Posted 2015-January-21, 02:31
Let's start by assuming we only know about the ♦ suit. Assuming standard leads, mud, and 2nd best from 4 cards W possible diamonds holdings are
kt5432, k95432, kt543, kt542, k9543, k9542, kt54, k954, kt4, k94, k54 with k onside and
9432, t54, 943, 942, 543, 542, 43, 42 and 4 with k offside.
3 6/1 splits 2 on 1 off * 7%/14
6 5/2 splits 4 on 2 off * 17%/56
11 4/3 splits 5 on 6 off * 62%/70
1.00% 0.50%
1.26 0.63
4.43 5.31
__________
6.69% 6,44% 6.69+6.44= 13.13 6.69/13.13= 51.0%/49.0% in favor of finesse. 51/49=1.04
But we also know that W didn't choose a spade so we reduce our probabilities due to restricted choice. If W had Hxxx or Hxx in both Spades and diamonds he would choose either suit only 50%, similarly if he had only low cards in both suits he would choose either suit 50%.
Assuming the spades split 4-4 and only counting akqj as honors there are 32 cases has a 2 or more card sequence, 20 cases of a A with no K, 17 cases of Hxxx or other HHxx, and one txxx. Since W led the 4 of ♦ we deduce that he did not have a spade sequence, we must reduce the diamond % of Hxxx and Hxx leads by 50% * percentage of spade Hxx(x)(x), and the diamond % of x(x)(x)(x) by 50% the percentage of possible spade xxxx leads, It is presumed that the opponents would overcall if they had 5 cards or more with 10 hcps.
6.69% *( 1- 1/2 (17/50))= 5.55%
6.44% * (1- 1/2 (1/50)= 6.38%
finesse now on 5.55/11.93= 46.5+ or 47% So we would not take it. odds 46.5/ 53.5 = .87
Next we can compare the a priori odds based on the cards with our knowledge of our customers. If we know W is a good student of US Bridge World, we can deduce that he prefers Kxx(x)(x) lead to quack xx(x)(x) leads. If they are European we will go with the odds that they would follow Bird-Anthias, and avoid higher Hxx(x)(x) leads. Otherwise, we will weigh our bets based on probability the a priori case that he will not care.