When is 50% better than 50% or is this percent evaluation erroneous?

[ggriffin0]

Getting right down to it. There appears to be only two options, what is the best line and why?

The Hand

Opener:

AJ1098753
J8
K83


Responder

K86
AQ73
95
AK62

Sequence

All White
4 -p -6 -all pass

*some bidding issues but this is what we get...

LHO leads a helpful 4 ...your line?

[kenberg]

There was a typo, I kept speaking of dropping a Q, I meant dropping the K. There is no Q to drop. We want to drop a K and cash a Q.  I hope I have fixed all of these. Thanks Todd.

Although we try to take the lead into account to see if we should modify our calculations, it is not always easy to do.  Would Lho have led the A if he had it? Maybe. Maybe not. If he had a diamond sequence, QJT, he surely would have led that. Or a club sequence. But I don't see that any of this leads to any good conclusion.

So let's forget inference from the lead, at least for the moment, and see where this gets us. Can we improve our chances beyond just guessing which finesse to take? Yes, I think so.

We have 11 tricks. We can up this to 12 with a successful finesse. We can also up it to 12 if we can bring down the heart K by ruffs. Let's trust that hearts are no worse than 5-2.

T1  Discard a heart on the club, win in hand.
T2 A
T3 3, ruffed with the J. Assuming that Rho started with at least 2 hearts we are ok, and maybe he cannot overruff even if he did start with 1.
T4 A
T5 2 to the K.
T6. 7 ruffed
T7 small back to hand.


Ok. Now we see what has happened. If the K has fallen we cash the Q and the other high throwing two and claim 12 tricks. If the K has not fallen, we cash the remaining throwing a and we lead a small toward the K8, making 12 if the A is on our left.

So now we calculate percentages, again ignoring any inferences from the lead. A reasonable approach is to calculate the probability of failure. We fail if the K fails to come down and if the A is wrong. These are roughly independent events, so we calculate the probability of each of them and then multiply.



Hearts will be 4-3  62% of the time. The K will be in the long hand 4/7 of the time. 4/7 of 62% is about 35%.
Hearts will be 5-2 31% of the time. The K will be in the long hand 5/7 of the time. 5/7 of 31% is about 22%
That still leaves 7% for a split of 6-1 or 7-0. On those the K is probably not coming down so let's put that all into the failure column.
So that's 35+22+7 =62 % that the K is not coming down.  Maybe only 61% if we are more careful, but that's good enough/

The finesse fails 50% of the time.

So we multiply 0.50 times 0.62 and get 0.31. 
So this plan, bring down the K or, failing that, take the finesse, at first glance has a failure rate of about 31%.

Actually, it is a little worse. This is because if hearts are 7-0 my A gets ruffed and if hearts are 6-1 the second heart might get overruffed. These things would happen before I ever get to the finesse. But they are quite unlikely.


So, if I were to do this with more care, my guess is that the failure rate would come out to around 35% or 36% Maybe 37. Under 40 anyway. So the probability of success is better than 60%.

Now: Nobody, nobody that I play with or against, calculates this at the table. Of course not. But bringing down the K by ruffing has some decent chance. For example a person could think "Hmm. 4-3 splits are something like 60%, the K will be with the short hand somewhat less often than it will be in the long hand but still it has a decent chance of happening and if that fails I can still take the finesse."
That reasoning alone could suffice to guide the play.

But that guidance is only if we ignore whatever inference we want to draw from the non-lead of the A. Some opponents would always lead an A if they have it. If so, the probability of the finesse failing is 100%, not 50%. That question, what would this particular opponent do if he held the A,  is beyond the reach of probability calculations.

Which is why bridge is more fun than probability calculations.

[Masse24]

Good summary on the math, Ken.

(Small typo on referring to dropping the K.)

Though we end in with the same number, I go about it differently. You're the math dude--I am not--so correct me if I'm off here. 

I add the 4-3 and 5-2 and 6-1 splits with the K dropping. So . . .

4-3 split is 62%. Honor dropping is 3/7*.62=26%
5-2 split is 31%. Honor dropping is 2/7*.31=9%
6-1 split is ~7%. Honor dropping is 1/7*.07=1%

Roughly 36%. (Bear with my rounding as this was done in my head.)
So we take the failure rate of that calculation, which is 64%, and multiply it by the finesse for K calculation (50%) to arrive at 32%.
Adding the 36% to 32% gives us 68%.

None of this accounts for the bizarro 7-0 split or 6-1 with an overruff. As you stated, highly unlikely. Subtracting those percentages would still give us well over 60%. Attempting to calculate that at the table would hurt my head.

So combining chances by first attempting to drop the honor gives us a higher probability of success than the simple  finesse. Knowing that, though, should lead us to the line you chose.

[kenberg]

I will say a bit tomorrow. Too bushed right now. There is one point I want to look at.

[kenberg]

Mpre typos, corrected I hope.

Feeling better, second wind.  I had just been to the 8th grade graduation of twin grandsons.  I love them dearly but the event went on forever! I was in the 8th grade class of 1952 and I don't recall that we did anything. I have a nice picture of the 40 or so kids that graduated with me, taken during the lunch hour. This ceremony tonight seemed endless. Some post-ceremony wine has helped. It cures a lot.


Anyway. Yes, you are correct. Let me do a pie chart for those who perhaps are mystified.


Let's agree for the moment that we just forget about the two kickers:  A being ruffed (the 7-0 split) or the second heart being ruffed and over-ruffed (6-1 split, stiff  on the right, and Rho holding the trump Q). These are both unlikely but at the end we can acknowledge that a slight adjustment is needed. So we first just ignore this, as both you and I  already did.

So now we go with your 36% chance of success, and corresponding 64% chance of failure, in dropping the K.


We construct a pie chart.

Take a circle,  draw a sector, a piece of pie, that takes 36% of the whole pie. So it's a little more than a third of the whole pie.  Color it red. That's the "drop the K sector"


The remaining 64% of the pie is the "failed to drop the K" sector.  Divide it in half, as if we were giving equal shares to each of the twin grandsons. So each of these halves is 32% of the pie.


One of these is the "failed to drop the K but the   finesse worked" sector, the other is the "failed to drop the K and then the &^#* finesse also failed" sector. Color the first blue, the other purple.


Only the purple is bad. Red is good because the K dropped. Blue is good because the K did not drop but the finesse worked. Purple is bad because the K didn't drop and the finesse failed.

So, you can add, as you did. The red is 36%, the blue is 32% , the sum is 68%. That's the chance of success.

Or you can just look at the purple, as I did. That's 32%, and it's the chance of failure. 32% chance of failure is a 68% chance of success.

So the plan of "drop the K and if that doesn't work take the finesse" has approximately a 66% chance of success.

We have to downgrade it a bit because of the possibility of a ruff of the A or an over-ruff of the ruffed second . It would still be more than 60%, my guess is that it would be around 66% but I didn't bother to work it through. I also just, for a rough estimate, skipped over the possibility of dropping a singleton K when the hearts are 6-1. Of course 1/7 of the time the stiff is the K. Call it 2/3.


That's not bad. How to weigh it against the thinking "Lho did not cash the A so he does not have the A, so therefore I need to take the 50% heart finesse"?  I have no idea. Depends on your opponents, no doubt.

[kenberg]

Above we have mostly ignored inferences from the lead. But there are interesting possibilities.

As noted, if we assume that Lho would always lay down the A if he had it, then when he does not lay down the A he doesn't have it, so the diamond finesse is doomed.
So: If we accept that he would always lay down the A if he has it, then when he  doesn't lay down the A we should take the heart finesse.
Playing against this opponent, you win if he has the diamond A because he lays it down, and you win when the K is on your right unless they take the first two tricks.

Let's try the other extreme. Let's suppose Lho would never lay down the A when he holds it. Now he leads, as he did, the 4. Why not a spot instead?  A very tricky/daring/crazy defender might underlead  the A but let's assume that he would not. In this case we can apply something like Restricted Choice: If the A is one our right then, from Lho's point of view, the club and diamond situations are pretty much the same. He holds neither the A nor the K in either suit so he is equally likely to have a club holding that inspires a club lead or a diamond holding that inspires a diamond lead.  So a club lead and a diamond lead are equally likely from a Lho who lacks the A.  But of he has the A, the diamond lead is ruled out since we are assuming that he would neither lead nor underlead the A.  So: Without the A to the left, a club lead and a diamond lead are equally likely. With the diamond A to the left, the diamond lead is ruled out.The conclusion is that the club lead makes it more likely that the A is to the left.

This is all a little iffy, not nearly as clear-cut as the usual RC situations. We haven't considered that Lho might lead neither a club nor a diamond, and saying that he would never lay down the A if he had it seems like a stretch. But there is something to this thinking. Lho has heard the auction, he expects declarer's values to mostly be in spades so he can expect the lead of a side A to set up a K on the board. Moreover, he expects declarer to have shape, maybe a void.  So there are a lot of reasons for not laying down an A. Never!?  Hardly ever!

The strictly math part about the odds of dropping the K and then, failing that, taking the finesse can be done, even of Todd and I skipped over the unlikely dangers of the 7-0 and 6-1 splits, and even if I, not Todd,  was careless with the arithmetic.. But judging what conclusions we can draw from the the 4 is, imo, a lot more subtle.

[ggriffin0]

Yes indeed 50% is NOT 50%. It is not simply a matter of on or off We delved deep into probability and for any perplexed about Ken’s comment about events being relatively independent of one another. The fact is whatever the probability of a certain action, the actual instance is just that, an event.

As for the line we chose, if there was one more in dummy we could see a more dynamic line, by ducking the first and trumping it. As is, Ken’s line seems right.

The K is off and the A is “on” for what it matters.