Playing The Odds

From a recent Wydaho online pairs game:

♠ A 8 4
♥ J 10 8
♦ Q 10 8
♣ A Q 9 7

Partner passes as dealer, then the auction proceeds 1 NT (15-17) on your right, all pass.
What do you lead? Since this is featured in a blog post, it must be some sort of "problem hand"; take an extra moment to think it over.

If you are like most players in that event, your fingers (or mouse) are hovering over the ♣ 7. Let's take a closer look at the ♣ 7 lead.

Whether the ♣ 7 is an effective lead depends on where the ♣ K is!

• If partner has the ♣ K, wonderful, you struck gold: At least three, possibly four club tricks for our side.
• If dummy has the ♣ K it will always win a trick, but the question is who has the ♣ J? If the king is in dummy and declarer has the jack, declarer will play low from dummy and win with the jack. That's not so good for you; declarer will score two club tricks, and your queen is in jeopardy. Ugh.
• If declarer has the ♣ K, ugh! The club lead lets declarer score a trick with the king, a trick that she cannot win without your help. Consider instead that if declarer has to lead clubs, or if partner leads clubs, your ace-queen will trap declarer's king; two club tricks for you, zero for declarer.

We don't know where the king is (without peeking, anyway); the best we can do is estimate what the chances are. Here's how:

There are 40 high card points in every deal: the AKQJ in each of four suits. The relative odds of any particular honor card in any particular hand roughly correspond to the relative distribution of high-card points in the four hands. (It gets more complicated if you know anything about the suit distributions.) 

Here you have 13 points, which leaves 27 points in the other three hands. Declarer has at least 15 points for her 1NT opening. So there is a 15 of 27 chance that declarer has the king; that's more than a 50-50 chance that the club lead will blow a trick (55.5% to be exact, but who wants to be exact with bad news like that).

But partner could have the king; what are the chances of that?  We don't know partner's point count, but we can make a rough estimate. There are at most 12 HCP outstanding (you have 13, declarer has 15-17). If dummy had 8 or 9 points, dummy would have bid over 1NT; so dummy is limited to 7 points. If partner had all the missing honors, partner might have bid over 1NT. So we guess to call it even, 6 points for dummy, 6 points for partner. A 6 in 27 chance is a bit less than 1 in 4, 22% to be exact. 

So, our chances on a club lead are:

• Partner has the king, 6 in 27 chances. 
• Dummy has the king, 6 in 27 chances. And if so, then 15 of 24 that declarer has the jack.(That works out to roughly 2 in 27 are break-even, 4 in 27 blow a trick).  Ugh. 
• Declarer has the king, 15 in 27 chances. Ugh.

If we lead a club, we have a 22% chance of something good, 8% of breaking even (king in dummy, partner or dummy has the jack), and a 70% chance of blowing a trick to declarer's king or jack.

PRINCIPLES:

• The distribution of high card points around the table is a rough estimate for the chances of any specific honor being in any particular hand.
• When most of the high cards are on your right, leading fourth-best from a broken honor holding is likely to lose a trick that you could win by just waiting.

Still want to lead the ♣ 7?  No, I didn't think so. Me neither. So what should we lead? Stay tuned...

Bonus Minute:  If you had a weaker hand, the chances of finding the ♣ K (or any other honor) in partner's hand are greater because partner will have a share of more missing high cards. Declarer will have a correspondingly lower chance of holding the ♣ K. Even so, the ace-queen holding is almost always dangerous to lead from when declarer has the strongest hand; with a 1NT opener, declarer will have at least 15 of 34 chances to hold the king. Figuring out why would be a good homework exercise...

-- Ray
Better Bridge in 5 Minutes. Guaranteed! (or the next one is free)