Combining Your Chances

The relationship between brothers is complicated.   My parents promised me that I would be as tall as my 6′ 2″ brother, Sam.   I topped out at a runty 5′ 9″ (in the morning) and have been looking for payback ever since.  I thought I had an opportunity for some payback when Sam joined us for our Thursday lesson series on various bidding conventions.  While we typically only bid the hands, Sam, sitting South, wanted to try playing the following diamond slam:


A note on the bidding:  The topic of that day’s lesson was Lebensohl over reverses.  North’s bid of 2NT over 2♥ showed in this case a weak hand, and demanded that South relay to 3♣, at which point North would typically sign off in South’s minor.  But South “breaks” the relay with his 3 bid showing 6-4 in diamonds and hearts and a maximum reverse.  This sets up a game force. North has perfect cards for slam try. Since the relay break sets up a game force, now North raises to 4 bid as a  slam try. South is more than happy to oblige and, after checking for aces, goes to 6, trusting his partner to have values to justify a slam try after trying to sign off using a lebensohl 2 NT. 

How would you play the hand after the lead of the ♠8, East playing the ♠K (promising the ♠KQ) under the ♠A?  Diamonds are not 4-0.  Click on the link below to see the answer.

I was happy to have Sam play the hand because I expected him to misplay it, allowing me to lord it over him once again.  He disappointed me, however.

The problem, Sam quickly assessed, was that in addition to the sure trump loser, there was a risk of a heart loser.  A 3-3 heart split is only a 36% chance.  If trumps break 2-2 (a 49% chance), the 4th heart can be ruffed in dummy, if necessary, but on a 3-1 break this may not be possible. 

On another day, there might have been the chance of setting up the 5th spade for a heart pitch (hoping that spades split 4-3); but the lead of the ♠8 (a high spot card in dummy’s suit) suggested that West might be short in the suit and looking for a ruff.  So Sam rightly stayed away from trying to set up the spade suit, and instead, proceeded to draw trump, leading low.  

East won the A tried cashing the ♠K.  Sam ruffed high to prevent an over-ruff, and tried drawing a 2nd trump, East showing out.  Diamonds were 3-1.  Now what?  It appears that he would have to rely on a 3-3 heart split – only a 33% chance.  Are there any other options?

To the unobservant, it looks like there is nothing more to do other than to finish drawing trumps and hope that hearts split 3-3, but my brother did not fall into that trap.  Instead, he stopped drawing trumps and played three rounds of hearts.  His reasoning was this – if hearts broke 3-3, there was no risk in playing them; but if hearts break 4-2, then he is going down no matter what unless the defender with the 3rd diamond had the 4th heart.  In that case, Sam is safe in playing all 3 hearts and then ruffing the 4th heart in the dummy.  

My brother was perspicacious enough to realize that, after the 3-1 trump break, he might as well play 3 rounds of hearts before drawing the last trump.  If hearts were 3-3 this would do no harm but leaving the 3rd trump in dummy gave him an extra chance in the case that hearts broke 4-2 since it was entirely possible that the hand with the 3rd trump also held the 4th heart.

So he played 3 rounds of trump, knowing that if he was ruffed on the 3rd heart, there was no way to make the hand.  But instead of ruffing, East pitched a club.  So the exact alternative distribution he hoped for had occurred.  These were all four hands.


I was in fact proud to concede “Well done, brother!”

On a less familial note, this hand is a very good illustration of the principal of combining your chances when playing a hand, a necessary feature of advanced card play.   There may be several different ways to play a hand and a clever declarer will do what she can to “combine chances” — in other words, play the hand in a manner so that the hand will make if any one of several ways of playing the hand works out.  In this hand, there were 4 distinct ways that he hand might make:

1.  If diamonds are 2-2, the hand always make (49% chance); but even if diamonds are not 2-2, then the hand still makes if …

2.  Hearts are 3-3, but even if hearts are not 3-3 (36% chance), the hand still makes if

3a.  Spades are 4-3 (62% chance) (note that declarer has sufficient board entries to ruff out and establish the 5th spade so long if diamonds are no worse than 1-3); or

3b.  The defender with 3 diamonds has 4 hearts (~35% chance), as was the case here.

Properly combining your chances makes this slightly under a 90% slam! 

Here, chance 3a is rejected immediately because the opening leads makes it appear that spades do not break.  But the proper order of play allowed my brother to combine chances 1, 2 and 3b, the critical play being to not draw dummy’s 3rd trump.

It is impossible to summarize all the different ways the declarer can combine chances.  But obviously, at the very beginning of the hand, the declarer must: a) identify ALL possible ways to play the hand; and then b) assess the probability of each succeeding, given the play of the cards so far; and finally c) determine the proper order of play to permit the declarer to try a second chance if the first one fails to pan out.  Not easy, for sure, but this is yet another reason why bridge is such a difficult, but fascinating, game.

— Tom Hunt



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