This is actually a very complicated problem (and a very popular one amongst bridge columnists!). The way I see it, the 2 lines of play go like this:

1) Play 2 rounds of diamonds, ending in dummy. This works if a) the diamond ace is doubleton, b) clubs are 3-3 or east has king-small or stiff king of clubs, or c) an opponent errs and takes the diamond ace too soon. This amounts, by my reckoning, to around 60% + the chance of the opponent making the indicated mistake.

2) Play to finesse clubs twice. This works if a) east has club king, b) west has club king, but east has diamond ace doubleton, c) west has club king and diamond ace doubleton and spades are 4-4, d) west has club king, east has diamond ace third but clubs are 3-3. I make this at around 70%.

So, questions: a) do I have the percentages (approximately) correct & b) if so, doesn’t the answer as to the correct line of play depend quite a bit on one’s estimate of the skill of the opponents?

Hi David. I also suspect Mr. Wolff oversimplified this. Space constraints? A key point is that when the club finesse loses , and a spade comes back, you’re out of stoppers. So you go down when left has the Ax diamonds. So line #1, in the column, doesn’t really “go after them both”, but rather goes after one and half 🙂

This relates to line 2, cases b & d in your analysis, which fails when the lead is from three spades. On the other hand, line 2 also works when the A of diamonds is singleton (good thing the 9’s there).

My numbers come out close to yours. Neglecting the “vacant spaces” considerations (tiny on this hand, as we know so little about the opponents distributions), I get 62.6% for line 1, and between 0.58 to 0.68 on line 2, depending on how likely you think the lead is from 3 (“x”) or from 4 (“y”). It’s roughly 0.58+0.1*(y-x). 4 card leads are much rarer than 5 card leads, so I think your bottom line is right on: it depends quite a bit on one’s estimate of the skill of the opponents.

Hi David and Amnon,

It is now 4:45AM in Las Vegas (PST) so it is obvious I need to get up very early in the morning to discuss bridge and especially percentage plays.

Between the two of you I would not dare try to impose my thoughts and will to dispute your calculations so I am left with only vague generalities. As both of you allude to, the skill of the opponents is important, and also necessary to be familiar with their count convention. After the 1st diamond is led to the 9, a clever declarer may be able to have a good idea of which opponent has the Ace and the number of diamonds (even or odd) his partner is signalling him.

As the three of us already know, the declarer has the strategical advantage of 1. seeing his 26 cards, 2. leading at his tempo, not theirs, and thus being able to glean which opponent has how many (It is quite often not signalled by (in this case) the ace holder, but rather almost always by the opponent who does not have the ace). At least to me, this advantage supersedes some exact percentage calculations and should definitely have an important bearing on the chosen line of play. Yes, David, the particular skill of the opponents and the liklihood of getting a false read very much enters the equation.

Also, yes, Amnon, oversimplification is often required when dealing with space considerations for writing the column, and that fact constantly occurs when my extremely valued co-writer and I debate what is necessary. This is not to say that we are not capable of awful mistakes, which unfortunately does happen, it is only to agree to Amnon’s perceptive comment.

Perhaps I should also mention, that in my very long career I have had the glorious opportunity to play against the world’s best and, at least for my money, Benito Garozzo of Italy is still the most difficult opponent to read, with some others fairly close behind, in being able to guess his cards, based on the bidding, opening lead, play up to the key point and tempo considerations. These electric moments when a contract was up for grabs in a hotly contested World Championship, will forever be an indelible memory.

Thanks for writing. Without your super expert comments, challenges and arithmetical calculations I, and all who are able to read and understand them, would be worse for not having done so.

I love the math. Barry Crane, when given a choice of leading from a four card or five card suit, required that teammates lead the four card suit. Against a weak player one could lead the Q of diamonds and overtake with the king. This makes for a pretty picture of “meat on the table.” This is a tempting offer to a defender who is dazzled by being able to win the king and queen with his ace. The reason that you take the line of play given by Bobby is because of the 9 of diamonds. This card gives you a second dummy entry, which allows you to wisely take full advantage of the wealth of your club J.