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The Percentage Line.

Today’s story involves percentages, well at least if you want to understand why it is correct to take a rather unusual action in declarer play. Even if you are not into percentages, it’s worth looking at the final percentage as to why the action taken is the winning one. Well, it would have secured you a making slam this time and the situation could well arise in the future.

To tell the story, we welcome Auckland’s Patrick Carter, a fine director, player and  bridge analyst.

 

“You pick up a very interesting hand:

 

 

After South passes, your partner opens a strong 1NT (15-17) but then North bids 4, vulnerable. What do you do?

At the table I was watching, East took a very reasonable punt at 6. The lead was the 4 and partner put down this very agreeable dummy:

 

 

 

It certainly looks like we will just lose one spade trick and wrap up the slam. North wins the first trick with the A and plays another spade. Clearly, they must have 8 spades for the high pre-empt and the 4 lead is so small it almost screams singleton. So, obviously, you ruff high and South discards. What next?

 

You play a trump to the king as North shows out and you have to lose a trick to South’s J83, down 1. 

 Unlucky, you think, and then you see the hand record which says you can make the contract. Wow! You can make the slam if you run the 10 through the South hand on the first round of trumps. Who would think of that play? And is it a good play? How likely was it that South would have all 3 trumps?

Vacant Spaces

To answer that last question you need to understand the vacant spaces principle. We knew that North had 8 spades and South had 1 spade. That means that North had 5 vacant spaces in their hand for cards which were not spades and South had 12 vacant spaces.

Imagine dealing the hearts out one at a time. Here are 17 vacant spaces for the first heart. 12 of them belong to South and so there are 12 chances out of 17 that South will be dealt that first heart. Now we deal the second heart and if South got the first one now has 11 vacant spaces out of a total of 16. Give that one to South as well and when we deal the third heart, South has 10 vacant spaces out of a total of 15.

If you have followed all that, then you will realise that the chance of South getting all 3 hearts is 12/17 x 11/16 x 10/15. You get out your calculator and find that is about 32.3%. So, it sounds like your line of playing a heart to the king would be successful 67.7% of the time. Unlucky!

But how often will you be successful by leading the 10 and running it unless South shows out? (If North has all 3 hearts you can put up the king and finesse the 9 on the way back).

South will have 3 hearts 32.3% of the time 
South will have 2 hearts 48.5% of the time 
South will have 1 heart   17.7% of the time
South will be void in hearts 1.5% of the time
(If you want the method of calculation of these figures, an email or call to Patrick will provide the answers. Please contact rksolomon@xtra.co.nz for details)

On this new plan, by finessing, you will make whenever South has 0 or 3 hearts, but you will fail sometimes when South has 1 or 2.

Although it is a 48.5% chance South has 2 hearts, that means that North has either the singleton Jack, or the singleton 8, or the singleton 3. Only 16.2% of the time will North have specifically the singleton Jack.

When South has only 1 heart, then North has 2 cards so will hold the Jack 2/3 of the time. 2/3 of 17.7% is 11.8%

16.2% + 11.8% adds up to 28%

In other words running the 10 will be successful 72% of the time and is the better way to play the hand. 72% comes from

32.3%     South has all 3 hearts

32.7%     South has 2 hearts and North’s singleton is either 8 or 3

  5.9%   South has 1 heart, the J

  1.5%    South has no hearts

………

72.4%

This is a higher % than the 67.7% of playing a heart to the king.

 The full deal:

 

 

What a play that would have been! But nobody found it. Every single declarer from the East seat made 11 tricks, whether they were playing in game or slam. The only declarer to make 12 tricks in hearts played it from the West side after ace asking meant they bid the suit first. North led the A but decided not to continue the suit.”

 

Thank you, Patrick. It is not unusual to have to use a high trump to prevent a defender from scoring a smaller trump. You are then likely to have to take a finesse in the trump suit. The above argument explains why it is thus better to take an otherwise somewhat risky finesse.

If nothing else, the above deal proves there is so rarely the perfect dummy. KJ doubleton next time, partner, please!

No percentages tomorrow as we write for JIN Club members…and others!  

Richard Solomon

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