The Icarus File

The Greek legend of Icarus should remind players that it is dangerous to aim too high in a world characterized by uncertainty and doubt. At a recent club game a pair of visitors handed us a shared top on the last round when the lady sitting South playing in a cold 5♥ contract won the first trick in dummy with the singleton ♣K and saw a route to 12 tricks through entering her hand immediately in spades in order to discard diamond losers in dummy on her ♣AQ. Her punishment was instantaneous as the spades were split 5-0 against her. She tied for a bottom with a declarer who went down 1 in 4♥, which should be worth an extra a quarter matchpoint for her, don’t you think?

‘Sorry, Dear,’ she apologized to her frowning spouse, ‘that is sure to be a bottom.’

‘Don’t be so sure,’ I quickly consoled her, ‘you haven’t played here before. It is hard to get a bottom in this club with 13 tables in play. You need to play a complex system like Precision to achieve bottoms with any consistency.’

This is true. Our best result that day was playing in 6♦ after 6 rounds of bidding missing 2 cashable aces while most EW pairs were mired in 3NT making 4. My 4NT on the 5th round was a sign-off not RCKB, so when partner bid 5♦, I felt I had to bid 6♦ to get a decent matchpoint result. The opening leader cashed her ♥A which caught an encouraging ♥9 from her partner. With the ♥K visible in the dummy she continued with a low heart whereupon I claimed. It seems in a perfect world the ♥9 was suit preference for spades. You see, both sides were trying too hard for perfection which was, temporarily at least, beyond their abilities to achieve.

Priebe’s Problem

The following problem was suggested by Jim Priebe, the well-known author of bridge mysteries, who in a team game defeated a slam which declarer could have made. The problem illustrates the difference between a priori probabilities and a posteriori probabilities that involve vacant places. In theory the correct play depends on what information has been provided during the bidding and early play.

In order to calculate probabilities in 2 suits after something is known about the other 2 suits from the bidding and play, we assume a random distribution in the unplayed suits. This means that probabilities can be calculated exactly form the numbers of possible card combinations. Sometimes a refinement must be added that complicates matters as on the following example where the distributions of spades and hearts become known early, and a decision had to be made on best play in clubs and diamonds.

North leads a heart, ruffed in dummy, and West takes his time to access the possibilities before him. The club finesse has obvious appeal as even just 4 club tricks will allow him to discard diamond losers from his hand. That has an a priori probability of 89%. The double finesse in diamonds has its attractions because of the intermediate spots. A double finesse gives a 76% chance of success with the 4th diamond providing the discard of a potential loser in clubs. If South should win the first diamond, he cannot safely return a club.

Eventually declarer leads the ♠7 towards his hand and is much surprised to see South show out. This means North has preempted most unexpectedly with 6 hearts and 4 spades. The vacant places remaining to accommodate 11 minor suit cards are 3 in the North and 8 in the South. Suddenly the previous analysis must be tossed out of the window. The a priori odds are no longer applicable, as they are based on an assumption of symmetry. Even an imbalance of 2 in vacant places is cause for a re-evaluation.

West must win in hand, finesse in trumps, cash the ♠A and return to hand with a club to the A in order to draw the last trump. South discards hearts. Declarer must now play for 1 loser in the minors. He cashes the ♥A looking to discard from dummy to one of the following 6-card positions.

West	Position I	Position 2
♠6	♠ —	♠ —
♥ —	♥ —	♥ —
♦ T8	♦ A9	♦ A973
♣ 75	♣ KJ96	♣ KJ

In Position 1 declarer plans to establish clubs by finessing for the queen. Even if the finesse loses and a hearts come back, he may ruff, return to the dummy with the ♦A and discard diamonds on winning clubs. In Position 2 the plan is to run the ♦J and if this loses to run the ♦T next. He must decide immediately on one position or the other as South is posed behind the dummy to discard appropriately. A facile argument might go like this: once it is known that North holds 3 cards in the minors, he is more likely to be short in the suit in which EW hold the most cards, clubs, so declarer should play on diamonds. One should try to be more exact than that. There are 3 apparent initial distributions:

	Case I	Case II	Case III
Spades	4 – 0	4- 0	4 – 0
Hearts	6 – 5	6 – 5	6 – 5
Diamonds	0 – 6	1 – 5	2 – 4
Clubs	3 – 2	2 – 3	1 – 4
Combinations	10	60	75

On this basis if North is short in clubs (Case III), declarer should play on diamonds, whereas if North is short in diamonds, he should play on clubs (Cases I and II) keeping ♦A9 in dummy. The total combinations favour the diamond play keeping ♣KJ in dummy by 75 to 70. This is a crude estimate as in Case II a singleton honor in the North allows the diamond play to get through successfully.

The single most likely distribution is Case III, and one might choose to play for that situation as this is the easiest to calculate at the table. However, one round of clubs has been played with North following with the ♣2, South, with the ♣4, the remaining combinations are now as follows.

	Case I	Case II	Case III
Spades	4 – 0	4- 0	4 – 0
Hearts	6 – 5	6 – 5	6 – 5
Diamonds	0 – 6	1 – 5	2 – 4
Clubs	2 – 1	1 – 2	0 – 3
Combinations	3	18	15

The effect of one round of clubs is that Case II has become single most likely distribution. One sees the attractiveness of playing for the club finesse. North has followed with a low club, but not just any low club, but with the 2 specifically and South with the ♣4. From an original split of ♣2 opposite ♣ QT84 there were just 2 plausible plays, the ♣2 from North and either the ♣4 or ♣8 from South. This assumes that the defenders would not play the ♣T or ♣Q unless forced to do so. We are in a restricted choice situation, so what I have called the Extended Kelsey Rule can be applied. Specifically, it is correct to compare directly the reduced combinations from differing splits when there is equality in the number of plausible plays across the board. This is a consequence of Bayes Theorem.

What Works Best?

Playing for the most likely case is a shortcut that often works, but to obtain the probability of success of a method accurately one must add up the number of success over all remaining combinations. Under Case II there are 2 situations where playing on diamonds succeeds, namely, when a singleton honour sits in the North. So the successes of the diamond play are increased by 6 combinations. On the other hand under Case III the diamond play is successful only 3 times out of 5, a reduction of 6 combinations. The success rate for the club play is 21 out of 36 (58%) and for the diamond play 15 out of 36 (42%).

When the Opposition is Silent

In the actual situation encountered at the table by Jim Priebe, there was no opposition bidding, so the heart suit could not be placed with confidence. When South showed out of trumps, it was most likely that the hearts were split 5-6, not 6-5 as assumed for the analysis above. The vacant places to accommodate the minors are most likely to be 4-7. One may ask how much better is the club play than the diamond play under that condition.

	Case IV	Case V	Case VI
Spades	4 – 0	4- 0	4 – 0
Hearts	5 – 6	5 – 6	5 – 6
Diamonds	1 – 5	2 – 4	3 – 3
Clubs	2 – 1	1 – 2	0 – 3
Combinations	18	45	20

Assuming from the absence of opposition bidding that South would not hold a 5- or 6-card minor to go along with his 6 hearts, we are left with just 2 cases. If declarer decides to play on clubs, he will be successful when clubs split 2-3, 45 combinations out of 65, a 69% chance. The diamond play will be successful in 27 combinations out of 45 when diamonds split 2-4 and 16 cases out of 20 when diamonds split 3-3 for a total of 43 combinations in total, a success rate of 66%.

I need only add that, of course, the winning play at the table was to bank on the diamond play. Against the odds clubs were dealt QT84 in the South, the losing combination for the club play. Here are the hands in full.

It could be said that, like Brutus and Cassius at Philippi, both declarers fell honourably upon their swords.

Vacant Places when Spades are 2-2

If the spades split a normal 2-2, the distributions after one round of clubs could be as follows:

	Case VII	Case VIII	Case IX	Case X
Spades	2 – 2	2- 2	2 – 2	2 – 2
Hearts	6 – 5	6 – 5	6 – 5
Diamonds	1 – 5	2 – 4	3 – 3	4 – 2
Clubs	3 – 0	2 – 1	1 – 2	0 – 3
Combinations	6	45	60	15

With the vacant places being 5 in the North and 6 in the South the imbalance is only 1.

The success rate for the diamond play is up to 72%, close to the a priori estimate of 76%, but the club play is much more successful at 88% (111 out of 126.) The probability of a singleton club remaining far outweighs that of a void.