Intrafinesse - this time with the eight

But what, I hear you ask apropos of my previous post, if the suit is A852 opposite J943, also to be played for one loser.  The odds are a bit different, because declarer can succeed against either 107 or 106 in front of the J9.

As before, let's suppose that when declarer leads from his Axxx, the next hand plays the king from K10 with probability p, similarly from Q10; he plays the 10 from 107 or 106 with probability q, and the king from KQ with probability half.

The critical holdings for second hand are KQ, K10, Q10, 107, 106, and 1076.

Declarer's options are:

a) LHO plays the king or queen.  (a0) declarer tries to pin the ten: for every six times there's a critical layout this succeeds 2p times. (a1) declarer tries to drop the other top honour: this succeeds 1 time.

b) LHO plays the ten, and the jack loses to the queen or king.  (b0) declarer finesses against the remaining honour: this succeeds 2q times.  (b1) declarer tries to drop the other honour: this succeeds 2(1-p) times.

c) LHO plays low, and the nine loses to the queen or king.  (c0) declarer tries to pin the ten: this succeeds 2(1-q) times.  (c1) declarer tries to drop the other honour: this succeeds 1 time.

So the combined successes per six critical layouts are:
a0b0c0: 2 + 2p
a0b0c1: 1 + 2p + 2q
a0b1c0: 4 - 2q
a0b1c1: 3
a1b0c0: 3
a1b0c1: 2 + 2q
a1b1c0: 5 - 2p - 2q
a1b1c1: 4 - 2p

If the defender adopts p = q = 0.5, each of these combinations succeeds three times.  If he adopts some other strategy, declarer can do better than that by making the appropriate choice.  If the declarer wants to save his energies for something other than divining the defender's habits, he can lock in three chances of success by adopting a0b1c1 or a1b0c0.

In my previous post, with the defence holding the eight, I recommended rising with K10 or Q10 at least half the time.  Since the defender cannot tell whether declarer has the eight, he should in practice rise exactly half the time.

How do you make this sort of choice?  It's easy to come up with ways to do it: my method is to look at the sum of dummy's spot cards in the suit and the board number; I rise if the sum is even.  But now I've told you all, I'll have to change it to something else.