[cherdano]

As a mathematician, it is easy to answer this question:
Impossible problem without specifying what LHO would do with QTx...

[cherdano]

T was played on the left, not on the right. I was asking because when LHO has QTx, then there is still the diamond extra chance, but only if we go for the drop. So it does matter what LHO plays from QTx...

If we assume that RHO randomizes his spot cards, then we can actually ignore them in the calculations.

The odds in the heart suit favor the finesse strongly. You pick up 3 original holdings (Tx) instead of one (QT).
[But this strongly comes from the assumption that LHO always plays T from Tx.]

Now when you play for the drop, you get the extra chance that the opponent with Q-3rd has exactly four diamonds. When you play for the finesse, you get the same extra chance with Q-4th onside.

Q-3rd are 6 cases (Tx or QTx with LHO), so you have a net gain of 5 cases with the extra chance. However, if we assume that LHO randomizes from QTx, we should count them only with 50%, or in other words a net gain of 3.5 such cases.
Ok, then in each of these cases, the likelyhood to get the actually needed 4-1 diamond break is 11.3% (from Pavlicek's suit break calculator). So we gain less than half a case (3.5 * 0.113), and it's not even close to making up for the disadvantage in the heart suit itself.

It all gets much closer if we assume that LHO treats the T as a spot card and randomizes. In other words, if we assume the trump suits is Ax opposite KJT9xx. I am pretty sure the odds will still favor the finesse, but I may do the math in another post.

[EricK]

Given LHO has a doubleton then there is a 75% chance it is Tx and a 25% chance it is QT (as there is one Q and 3x's).

[brianshark]

I would hazard a guess and say that because the T means nothing about the location of the Q, that the odds remain unchanged from the pure odds which are approximately 51%-drop, 50% finesse.

http://www.automaton.../en/OddsTbl.htm

Table: (apologies for the layout)
No W E Probability Times Total
1 - QTxx 4.783 1 4.783
2 x QTx 6.217 2 12.435
3 xx QT 6.783 1 6.783
4 T Qxx 6.217 1 6.217
5 Tx Qx 6.783 2 13.565
6 Txx Q 6.217 1 6.217
7 Q Txx 6.217 1 6.217
8 Qx Tx 6.783 2 13.565
9 Qxx T 6.217 1 6.217
10 QT xx 6.783 1 6.783
11 QTx x 6.217 2 12.435
12 QTxx - 4.783 1 4.783

1, 2, 3, 6, 7, 8 9 and 12 are now impossible after the first round of the suit, based on your assumptions.


That leaves:
No W E Probability Times Total
4 T Qxx 6.217 1 6.217
5 Tx Qx 6.783 2 13.565
10 QT xx 6.783 1 6.783
11 QTx x 6.217 2 12.435

(Total points = 38.000)

With layout 5, you will make and 11 you will lose, regardless of the play, so...

No W E Probability Times Total
4 T Qxx 6.217 1 6.217
10 QT xx 6.783 1 6.783

So that leaves odds of 6.783 : 6.217 in favour of playing for the drop.

Also note that for finessing, 4+5 = 19.782 / 38 = ~52%
And for dropping, 5+10 = 20.348 / 38 = ~53.5%

That seems a little off. I'm not sure why the finesse isn't still 50%. Maybe I'm missing something. Perhaps it's a weighting issue regarding case 11, because as was said earlier, we don't know the frequency LHO would play T from QTx. However, I'm convinced it only changes the overall probability of succes, not the relative probabilities between the 2 relevant lines.

EDIT: Ingore, I have the wrong number of missing cards so the analysis is redundant.

[FrancesHinden]

The RHOs and LHOs seem to have got confused between various of your postings.

I think we are playing Ax in dummy opposite KJ9xxx in hand where we started with low to the Ace, and LHO played the 10. We assume that RHO plays his spots at random, and that LHO

- always plays the 10 from 10x
- never plays the 10 from Q10
- plays the 10 50% of the time from Q10x

Then the a priori chance that he has these various holdings is:

10: 2.83%
10x: 10.2%
Q10: 3.39%
Q10x: 5.1% (half of the total chance that he started with this)

If he has Q10x or singleton 10 we also make if he has 4 diamonds with Q10x, or 1 diamond with singleton 10. The second of these is wildly unlikely (although you don't give us the auction). The first of these is just over 11%.

So the finesse is right 10.2% of the time plus a tiny amount of 2.8%
The drop is right 3.4% of the time plus a another 0.6% or so

(or in summary: what cherdano said, and what EricK said more concisely)

[this is a way more interesting problem than as stated here because I don't think that given strategy for LHO is best: question 2, assuming LHO knows the layout, although not in time to lead a spade, what is LHO's best carding strategy?]

[FrancesHinden]

brianshark, on Jan 29 2007, 09:12 AM, said:

I would hazard a guess and say that because the T means nothing about the location of the Q, that the odds remain unchanged from the pure odds which are approximately 51%-drop, 50% finesse.

You only have an 8-card fit, not a 9-card fit.

[brianshark]

Oops, I thought it was 6-3...

[Halo]

EDIT : Post removed - read the problem wrong!

[Halo]

Back to the fray:

I guess the issue here is what it means to say the ten is not significant. If it isn't significant, then it is just an x and we can't 'see' it being played on the first round.

Now RHO started with (relevant distributions):

Qxxx
Qxx
xxx
xx

And this time it is distinctly better to finesse, because there are more Qxx distributions than xxx, enough to outweigh the fact that there are more Qxx distributions in LHO with four diamonds than there are Qxxx in RHO with four diamonds.

[Edited 3rd last line to make sense - get it right sometime).

[cherdano]

Jlall, on Jan 29 2007, 05:37 PM, said:

You just need 3 diamonds to cash, not 4 diamonds

No, you need to pitch both the spade and the club loser on the diamonds.

[FrancesHinden]

Jlall, on Jan 29 2007, 03:37 PM, said:

You just need 3 diamonds to cash, not 4 diamonds

What are you doing with your club loser?

[FrancesHinden]

Jlall, on Jan 29 2007, 03:36 PM, said:

BTW I MEANT TO SAY LHO PLAYS THE T 100 % of the time from QTx SORRY. Essentially I meant to say LHO always plays the ten, and RHO always varies his spot play.

OK, well in that case

Q10x and 10x are equally likely on our left.
So are comparing the chance of Q10 doubleton OR Q10x with 4 diamonds against 10x which is

10.2% v. 3.4% + 11% of 10.2%. Still much better to finesse.

Change the hand to give declarer AKQ tripleton and it will be much much closer.

[kenrexford]

I'm curious as to the vulnerability and the auction. It seems that there may be some subtle psychological inferences at work here, and some additional mathematics might be done with this.

One simple and not so useful mathematics number is the location of the spade Ace. This seems to be odds-on to the right of Declarer, making the location of the heart Queen initially 13:12 behind the Jack. On all deals where both sides follow to the first heart, the Queen is off 12:11. On all deals where a second small heart is played by Declarer's RHO, the odds of the Queen being behind the Jack are now 12:10. This, of course, looks at the problem too simplistically.

However, some other assumptions may be fair, like why no bidding was done if no bidding was done, why no attempt at finding a void was tried, and the like.

[Halo]

Honesty requires a 3d/4d post from JLall.

[FrancesHinden]

kenrexford, on Jan 29 2007, 07:13 PM, said:

I'm curious as to the vulnerability and the auction. It seems
One simple and not so useful mathematics number is the location of the spade Ace. This seems to be odds-on to the right of Declarer

Why? People generally don't lead an ace in dummy's first bid suit against a slam.
The only possible inferences I can see from the auction/lead are that
- LHO didn't lead trumps which might make the finesse more likely wrong (depends if it's know we are off the HQ or not)
- LHO didn't lead a singleton diamond

[kenrexford]

Maybe, maybe not (Ace lead). It depends upon the auction, of course. However, if the auction suggested, say, a doubleton spade in Opener's hand (not likely, but possible), then we can rule out spades being 5-1 or 6-0. This gives RHO at least two spades. Each little piece changes the odds slightly.

Most "odds" analysis assumes that all patterns are possible. 7-0 in spades, either way, and the like. If you can eliminate out some holdings as no longer possible, or highly unlikely psychologically, then you can approach better "odds" analysis.

My point is that I dislike declarer play problems outside of the context of an auction. It is bad enough to lack info on table feel aspects, especially when the "odds" issue might be tight.

Normally, this is not particularly relevant. However, my gut leans me on this one to table feel and inference over pure mathematics. As a simple "proof" of this, playing for diamonds to be 4-1 begs the question of why LHO would opt against a diamond lead with a stiff.