I thought a little more about the percentages. I think my line works about 58% of the time.
Outline of the argument:
The
split 3-2 about 68% of the time, I will argue that when
are 3-2 my line works 65% of the time. Multiplying these percentages gives about 44%.
The
split 4-1 about 28% of the time. I will argue that when
are 4-1 then my line works 50% of the time. Multiplying these percentages gives 14%.
Adding the two cases gives 58%.
I think it is all correct.
Details:
in those cases where the
are split 3-2, it appears that my line works 65% of the time. Here is the reasoning:
I will assume that E never covers when he holds Hx or Hxx, if he does cover, that helps me. So assume he doesn't.I also assume that
are 4-3. And I call the spots x, y and z, their exact value does not affect my play.
There are twenty ways, all equally likely, for the suit to split 3-2. The plan is to go to the board in
and lead the
T. If it loses, I try to get back to the board with a
. If I can get back to the board, I again finesse in
. If the
is taken by the A, I will, when next in, cash the
A and hope.
Here are ten ways that I succeed no matter where the
A lies. I will give the E holding:
KQ
KQx
KQy
KQz
Kx
Ky
Kz
Qx
Qy
Qz
In the first four of these I don't care whether E covers or not. If he does I take the A and then play the J, if he ducks the T wins, I repeat the finesse, winning and then I play the J.
in the remaining six cases the T loses. Either I get back to take the finesse or if that fails I just lay down the A dropping East's remaining honor. So in these ten cases I make it, regardless of where the
A is.
Now there are six cases where I will bring in the contract if the
A is with W:
Kxy
Kyz
Kzx
Qxy
Qyz
Qzx
In these six cases I will make it providing I can get to the board with the
A. The T will lose to a doubleton honor. I get back in and lead a
toward the board. If the K holds I repeat the finesse and pick up the suit.
The first ten are ten out of 20 for 50%, the next six are six out of twenty which would be 30% but it only works when the
A is onside so I cut that in half to get 15%.
I add 50% and 15% to get 65%.
Note. I have listed 16 possible holdings for E. The other four are xy, yz, zx, xyz. That's the twenty possible holdings that I was referring to. For these four, my line fails.
Ok, this is 65% of the 3-2 cases and I guess the
are 3-2 only 68 % of the time. That's about 44%. But we still have to look at the 4-1 splits. We get 4-1 spits 28% of the time. There are ten ways to split the cards 4-1. Here are ways for that to work out, again I give the E holding. Note that it's good that we have the
8 in our hand. If the T is covered we win and play the J establishing the 9 and 8, if the T is not covered we lead again toward the hand etc.
K
Q
KQxy
KQyz
KQzx
That's five. So when the cards are 4-1 I think half the time we make it. Again we don't care where the
A is.
So that's another 14%.
So I get:
I think this makes about 44% plus 14% for 58%.
Added: I mentioned that when E holds xyz, so W holds KQ tight, I will go down. But I might not. The T will lose to an honor and then, if I can get back to the board, i will repeat the finesse and it will lose. Think about it. This means that when E is holding
xyz and also the
A, he should not take the
K with his A. If he does, he leaves me with no option but to lay down the
A and I am delighted to see the fall of the other honor. He must duck the
K and then, after I take the second finesse, losing, then W cashes his
and leads a
to his partner's A. Since I doubt this duck would occur to most E players, this slightly, very slightly, ups the chances for success.