Fortune Favours the Logician

It may be simple stuff for some, but it took a while for me to get the right answer for this question. So, penning it down.

Say you have this situation while declaring a hand.

♥ Axx
♣ Kxx

♥ Kxxx
♣ AJxx

You are playing a suit contract. You can afford to lose only one trick in these suits put together. There are sufficient trumps in dummy to ruff losers.

What’s the percentage play?

There are three lines that come across as logical ones:

Line 1: Combine the chances of finesse and 3-3 in clubs
Club Ace. Finesse Club J. If it fails, fall back to Clubs 3-3.  

Line 2: Combine the chances of finesse in clubs and 3-3 in hearts
Duck a heart. If hearts 3-3 you can throw club loser on 4th heart.

Line 3: Combine the chances of finesse and 3-3 in clubs but with a variation
K of C. A of C and club to J. Works when Q is with RHO, or C 3-3.

Initially, the three lines looked similar to me as both of them are combining chances of 3-3 in a suit and finesse.

And then there was a seed of doubt in mind, and I decided to actually calculate percentages by thinking what would happen if we had 100 deals rather than simply adding percentages.

Line 1:

Success percentages here would be:

  • 1.21 (Singleton Q falling offside)
  • 50 (Q onside)
  • 17.75 (Q offside but Clubs 3-3)

Total: 68.96%

Line 2:

This line had initially looked best to me because it seemed to combine the chances of two suits independently. It seemed I can combine 31% of 3-3 with 50% of finesse totally it to 81%. Also, it sounds logical when you say, “You can try hearts 3-3 first. If that doesn’t work you can fall back to finesse later too, as that chance is not gone”.

Let’s see what happens when we assume we have hundred deals.

35.5 deals would have hearts 3-3
64.5 deals would have hearts break bad

50 deals totally have club Q right
50 deals totally have club Q wrong

You try heart 3-3 first. It will work in 35.5 deals.
In 64.5 deals it will fail and you’d try club finesse. Now while we did have Club Q right in 50 deals originally, some would have gone in hearts 3-3 scenario.

So, in rest of the deals, there will be 0.5121*64.5 deals in which we’ll be able to catch club Q with finesse. i.e. 33.03
(0.5121 and not 0.5, as you would play Ace first and offside singleton Q may fall)

(So this was the realisation point. I can’t really add 35 with 50. That 50 is applicable to rest of 65 and not to whole 100).

Hence in total you’d get it right 35.5+33.03 i.e. 68.53% times.

I realised that I was wrong in adding up percentages of hearts 3-3 and club finesse. That’s because some scenarios of club finesse working are also covered under the scenarios where hearts are 3-3.

Line 3:

Success percentages here would be:

  • 1.21 (Singleton Q offside falling)
  • 8.1 (Doubleton Q offside falling)
  • 50 (Q onside)
  • 17.75 (Q offside and clubs 3-3)

Total: 77.06%
We have a winner!

The reason this line works better than others is that it also covers the chances of Qx falling from LHO, without compromising any other chances. That’s a significant improvement.

Also, heart suit is really a mirage. One can combine these chances better in one suit. When we try hearts 3-3 first and it doesn’t work, we lose out the scenarios where hearts 3-3 didn’t work, but club finesse was working. When we try these all in one suit, we don’t lose any chances.

It was from a hand played by Uttam Gupta, one of the leading players of India. He took the right line, but for over a week, I was convinced in my mind that Uttam had taken non optimal line.

Then I did calculation, and my calculation proved me right.

And then I realised my mistake today, did calculation again, and this time it proved Uttam right.

It took me so many days to come to right conclusion, and Uttam did in a minute or maybe less.

Funny part is that in actual hand, QTxx of clubs was offside. And hence Uttam would have gone down.

But yours truly, the great defender sitting West, played C T on second club trick. 9 fell from East. Dummy hand 8. Sharp Uttam let the 8 run to my Q and his fourth club became winner.

Fortune favours the logician?

Note: I have considered only the priori probabilities. Depending on the way bidding and the game went till this point, of decision, will affect the probabilities. I’ll see if I can analyse those and update the probabilities with those factors considered.

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