Solution to The Percentage Line- February 1999

Solution to The Percentage Line February 1999

Vul: You
Dlr: You
Scoring: IMPs

PARD (dummy)

10 6 5 4 2

A 4

K 7 4

Q J 6

It is clear that after you draw the trumps, you have a crossruff for 12 tricks. The

A is a sure loser, and so you need to find the best play to avoid two losers. After winning the

A, you lead a spade towards your king, and righty follows with the

Q. Your play.

If righty holds AQJ or if lefty holds AJ, it doesn't matter what card you play. The relevant combinations appear to be stiff A on your left or stiff J on your left, both of which are equally likely holdings. If this was the full analysis, then you would have an equal chance to play the right card, either the K or the 3.

However, there is one more relevant combination that should be considered: when lefty has stiff Q.

The correct analysis comes from what we call the Law of Restricted Choice. The typically referenced resticted choice situation is the following.

Contract: 6

Opening Lead:

YOU

K 9 8 7 3

K Q 8 3

A Q 10

Upon cashing the

A, lefty plays the

3 and righty plays the

J or the

Q. You next cross back to hand in diamonds and lead the

9 and lefty plays low. What card should you play from dummy?

The only relevant distributions are shown at right. The 4-1 divisions will happen approximately 2.8% of the time and the 3-2 division will happen 3.4%. It may seem like this means righty is more likely to hold a doubleton, because the honour he has played is known. This is not the case, however.

PARD

A K 10 8

LEFTY

J 5 4 3

Q 5 4 3

5 4 3

RIGHTY

Q J

YOU

9 7 6 2

On any given day righty holds the

QJ, he will play the

J half the time and the

Q the other half of the time. So, once we know the card righty played to the first round of spades, the percentages are 2.8% against (roughly) 1.7% to play righty for the stiff honour. The "choice" righty had with his doubleton lowered the probability by a factor of two.

That's The Law of Restricted Choice in a nutshell.

On this hand, righty played the

Q (or

J) on the first round of the suit. Using the Law of Restricted Choice, if he started with

QJ, then he had two choices of play, both leaving you with the same problem. With doubleton ace, he had no choice but to play his honour.

This means that you have (roughly) 2 to 1 odds if you play the K to this trick. Who ever knew restricted choice would be so simple!

PARD

10 6 5 4 2

LEFTY

RIGHTY

A Q

A J

Q J

YOU

K 9 8 7 3

This situation originally came up in actual play against Shiela Dickie and then against Dan Watson. Each time, I was defending holding the AQ of trumps in a known 10 card fit and played the queen in the hopes that partner held the jack. Each time, partner held the jack and declarer played the king. After some analysis, Dan Watson and I concluded that this was restricted choice.

Larry Meyer submitted a perfect restricted choice analysis. (Is Larry ever wrong?) Richard Stattelman told me he wanted to play the K, but didn't give me a reason why. Randy Corn said that nobody plays the queen from AQ, which I felt was a good reason for playing low.

This Month's Winners

Dan Watson (Vancouver, British Columbia)
Larry Meyer (Vancouver, British Columbia)

Honourable Mention

Richard Stattelman
Randy Corn (Seattle, Washington)

Congratulations to everyone who submitted solutions!

The Percentage Line - a quiz show by Brad Bart.
Please send your comments to bbart@cs.sfu.ca