Two Chances Aren’t Always Better than One

Eddy Kantar, a favorite author, has been awarded a prize for his 2009 book, Take All Your Chances in Bridge. A recurring theme is that in declarer play two chances are better than one, if one can take the first chance for success without losing the advantage of a second chance should the first one fail. Commonly, one is advised to play for the drop of an honor in one suit before finessing for an honor in a second suit. To finesse first is to risk immediate defeat.

Kantar’s book is basically a workbook on the theme presented in various settings. He takes the sensible approach of leading the student repetitiously to the correct approach through a natural logical process without piling on abstract mathematical arguments. However, being a contrarian, and a mathematician to boot, I love to pile it on to see if it is in fact true that 2 chances are always better than 1. If not, why not?  How come?

Tim Bourke, a man who has cast a critical eye on many a declarer play, brought to my attention to the following hand presented by Kantar in a 1981 book entitled Test Your Declarer Play Volume 2. Tim asks whether Kantar’s suggested line is always the correct one.

♠ A2	♠ Q3	1♦	2♣
♥ J65	♥ QT72	2♦	2♥
♦ AKJT98	♦ 2	2NT	3NT
♣ 62	♣ AKJT43	Lead ♠5

The spade lead is not surprising. West puts up the ♠Q losing the ♠K. Kantar’s suggestion is to play off the AK in the longer minor, clubs, to see if the ♣Q falls. If so, declarer has 9 tricks to cash. If the ♣Q doesn’t fall, declarer is in the dummy to take the finesse in the shorter minor, diamonds, and may still come to 9 tricks that way.

Mathematically we can express this idea in the following equation:

P1   + (1-P1 ) x P2 > P3,

where P1 is the probability the drop play will succeed, (1-P1) is the probability it will not succeed, P2 is the probability the subsequent finesse will produce 6 tricks, and P3 is the probability that a double finesse in clubs will succeed.

The A Priori Odds

The probabilities depend on what one assumes about the distribution of the sides. Let’s calculate the odds for the case of maximum uncertainty which is represented by the a priori odds.

The A Priori Odds of the NS Club Splits

Split	4 – 1	3 – 2	2 – 3	1 – 4	Total
Probability (%)	14.3	33.9	33.9	14.3
P1 Component	2.8	13.6	13.6	2.8	33.8
P3 Component	11.3	20.3	13.6	2.8	48.0

The A Priori Odds of the NS Diamond Splits

Split	3 – 3	4 – 2	5 – 1	Total
Probability (%)	35.5	24.4	7.3
P2 Component	17.7	8.1	1.2	27.0

Under these conditions:   P1 + (1-P1) x P2   =  52%, and P3 = 48%.

Clearly it is better to take the 2 chances as described rather than stake everything on the double finesse in clubs.  We conclude that Kantar’s line is better by 4%, but only if the a priori odds can be applied with sufficient accuracy. An essential characteristic of the priori odds is that the numbers of vacant places are equally divided. What happens if there are more vacant places in the North, which will tend to favor the club finesse and be detrimental to the diamond finesse?

When South Preempts

Let’s go through the mathematics when South enters the auction with a 2♠ preempt. West still plays in 3NT for the lack of a better alternative. The play to the first trick is the same, but now declarer must take into account that the spades are split 3-6. The a priori odds no longer apply, instead the club splits have the following probabilities.

The A Posterior Odds of the NS Club Splits*

Split	4 – 1	3 – 2	2 – 3	1 – 4	Total
Probability (%)	23.8	40.7	25.5	5.7
P1 Component	4.8	16.3	10.2	1.1	32.4
P3 Component	19.0	24.4	10.2	1.1	54.7

*from J.P.Roudinesco’s The Dictionary of Suit Combinations

The odds of the success of the double finesse in clubs has improved to over 50% as the ♣Q is more likely to be in the North hand in the ratio of 10 to 7. The same is true of the ♦Q, so we expect as lesser probability for the success of the diamond finesse.

The A Posteriori Odds of the NS Diamond Splits*

Split	3 – 3	4 – 2	5 – 1	Total
Probability (%)	33.9	12.7	1.7
P2 Component	17.0	4.2	0.3	21.5

* from J.P.Roudinesco’s The Dictionary of Suit Combinations

Based on these odds, the chance of success of a second round finesse for the ♣Q is 55%. The chance of the alternative procedure is 47%, so the odds now favour the double finesse by a substantial margin. The one chance in the club suit is significantly better than the combination of 2 chances, the drop in clubs and the subsequent finesse in diamonds.

The Effect of the Opening Lead

If the only information declarer has concerning the distribution of the defenders’ sides comes from the opening lead, the situation is uncertain. As there are 9 spades in the opponents’ hands, the suit cannot split evenly. In Kantar’s text the lead was the ♠5 from a 5-card suit, so the imbalance in the spade suit made the double finesse in clubs an even worse proposition than the a priori odds indicate. The normal lead is in the unbid suit, so it is possible that North led from a 4-card suit, in which case the probability of success of the double club finesse is 50.8% and that of the 2 chances, 51.4%. Thus, whether the imbalance of 1 vacant place is in the North or the South, it is better to play for the drop in clubs.

Some would argue that playing to drop the ♣Q is a safety play of sorts which does not risk immediate defeat on this particular hand. If the difference in the probabilities of success is a few percentage points, consistently choosing the lower percentage play may not entail a significant cost over the short term. However, in the case of a 3-6 spade split, the margin of superiority of the double finesse is substantial, so declarer should believe the bidding. He won’t be far wrong even if the spades are split 4-5.

For some the lesson is ‘don’t put all your eggs in one basket.’ For others, ‘always pursue the line that offers the greatest chance of success, given what one knows at the time of decision.’