Ok, I'll word it differently.
Denote as x the probability that partner has 98xx,
and as y the probability that partner has 9x (including 98), and
as z the probability that partner has 8x (but not 98).
3z=2y, as here there are two 8x doubletons and three 9x doubletons.
I am ignoring the possibility of 98x.
I then denote p the probability that declarer plays the
K from S Kxx.
When you have an agreement to alway play the 9 from 98xx,
o when partner plays the 9, you have all of x and p of y (but no z)
o when partner plays the 8, you have z
When you have no agreement on whether partner will
play the 8 or the 9 from 98xx
o when partner plays the 9, you have half of x, and p of y (but no z)
o when partner plays the 8, you have half of x, and p of z (but no y)
So, lets take fake numbers. K vs. xxxx a priori is one
of ten 4-1 splits, 0.02826. Each of those doubletons
a priori is one of 20 3-2 splits, each 0.0339.
Then, lets say you judge that from Kxx in spades, this
particular declarer is very clueless, and
would play the K from S Kxx one third of the time, p = 1/3.
When you have an agreement to alway play the 9 from 98xx
o when partner plays the 9, 0.02826 vs. 1/3*3x0.0339; 46.01% for 98xx
o when partner plays the 8, 0% for 98xx
When you have no agreement on whether partner will
play the 8 or the 9 from 98xx
o when partner plays the 9,
0.5*0.02826 vs. 1/3*3x0.0339, 29.42% for 98xx
o when partner plays the 8,
0.5*0.02826 vs. 1/3*2x0.0339, 38.47% for 98xx
Thomas