 19 represents a slightly optimistic estimate on average.

Subjectivity These probabilities are based on the dealing of the cards, a situation of maximum uncertainty with regard to the placement of the cards. The auction may provide clues that constitute information that gives greater weight to one condition over another.  There is another factor to be taken into account, which is: with which shape is partner most likely to have doubled? The most advantageous situation for takeout is represented by ConditionVI for which the doubler holds 9 cards in the majors with longer hearts that spades. If partner takes out to 5♥, all is well, as from his point-of-view that should represent the best fit. Not surprisingly this is also the condition that represents the greatest number of total tricks. Thus, it is also the condition under which it is most dangerous to pass the double for penalty.

The degree to which one might wish to adjust the probabilities depends on the known behavior of one’s partner. If his doubles are generally penalty orientated and send the message, ‘they can’t push us around’, then one might favor leaving the double in as penalty. If his message is more likely to be, ‘we should play this hand’, then Condition VI becomes more likely and one would be more inclined to take out to 5♥.

Theoretically there is nothing wrong with adjusting probabilities according to what you expect from the known inclinations of one’s partner and/or one’s opponents. To be realistic, probabilities must reflect the current state of partial knowledge. That is ‘un-mathematical’ unless we can assign some numerical percentages to our subjective bias. So, in the above example, one should make an estimate of how likely it is that the doubler is operating under Condition VI rather than Condition I. Probabilities are adjusted accordingly. If one is maximally uncertain about partner’s action, one accepts the weights as shown which represent the probabilities of the deal, the a priori condition of maximum uncertainty. Partners being partners, that may be the best policy overall. However, Larry Cohen has given us many examples where experts bid one more than the rest of us, not always best.