The Paradox of Percentages

This hand from the weekly pairs session provoked a bit of lively discussion between Mike Seaver and myself.

First, the bidding:

West	North	East	South
Pass	1♠	Pass	1NT(1)
Pass	Pass(2)	Pass

(1) In our system two over one responses are forcing to 2NT, so are usually based on a minimum of eleven HCP

(2) Conservative. Partner figures to hold at most two spades most of the time and ♥KQ may well not be pulling their full weight

A non-forcing 1NT response to one of a major opening is a definite weakness in Acol-type systems. On a (very) good day, you avoid getting too high with misfits - on a bad day, you play in a totally unsuitable contract. I think on my actual hand, passing was too unilateral and it would have been better to rebid 2♣ to keep my options open. Of course, if we would then have stayed out of 5♣ is  different question......

Anyway, how should Mike have played the hand on the lead of a the diamond deuce (fourth best) to the queen and ace?

In isolation the best way of playing this combination, according to the percentages, is to finesse twice, so he led the ♣10 and West played low.

Mike felt that the correct play now is to win with ♣A and lead a second club, making the contract whenever clubs were evenly split and also for those 3-1 breaks when the singleton was an honour.

I didn't agree: I favoured an initial finesse;

(1) If the ten holds but East follows suit, you can follow with a second round to the ace and then a third round of the suit. West will win and play a heart which you must win in hand, overtaking one of dummy's honours. You can then cash your three club winners, discarding dummy's small spades and play a diamond (or a spade), generating a ninth trick.

(2) But if the ten loses, say to the queen, the defence will come back a heart or a diamond and you lack the two entries necessary both to take the second club finesse and enjoy the long clubs.  You therefore have no choice but to cash the club ace and if clubs were originally 3-1, making only eight tricks. If the clubs started 2-2, you make your contract easily.

(3) The third possibility is that the ten holds but East shows out. You follow with a second club and West plays low. You can win with the knave and play a diamond on the (fairly safe) assumption that each opponent started with four. The opponents can generate a second diamond trick (by playing back a diamond) or a club trick (by playing a club) but not both.- you thus have time to generate two spade tricks before the defence have five tricks.

(4) The final option is again that the ten holds and East shows out. This time when you follow with a second club, West puts in a an honour card.and you win with the ace. You then follow with a third club and if West takes this, the hand reduces to hand (1) above. If  West ducks, again you have time to develop a second diamond trick and two spade tricks.

In summary, this line works when clubs are 4-0 onside, 3-1 when the singleton is not an honour, and all the two-two breaks. It therefore seems to gain approximately 5.0% of the time over the alternative of simply cashing the ♣A (but see below).

This got me thinking.

Under normal circumstances, finessing the first round - losing to an offside queen - and then playing for a 2-2 break after all is against the odds. The play of the queen affords an increased likelihood that the player does not hold the king-queen doubleton (this is the principle of restricted choice). Indeed, once East has produced the queen, the odds are two to one that he does not have the king.

However if you don't have the opportunity to lead a second round of the suit and see West follow, surely the odds favouring a finesse do not apply? When you are in dummy considering the likely division of the remaining two cards, the odds are just better than evens that cashing the ace will draw the remaining cards. It is only in crossing to hand and leading a club to take the finesse that the odds change.

One final thought: if you have determined to play the clubs by cashing the ace and then playing another club, are you not maximising your chances of an overtrick to lead a spade at trick two in case West has the spade ace and cannot afford to rise with it?

By the way, I may have won the argument, but I would have lost the war for the full deal was: