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.