Inference after a Weak Two

In the USBF Women’s Final the Baker team lost a close match to Sonsini partly due to their losses on boards in which the opening bid was a weak 2. As originally conceived by Howard Schenken the weak 2 augmented the standard US opening bid structure by the addition of a light, constructive call that showed a good 6-card suit without the normal requirement for an bid at the one-level. Many players still adhere to the ‘good-suit’ principle in first and second seat, while allowing for more freedom, and hence for more uncertainty, in third seat. Karen McCallum of the Baker team believes that strong suits can be opened light at the one level, so even in first seat there is no promise of a good suit being held.

There are players who allow themselves to open a weak 2 with a bad suit, but prefer to have some back-up in the form of a high honor in a side suit, maybe even with a king and a queen outside. This dubious practice is protected to some extent by the degree of uncertainty with which it operates. It is the task of a declarer to plan her play according to the tendency of the preemptor. Here is an example from the Baker-Sonsini match.

McCallum-Baker are vulnerable against not, so Radin’s raise of Soncini’s 2♦ opening bid can be expected to be weak and adding to the preemptive value in line with the ‘last-guess’ principle. When considering Baker’s options with regard to McCallum’s power double, one should consider the most likely number of total trumps available. Following the suggestions of Larry Cohen in this regard, Baker should assume her partner holds a 4=4=1=4 hand, making the division of sides 7=7=3=9. That turns out to be the exact division. The number of total trumps is 19, so if 5♣ makes, 4♦ can be expected to be down 2, doubled for a score of +300. This would be an insufficient return against +600 at the other table.

From Baker’s hand a conservative estimate of the number of tricks available against 4♦ is 5, one in each minor and 3 in the majors. If clubs are divided 2-2, there are 6 losers in 4♦ but no more winners in 5♣, making the Law suspect. Now 4♦* would yield +500 making the decision much closer.

Baker bid 5♣ and was faced with the task of avoiding 3 losers in the majors. The lead of the ♦4 to the ♦K was taken by the ♦A, a diamond was ruffed, and trumps were found to be split 2-2 with the ♣J on the left. Sonsini holds 5 cards in the majors and Radin holds 7. The hearts could be split 3-3 and spades 2-4, or the hearts could be 2-4 and spades 3-3.   Which is more likely? The success of the contract will depend on which view is taken.
If there were no bias to be applied to the major suits, the number of combination for each situation is 20 of a 2-2 split and 15 for a 2-4 split for a product of 300 combinations. In a case where the decision appears to be a toss-up, it behooves one to look for a clue that might tip the balance in one direction or the other. The clues come from the bidding and the opening lead.

Sonsini has opened a weak 2 on a bad suit. Suppose we assume she is the type who feels more comfortable opening a weak 2 with an outside top honor. Which major suit honor is she likely to hold? The missing majors are ♥ KQT963 and ♠ QJT853. Assume she holds the ♥K. How does that affect the numbers of card combinations? We have the following numbers of combinations.

Sonsini has opened a weak 2 on a bad suit. Suppose we assume she is the type who feels more comfortable opening a weak 2 with an outside top honor. Which major suit honor is she likely to hold? The missing majors are ♥ KQT963 and ♠ QJT853. Assume she holds the ♥K. How does that affect the numbers of card combinations? We have the following numbers of combinations.

Hearts Split	Sonsini	Radin	Hearts	Spades	Product
3-3	♥Kxx	♥xxx	10	15	150
2-4	♥Kx	♥xxxx	5	20	100

Under this assumption the odds are 3:2 that the hearts split 3-3. Carry it a step further: suppose Radin holds the ♥K.

Hearts Split	Sonsini	Radin	Hearts	Spades	Product
3-3	♥xxx	♥Kxx	10	15	150
2-4	♥xx	♥Kxxx	10	20	200

Now the odds favor the 3-3 in spades in a 4:3 ratio. Overall if one thinks there is better than a 50% chance that Sonsini holds the ♥K, one should play for hearts to be 3-3. To put it even more roughly, which holding looks more likely:
♠ Qx ♥ Kxx ♦ Qxxxxx ♣Jx  or ♠ Qxx ♥ Qx ♦ Qxxxxx ♣ Jx ?

Here is the full deal.

 

The winning decision is to play for the hearts to split 3-3. Duck a heart. The losing spade eventually goes away on the 4th heart in dummy.
The double dummy analysis tells us the number of total tricks was 18. In the same situation at the other table Beth Palmer passed Lynn Deas’ balancing double and collected a penalty of 500. This would have represented a loss on the board if Baker had played for hearts rather than spades to be 3-3. As it was Palmer’s pass resulted in a 12 IMP gain. In effect she took the sure plus rather than face what might be a tough play problem at the 5-level.

If the preemptor were a disciple of Karen McCallum, it may be assumed she holds one or the other heart honor to make up the minimum number of HCPs needed for a first seat 2♦. Here are the combinations for the condition of split honors.

Hearts Split	Disciple	Pard	Hearts	Spades	Product
3-3	♥Qxx	♥Kxx	6	15	90
2-4	♥Qx	♥Kxxx	4	20	80

Putting a restriction on both the LHO and the RHO reduces the odds of a 2-2 heart split to 9:8, but it is still better to play for the even split in hearts. This also covers the case where it may be assumed that the LHO must hold either the ♥K or the ♥Q, but not both.

The above analysis is a simplification to illustrate the method, the full implementation of which would require a computer program to weed out the impossible heart-spades combinations. At the table one can’t perform an exact calculation, but one can assume the preemptor holds at least one top heart honor whereas there is no restriction on the spade honor locations. As shown above that suggests hearts are more likely than spades to split 3-3.