Main Analysis 7A47 by Richard Pavlicek
|3 NT South|| A 5|
K 8 4
10 9 5 3
A J 9 4
|Lead: 3|| K 6|
A Q 7
A J 8 4 2
K 6 3
A preliminary analysis reveals two viable plans: (1) Win the A and lead another diamond, then fall back on the club finesse if diamonds do not behave, or (2) Win the A, then if no honor drops, win the A, K and lead a club toward the J-9. Either plan is strong and will succeed a great majority of the time. But which is better?
Regardless of the plan chosen, declarer should win the K and cash two top hearts (ace and king) in case there is a revealing show-out, then lead the 10 from dummy to the ace. Assuming everyone follows suit and no diamond honor drops, you are now at the crossroads: Continue diamonds? Or switch to clubs?
Problems like this are difficult to analyze, because the probabilities of the division of any particular suit are dependent on the distribution of other suits, a virtual merry-go-round. The best way to get an accurate assessment is to consider hand patterns that fit the scenario. First, lets assume spades are 5-4 (either way), not guaranteed because West could have led from three, but practical for analysis. From winners cashed, hearts must be 4-3 or 5-2 (either way) and diamonds 2-2 or 3-1 (either way). Further, Wests choice of a spade lead should preclude holding a longer suit. Based on these conditions, my Hand Pattern Analyzer lists 18 possible distributions with the relative percent chance of each.
Hand Pattern Analysis for West-East 9=7=4=6
Known: Ws4-5 Wh2-5 Wd1-3 Ws)Wh Ws)Wc
Contracts compared Vulnerable: 3N 3N
18 Distributions Sorted by Frequency
|Weight||Tricks Won||West||East||Factors||Percent||Make Pct||Trick Avg||IMP Avg|
|1||100||B9||5323 2||4423 1||126 35 6 20||13.33||100||100||11.00||9.00||2.00||0|
|2||100||CC5 B94 B86||5422 3||4324 1||126 35 6 15||10.00||100||60.00||11.33||9.60||5.73||0|
|3||100||CC B92||4324 1||5422 3||126 35 6 15||10.00||100||100||11.33||10.00||1.33||0|
|4||100||BB2 A9 79||5413 4||4333 0||126 35 4 20||8.88||75.00||100||9.75||10.00||0.25||3.25|
|5||100||BB2 A9 79||4333 0||5413 4||126 35 4 20||8.88||75.00||100||9.75||10.00||0.25||3.25|
|6||100||CC5 BB10 AA 994 794 786||5332 2||4414 3||126 35 4 15||6.66||66.67||80.00||9.53||10.00||0||2.33|
|7||100||CC5 BB10 996 A94 7A5||5314 4||4432 1||126 35 4 15||6.66||83.33||100||9.97||10.33||0.13||2.17|
|8||50||B9||4423 1||5323 2||126 35 6 20||6.66||100||100||11.00||9.00||2.00||0|
|9||100||CC B92||5224 3||4522 3||126 21 6 15||6.00||100||100||11.33||10.00||1.33||0|
|10||100||BB2 A9 79||5233 2||4513 4||126 21 4 20||5.33||75.00||100||9.75||10.00||0.25||3.25|
|11||100||CC5 BB10 996 A94 7A5||4234 1||5512 6||126 21 4 15||4.00||83.33||100||9.97||10.33||0.13||2.17|
|12||50||CC5 BB10 995 794 786||4432 1||5314 4||126 35 4 15||3.33||66.67||80.00||9.50||9.97||0||2.33|
|13||50||CC5 BB10 996 A94 7A5||4414 3||5332 2||126 35 4 15||3.33||83.33||100||9.97||10.33||0.13||2.17|
|14||100||CC BB5 99 89 884||5431 4||4315 4||126 35 4 6||2.67||58.33||66.67||9.67||9.75||0||1.00|
|15||40||CC5 BB10 AA 994 794 786||5512 6||4234 1||126 21 4 15||1.60||66.67||80.00||9.53||10.00||0||2.33|
|16||100||CC BB5 994 AA A9||5215 6||4531 4||126 21 4 6||1.60||100||100||10.25||10.17||0.08||0|
|17||40||CC B9 BB4||5521 6||4225 3||126 21 6 6||0.96||100||100||11.17||10.83||0.33||0|
|18||40||BB 88||5530 7||4216 6||126 21 4 1||0.11||50.00||50.00||9.50||9.50||0||0|
|Known||18 cases Totals||2.31||2.34||3970764||100.00||86.86||92.74||10.43||9.81||1.27||1.35|
The Factors cell in each row, when multiplied out, shows the number of specific layouts for that case. For example, in Case 1 spades can be distributed 126 ways (9c5), hearts 35 ways (7c3), diamonds 6 ways (4c2) and clubs 20 ways (6c3) to produce 529,200 specific layouts.
An adjustment should be made, analogous to restricted choice, in cases where West has equal length (4-4 or 5-5) in the majors. On average he would probably select a spade lead only half the time, so the weight of these cases is halved (50 instead of 100). Also, when West is 5-5, I reduced the weight even further to 40, because he might have bid on some of those hands subjective of course, but the amount is surely more accurate than no adjustment at all. Weight adjustments are reflected in the Percent column and all totals.
To compare two plans, it is necessary to determine the tricks won by each plan in each case. Cases 1 and 8 are simple: Plan 1 always wins 11 tricks (diamonds 2-2) and Plan 2 always wins 9 (fourth club sets up). Tricks won by each plan are given respectively in hexadecimal (to ensure a single digit for both) followed by the portion to which it applies. A portion of 1 is presumed if no number is given.
Case 2 is not as easy. When West has Q-10 or Q-x (5/15 of the club layouts) both plans win 12 tricks, because Plan 2 can safely switch to diamonds when the Q drops, so the first entry is CC5. When West has 10-x (4/15 of the club layouts) Plan 1 wins 11 tricks, but Plan 2 wins only 9, so the second entry is B94. Finally, when West has x-x (6/15 of the club layouts) Plan 1 still wins 11 tricks, but Plan 2 fails (8 tricks) so the last entry is B86. Note that the sum of portions (5+4+6) equals the number of club layouts.
Case 6 is more difficult, because with diamonds 3-1 there are two equally likely occurrences: an honor drops, or it doesnt; so the 15 club portions must be considered twice, effectively 30 portions in all. For the 15 club layouts with a stiff diamond honor, both plans succeed, winning 12 tricks if the Q drops (5 layouts) or 11 tricks if it doesnt (10 layouts), so the first two entries are CC5 BB10. When no diamond honor drops, both plans win 10 tricks when West has Q-10 (1 layout) but only 9 tricks when West has Q-x (4 layouts) so enter AA 994. Now for the bad news: When West has 10-x (4 layouts), Plan 1 not only fails but is down two ( A is lost as E-W scarf up the rest), while Plan 2 survives (794); and when West has x-x (6 layouts) both plans fail but Plan 2 by only one (786). Check the sum of portions (5+10+1+4+4+6) to verify it equals 30.
Results clearly show Plan 2 to be better. The totals of the Make Pct column show that Plan 1 makes 3 NT 86.86 percent of the time, which is certainly good, but not as good as Plan 2 with 92.74 percent, almost 6 percent higher. The most meaningful column, however, is the IMP Avg which indicates the average number of IMPs won per board if every possible layout were played once. Note that these are IMPs won (not net IMPs) so the crux is the difference in the two averages, which shows that Plan 2 gains 0.08 IMPs per board.
The Trick Avg column tells a completely different story, which is pertinent to matchpoints. As might be expected when overtricks are crucial, Plan 1 is the big winner.
© 2016 Richard Pavlicek