Total Trumps and 5-4-3-1

In this segment we consider the implications with regard to card distributions of doubling an opponents’ preemptive raise at the 3-level that implies a 9-card fit. In particular, we wish to determine the probabilities of the number of total trumps based on an assumption of a random deal of the unknown cards, a flawed assumption to be sure, but one that enables a reasonably accurate estimate as a first approximation. As in the case of a 4-4-4-1 hand, only the most likely shapes opposite are considered as extreme distributions are rare, and in such cases partner will take appropriate action independently.

	I	II	III	IV	V
	♠ 5 – 4	♠ 5 – 4	♠ 5 – 5	♠ 5 – 5	♠ 5 – 6
	♥ 1 – 3	♥ 1 – 3	♥ 1 – 3	♥ 1 – 3	♥ 1 – 3
	♦ 4 – 3	♦ 4 – 4	♦ 4 – 3	♦ 4 – 4	♦ 4 – 3
	♣ 3 – 3	♣ 3 – 2	♣ 3 – 2	♣ 3 – 1	♣ 3 – 2
Sides	9=4=7=6	9=4=8=5	10=4=7=5	10=4=8=4	11=4=6=5
Also	8=4=8=6	9=4=6=7	10=4=6=6	10=4=5=7	7=4=10=5
	8=4=7=7	7=4=8=7	8=4=9=5	9=4=9=4	7=4=7=9
			8=4=6=8	9=4=5=8
			7=4=9=6	6=4=9=7
			7=4=7=8	6=4=8=8
Trumps	18	18	19	19	20
Total Weights	257	169	231	90	37
Percentages	33%	22%	29%	11%	5%

¶

Here are the 5 most frequent conditions when the opponents have a 9-card heart fit and one holds a 5=1=4=3 hand.

It may be surprising to those who rely on a priori odds that a 5-3-3-2 shape opposite is more likely than a 4-4-3-2 shape. The above figures are estimates of the a posteriori odds under the given restrictions. There are 6 possible divisions of sides with a 5-3-3-2 shape and only 3 with a 4-4-3-2 shape. More conveniently, one should deal with the total number of trumps. The percentages are as follows: 18 trumps (55%), 19 trumps (40%), and 20 trumps (5%). In 11 out of 20 deals, a player holding 5-4-3-1 shape will be in a situation where the total trumps number 18.

The most important suit is spades in which it is assumed the player holds 5. One wishes to estimate what are the percentages for holdings of 7, 8, or 9 cards in the suit when consideration is limited to these 5 most common conditions.

Number of Spades Held	6	7	8	9	10	11
Percentages	6%	27%	34%	24%	8%	1%
Number of Diamonds Held	5	6	7	8	9	10
Percentages	3%	20%	32%	30%	13%	1%
(4 Spades)			42%	40%	17%	1%
Number of Clubs Held	4	5	6	7	8	9
Percentages	2%	14%	29%	33%	20%	3%
(3 Spades)				40%	38%	21%

¶

One has maximum uncertainty as to whether there are 6-7 spades, 8 spades, or 9-11 spades opposite, each with a probability of 1/3. Guessing 8 opposite is smack in the middle of the expectations, in which case the total number of trumps is 17. It would appear then that it is OK to bid 3♠ on the expectation of partner’s shape, or to stretch to 4♠ with appropriate controls.

There is a 53% chance that partner will hold 3 or fewer diamonds. What if one holds just 4 spades? One may convert the diamonds to spades and apply the percentages given above for diamonds, but if we assume the opponents would not preempt to 3♥ when holding as many as 7 spades, we can eliminate the lower two holdings when considering a 4-card spade suit. It would be better to double to show 4 spades and let partner decide whether to support spades or not. The odds are roughly 60-40 that you will find a good fit there, so a double is not aggressive since it is significantly on the right side of the odds.

There is only a 23% chance the partnership holds 8 or more clubs. What if one holds just 3 spades? One may convert the clubs to spades and apply the percentages. Normally it is not best to suggest playing in spades when holding 5-4 in the minors. If one assumes the opponents don’t hold as many as 7 spades for their preemptive action, the odds swing to roughly 60-40 that you will find a good fit in spades. It is acceptable to double with only 3 spades if the opponents are assumed short in spades, dangerous otherwise. After 2♥ -3♥, one may double hopefully, but after 1♥ – 3♥, one has to be more cautious.

The Mode and the Most Likely Distribution of Sides The mode is the most likely number of trumps for each suit. One sees the modes for spades, diamonds, and clubs are 8, 7, and 7, which is agreement with the single most likely side, 8=4=7=7. It is to be noted that given the 3-1 heart split, there are 22 cards remaining to be divided between 3 suits. The most even division is the most probable, that being an 8-7-7 split. In order of probability the component sides are 8=4=7=7 (13%), 7=4=8=7 (9%), and 7=4=7=8 (8%). One may wait a long time for a specific distribution to turn up at the table, but in total the most likely division of sides (8-7-7-4) occurs roughly 30% of the time.

Competing over 3♠ After the opponents have bid preemptively to 3♥, the odds favor taking action when holding 5-4-3-1 shape with a singleton heart, even with only 3 spades. The same percentages can be used to gain a rough estimate of the number of hearts held when the opponents preempt to 3♠. The difference is that one needs more to bid a level higher, so a 9-card fit is the norm for bidding at the 4 level, only a 33% chance with a 1-5-4-3 shape. At IMPs scoring, vulnerable against not, bidding 4H is not unreasonable when holding 4 hearts when the high card content otherwise justifies the risk.

Larry Cohen Defies the Law

The tradition in America is that a double of one major promises at least 4 cards in the other, but there are cases where an off-shape double is made on general strength. This can lead to problems later. The idea is that the doubler can get around to describing the nature of his call on the next round, but this method is fraught with danger as it is susceptible to preemptive action. The following example involves Larry Cohen and Dave Berkowitz playing in the finals of the 2004 Vanderbilt Cup against a famously active Italian pair. On Board 60, the opponents preempted, Cohen held a 4-4-4-1 hand, but it was Berkowitz who doubled first to show strength. It appeared they held at most 8 hearts between them, nonetheless, Cohen bid to 4♥, vulnerable vs not.

Dealer: North Versace Vul: E/W		Versace
		♠	10 8 7 6 3
		♥	7 2
		♦	5 4
		♣	J 10 6 2
Berkowitz				Cohen
♠	A K			♠	2
♥	K Q 4			♥	A J 8 5
♦	Q 3 2			♦	J 10 9 8
♣	A Q 9 8 5			♣	K 7 4 3
		Lauria
		♠	Q J 9 5 4
		♥	10 9 6 3
		♦	A K 7 6
		♣	—

Berkowitz	Versace	Cohen	Lauria
—	Pass	Pass	1♠
Dbl.	3♠	4♥	4♠
Dbl.	All Pass

Michael Rosenberg commented dourly on this deal in the Nov 2004 issue of The Bridge World. 4♥ is off 1 and 4♠ is off 2, so the total (major suit) tricks are 17 and the total trumps are 17, a perfect match. However, the best fit EW is in clubs, making 10 tricks, but the method of scoring dictates that EW play in the inferior fit. There is one other consideration: EW can make 3NT. The only alternative Cohen had in order to maintain the possibility of reaching the right contract was a space-saving double. That would make sense if the proper interpretation could be placed on that bid. It can’t be a penalty double, can it?

Cohen’s decision to bid 4♥ is wrong in theory, but the Italian pair could not afford to gamble that EW might make their game. 4♠ was ‘good insurance’. How could Lauria tell that the defense can take the first 4 tricks off the top? There is enough uncertainty that South makes his decision on general grounds. Cohen’s bold move has paid off.

This illustrates how the game is played. South opens light in third seat, West doubles on an off-shape hand too strong for an overcall, North preempts with a poor hand driving the opponents to an unmakeable contract, after which partner saves them by pulling to another unmakeable contract. Everyone feels they have routinely made the right decision. Only those who can see all 4 hands feel some improvement can be made.

Let’s consider Cohen’s decision to bid 4♥. As he holds a singleton spade, Cohen can place Berkowitz with at least 2 spades, possibly 3. Berkowitz may have a hand too strong for a NT overcall. Quite possibly the number of total trumps is 17 and the best fit is in a minor suit. One sees that the best game is 3NT. Of course, Cohen can’t know for sure the best available alternative. It would be convenient if 4♥ proved the best spot, but there is no reason for thinking it is, except on general grounds. This is a good situation for a doubtful double that leaves the decision to partner, who, after all, stands to have the best hand at the table. Double cannot be unilaterally for penalty, but all alternatives remain in place.

3 NT is To Play

The best answer to frivolous preempts is to bid an uninformative 3NT directly and see if the opponents can find the defence to beat it. After all, the opponents are reducing the information content of their bids, so why should the stronger side help them out by expressing doubt and giving away information unnecessarily? Bidding what you think you can make is the other side of the application of the doubtful double. A problem may occur if a thinking partner has other ideas, as in this example from the finals for the 2009 Vanderbilt Cup.

Dealer: North Vul: None		Elahmady
		♠	K J 10 8 7 2
		♥	K 9 8 7 3
		♦	2
		♣	3
Moss				Gitelman
♠	Q 6 5 4			♠	A 8
♥	5			♥	A Q 10 6 4
♦	K 10 9 7			♦	A Q 4
♣	A J 8 7			♣	K 5 2
		Sadek
		♠	3
		♥	J 2
		♦	J 8 6 5 3
		♣	Q 10 9 6 4

Moss	Elahmady	Gitelman	Sadek
—	2♥*	3NT	Pass
4♣**	Pass	4NT	Pass
5NT	Pass	6NT	All Pass

**Looking for a 4-4 fit?

The match was tied at the half-way point when Elahmady used a preemptive 2♥ opening bid that showed hearts and another unspecified suit. Obviously he was not trying to describe his hand accurately, but rather to disrupt the opponents. Gitelman bid what he thought he could make. He would welcome a heart lead, so his shut-out bid put the pressure on Sadek to find a better lead. It was akin to a ju-jitsu move where the aim is to use the opponent’s energy to work against them – a good strategy.

On such jumps to 3NT the bidder has a right to expect a few useful pieces to appear in the dummy, but some partners are very protective of their right to be heard from. Thinking, perhaps, that one doesn’t preempt after a preempt, Moss felt an ace and a king were enough to make an attempt to improve the contract, perhaps to 4♠. Elahmady’s uninformative 2♥ bid was working its magic. Gitelman signed off, but Moss was still not satisfied that a power slam wasn’t possible. 5NT was not forcing and makeable, but Gitleman could see extras for his previous 4NT sign-off, and went on to the hopeless slam, losing 11 IMPs, a prime example of two heads being worse than one.

One might imagine an auction where Gitelman doubles, Moss shows interest in a spade contract, and Gitleman signs off in 3NT. In this sequence Moss is given the opportunity to have his say, which can then be over-ruled graciously. If Moss persists with a 4♣ bid, 4NT reinforces the opinion that 3NT was best, and all’s well that ends well. This is contrary to the concept of a doubtful double and bidding what you think you can make when you have no interest in slams, but it massages a partner’s ego who no longer feels neglected. ‘Well, I tried’, he may say with a sigh of resignation.

If one has to bid psychologically to keep a partner’s ego satisfied, science takes a back seat. Second guessing becomes part of the system when neither player can assume the captaincy of the auction. In my opinion, the disaster on Board 33 led to further loss of 13 IMPs three boards later on this combination of hands.

Moss	Elahmady	Gitelman	Sadek	Moss	Gitelman
1♦	1♥	1♠	2♥	♣A K J 7	♣Q 9 8 5 4
3♠	Pass	4♠	All Pass	♥10 7	♥A 6
				♦A 10 7 6 5 3	♦K J
				♣3	♠Q J 10 9

 

In the other room on a similar auction Levin cue bid his ♥A and Weinstein proceeded to slam in the hope his diamonds would prove in the end a source of tricks. They did. Gitelman proved to be a bit shy despite his fine support for partner’s presumed lengthy diamond suit. The East hand is much better than might be expected from the initial free bid. It would cost to make a slam try only if an aggressive Moss would over-react to any modest move in that direction. The way I see it, the partnership psychology was wrong.

Total Trumps and 5-4-2-2

In the above example Gitelman held a 5-4-2-2 shape, and partner had advertised length in his short suit, albeit one with KJ doubleton. The 5-4-2-2 shape has gained a bad reputation over the years even though the a priori probability of finding an 8-card fit in one of the long suits is the same as for the 5-4-3-1 shape (74%). The total trumps are one less for the former shape, and the latter shape has the advantage of a built-in shortage.

One may analyze the distribution of sides for 5-4-2-2 in the same way as described above for the 5-4-3-1 shape when the opponents have advertised a 9-card heart fit. We spare the reader the details, only to show below the most likely distributions which illustrate the potential problems encountered with the 5-4-2-2 shape.

	Division	Partner	Occurrence	Total Trumps	Estimated T.T.
I	8=4=8=6	3=2=4=4	13%	17	18
II	8=4=7=7	3=2=3=5	12%	17	19
III	9=4=7=6	4=2=3=4	11%	18	18
IV	7=4=8=7	2=2=4=5	9%	17	19

 

The estimated total trumps are what partner would calculate if he assumed the standard 4=1=4=4 shape opposite. This would be a very bad estimate when he held 5 clubs, the suit in which his partner holds only 2. The mathematical reason why Cases II and IV have such a high probability of occurrence derives from the fact that the most even split between 11 unknown clubs is 6-5, in the ratio of 7:5 with regard to a 7-4 split. This argues against doubling for takeout with a 5-4-2-2 shape. The temptation is to try to hit the best fit in the long suits, while the danger is that partner’s best suit is the one in which one holds a doubleton.

 

If one simply bids one’s 5-card suit, there is a 20% chance that the 4-card suit provides the better fit. Of course, if the 5-card suit is spades, there is no concern about that when competing for a part score or a game. If the 5-card suit is diamonds and the 4-card suit is spades, it is a different situation. One may double and correct to diamonds if partner bids clubs. (Remind yourself that partner is not trying to be difficult, he is merely bidding his longest suit. Of course if he jumps a level to do so, one may question his judgment.) If the 5-card suit is clubs, one needs must correct 4♦ to 4♠, and partner must interpret this as suit correction and not an indication of slam invitational values.

 

Finally, one may ask whether over a preempt to the 3-level in spades one should bid a 4-card heart suit to the 4-level as Larry Cohen did in the above example. Even if you are wrong, the opponents may ‘save’. Otherwise, provided that the opponents don’t hold more than 6 hearts, there is roughly a 2 out of 3 chance that one will find an 8-card or longer heart fit in partner’s hand. As noted above it pays to have a good holding in the doubleton suit when that is probably partner’s longest suit. Here is a case where a doubleton honors do not constitute ‘wasted’ values, but, in fact, are an essential component in the hands taken as a complementary whole. (In terms of the losing trick count, one doesn’t like to gamble with 2 losers in the short suit.)