Bob Mackinnon

Baby Knows Best?

Last week on PBS, Charley Rose interviewed Alison Gropnick, a professor of psychology and author of the recently published book, The Philosophical Baby, about her research into the brain activities of infants. Her view is that we are all born with many innate mental facilities that are allowed to die on the vine, as it were, so growing to adulthood is not just a process of gain, but also a process of loss. For example, we know a second language is best learned earlier rather than later in life. Furthermore, according to Dr Gropnick babies have a grasp of the basic concepts of statistics, and can evaluate alternative courses of action taking probability into account, whereas, college students have difficulty absorbing the fundamentals of probability during their first academic exposure. I gather here she was speaking from a painful, personal experience. My conclusion is that children should be exposed to the study of probability at a young age, thus learning early how to cope with uncertainty in a rational manner.

There is more to this than a condemnation of current teaching methods. If one is educated to follow do-and-don’t rules rigidly rather than to think independently on the basis of evidence, natural abilities are stifled, and one becomes less adaptable to situations where uncertainty exists. Well, we can see how this affects performance at the bridge table. A bridge hand is best played with the happy expectation of a curious baby about to open a door and uncover the mysteries of a kitchen cabinet or a bedroom closet.

Don’t Worry, Be Happy In this segment in answer to a comment made by a reader of a previous blog, we consider a situation where a declarer in order to succeed in 3NT needs to develop tricks in an 8-card suit missing the AQ. Declarer can’t afford to lose the first trick in the suit to the queen on the right-hand side. There are conflicting emotions at work, the hope of succeeding and the fear of losing. Which should dominate?

The nature of the game is such that the optimum result is usually achieved by the taking of tricks rather than by the avoiding of losing them. This is true generally at matchpoint scoring, but it is also true at IMPs where there is no scope for a safety play, as in the deal to be studied, when one bids to a close game and merely hopes to make it. In these circumstances declarer’s plan should be geared towards maximizing the chance of taking tricks, rather than minimizing the chance of losing them. If the cards are as badly placed as one fears, then one shouldn’t have bid 3NT in the first place. To be consistent, declarer should assume that the contract is makeable, even if the odds are firmly against it.

Often declarers reach a decision point where the chances of success are less than 50% regardless of which choice is made. This is common in the ‘Eight-Ever’ situation when the player has 8 trumps and must decide whether or not to finesse for the queen. When the LHO follows to the second round, the finesse may be against the odds, yet we know that finessing is better than playing for the drop. So we follow through. The same principle applies to more complex problems. When 2 or more plays in a suit are required to achieve a goal, one must be patient and resist the temptation of achieving an immediate success on the first round at the expense of a reduction in the overall chances. We shall now investigate a situation where the constraints of an opening bid affect the decision.

CARDE’s Problem

West East
Jxx Axx
A109x Qx
8x KJ109xx
AQxx Kx


West North East South
1 2 Pass
2 Pass 3 Pass
3 NT All Pass
Opening Lead: 2 (low from odd)


The 2 is won by declarer with the J. The bidding places North with 5 spades and the bulk of the 16 missing HCP. Declarer’s hope is to establish 4 tricks in the diamond suit in such a way that North does not win a diamond after the spades have been established. The choices are: 1) to run the 8 on the first round and hope not to lose to the Q, or 2) to play to the K and duck the second round to North’s presumed bare A. In this latter case, East will take the third round of diamonds with the Q but will have run out of spades. As noted above, the establishment of diamond tricks generally requires more than one round to be played. First we look at the 5 most common distributions of sides for a 7=7=5=7 division of sides with spades split 5-2.

5 – 2 5 – 2 5 – 2 5 – 2 5 – 2
3 – 4 4 – 3 2 – 5 3 – 4 2 – 5
2 – 3 2 – 3 2 – 5 3 – 2 3 – 2
3 – 4 2 – 5 4 – 3 2 – 5 3 – 4
Weights 10 6 6 6 6


The 2-3 split in diamonds is nearly twice as likely as the 3-2 split due to the imbalance in the vacant places with a preponderance of spades in the North hand. When a problem is complex, one first solves a similar, closely related problem in order to shed some light on the path towards a decision. Here we shall assume that North holds the A, which is very likely needed to promote his hand to the status of an opening bid. There are 2 choices of play that will be successful under differing conditions:

Run the 8: AQx opposite xx Play to the K: Ax opposite Qxx


There are 2 attractive features of the K play: 1) initially a 2-3 diamond split is more likely than a 3-2 split, and 2) it would be embarrassing to go down in a makeable contract by losing to the Q on the first round. It requires a certain toughness of mind to reject going up the K. If one makes the correct play one needn’t feel embarrassed when it fails, as it often will. (Not many feel guilty when the wrong play succeeds, learning nothing from their mistake. It is futile to criticize a lucky play publicly, but some do.)

Placing the A with North sets the vacant places North to South as 7 to 11, so before a diamond is played the odds of the Q being dealt to the South hand are 11:7. Thus, a perfunctory analysis indicates playing to the K, but one should look more deeply into the situation. There may be further constraints that apply because, in theory at least, the North hand must fulfill the normal requirements for an opening bid.

The Effect of the First Play in Diamonds We denote the low cards in diamonds as u,w, and y, and assume North has played card u on the lead of the 8 towards dummy. The successful placements are now specifically 2 in number: 1) Au opposite Qwy, and 2) AQu opposite wy. We needn’t concern ourselves about the losing combinations; we want to discover which winning condition is more probable. This depends on the associated number of card combinations in the suits that have not been played, hearts and clubs.

The Bidding Constraint North holds KQxxx. Under placement #1 the K is needed in the North hand whereas as under placement #2 the K may be held by either defender without prejudicing the requirement for an opening bid. We shall now look at the effect of this constraint on the 5 most common distributions of sides and determine the number of combinations for each. The placement of the J and J are irrelevant so we treat them as x’s in the notation, denoting low cards that could be dealt to either hand without effect.

Au – Qwy Au – Qwy Au – Qwy Au – Qwy Au – Qwy
Kxx – xxxx Kxxx – xxx Kx – xxxxx ?xx – ?xxx ?x – ?xxxx
xxx – xxxx xx – xxxxx xxxx – xxx xx – xxxxx xxx – xxxx


Combinations Available:

15 35 20 21 6 35 35 21 21 35
525 420 210 735 735


The question marks under Conditions IV and V are used to convey the fact that the K is free to be placed on either side without violating the bidding constraint. Under Conditions I to III the number of allowable heart combinations is reduced by the requirement that the K must sit with North. The effect of this reduction is that Conditions IV and V are now the most probable and Condition I is demoted to 3rd place.

Requiring that a specific card be placed in a particular hand greatly reduces the number of available combinations in the suit thereby affecting the probabilities in other suits. As a result, under our restrictions, there are more possible combinations for which running the 8 on the first round is the winning play, roughly in the ratio of 5:4. There are other, much less probable, situations where running the 8 is correct:

Diamonds 2 – 3

Diamonds 3 – 2 Diamonds 4 – 1
Qu opposite Awy Quy opposite Aw AQuy opposite w
Quw opposite Ay AQuy opposite y


It would take a computer to calculate all the probabilities involved, but we have considered the most likely scenarios, and these situations give ample justification to the standard advice of choosing to finesse first against the lower missing honor.

Finally we note that through the bidding constraint the odds are strongly affected by the location of the K. In some situations it pays to gather information on the location of a high card in one suit before making a critical decision in another, say, by leading low towards the Q at trick 2. Discovery plays of this sort can be helpful especially at matchpoints, but here safety considerations preclude that operation.

Leave a comment

Your comment