Main Study 7Z75 by Richard Pavlicek

How Percentages Change

One of the perplexing areas in bridge theory is how the probability percentages of various suit divisions and honor locations change as the play progresses. Percentages calculated at the start of a deal — called a priori percentages — are not set in stone. As you garner information from trick to trick, percentages change — sometimes drastically. Exactly how much they change is often difficult to determine, not only because of subjective factors but because of misconceptions in probability theory.

This study is from declarer’s perspective. To simplify matters, I will assume there are no inferences from the bidding. In real life this is rarely true because even a lack of enemy bidding conveys some information, such as ruling out wild distributions. Nonetheless, I want to focus on how percentages change with card play, so bidding deductions will be ignored.

For example, suppose you have a straight finesse for a king. Before any cards are played, this is obviously a 50-percent chance; but as play continues it may become greater or smaller depending on what you learn about the enemy distribution. If you subsequently learn that East has four cards in the suit and West has only one, the finesse then becomes 80 percent (4-to-1 odds in your favor) if taken through East.

Top Main

Kinds of Information

During the play, information you obtain about the enemy distribution is of two basic kinds: (1) extracted information, which is extremely valuable and helpful, and (2) volunteered information, which is suspect and often useless.

Extracted information typically occurs when an opponent shows out of a suit, thus revealing an exact layout. The information about the suit led (not the suit discarded) then becomes a proven fact and can be used in full to determine subsequent odds and percentages. Defenders may try to deceive you, but they cannot hide the inability to follow suit.

Volunteered information is that which an opponent chooses to tell you, such as the cards played in following suit, and the choice of leads and discards. This information is tainted. Any meaning you might attach to it could be the deliberate plot of a defender trying to fool you. An experienced declarer learns to be skeptical of anything that comes too easily. It’s like listening to a sales pitch — you may get some facts now and then, but there’s a lot of bull in between.

Volunteered information in the form of the opening lead, while still suspect to some extent, is more reliable than a lead chosen after dummy is in view, or a discard. In general, declarer does well to take the opening lead at face value. At least it may help you from becoming paranoid if you believe something.

Top Main

Play this Slam

To illustrate how the percentages change, consider the play of the following 6

contract. (Thanks to David Burn of London, England, who posted this problem on the rec.games.bridge newsgroup.)

Q J 6 2
A 10 9 8 7
J
A 8 2
Table
A K 9 8 4 3
Q 6 3
A 8 4
5

West leads the 10, East plays the king, and you win the ace. You have nine top tricks, and it will be easy to ruff two diamonds for 11. Therefore, all you need is a second heart trick, and you consider the play of that card combination. Alas, there are six reasonable ways to play hearts:

A. Win the ace then let the 10 ride
B. Win the ace then lead to the queen
C. Run the 10; if it loses to the jack, run the queen
D. Lead to the queen; if it loses, finesse the 10
E. Finesse the 10; if it loses to the jack, run the queen
F. Run the queen; if it loses, finesse the 10

Quite a decision! In order to find the best play, let’s see how the missing hearts can be distributed. The following table shows the 16 possible cases and the percentage chance of each. Immediately below is a comparison of Options A-F.

A Priori Percentages
# West East Percent # West East Percent
1 KJxxx — 1.9565 9 Jxxx K 2.8261
2 KJxx x 8.4783 10 Jxx Kx 10.1739
3 KJx xx 10.1739 11 Jx Kxx 10.1739
4 KJ xxx 3.3913 12 J Kxxx 2.8261
5 Kxxx J 2.8261 13 xxx KJ 3.3913
6 Kxx Jx 10.1739 14 xx KJx 10.1739
7 Kx Jxx 10.1739 15 x KJxx 8.4783
8 K Jxxx 2.8261 16 — KJxxx 1.9565

A Priori Percentages
#	West	East	Percent	#	West	East	Percent
1	KJxxx	—	1.9565	9	Jxxx	K	2.8261
2	KJxx	x	8.4783	10	Jxx	Kx	10.1739
3	KJx	xx	10.1739	11	Jx	Kxx	10.1739
4	KJ	xxx	3.3913	12	J	Kxxx	2.8261
5	Kxxx	J	2.8261	13	xxx	KJ	3.3913
6	Kxx	Jx	10.1739	14	xx	KJx	10.1739
7	Kx	Jxx	10.1739	15	x	KJxx	8.4783
8	K	Jxxx	2.8261	16	—	KJxxx	1.9565

Option Winning Cases Percent
A 4-10, 12-16 69.2174
B 4-6, 8-16 69.2174
C 1-9, 13-16 76.8261
D 1-5, 9-16 76.8261
E 1-12, 16 77.9565
F 1-12 76.0000

Option	Winning Cases	Percent
A	4-10, 12-16	69.2174
B	4-6, 8-16	69.2174
C	1-9, 13-16	76.8261
D	1-5, 9-16	76.8261
E	1-12, 16	77.9565
F	1-12	76.0000

The total percent of each option is found by adding the percentages of each case in which the option succeeds. Option E (finesse the 10, then run the queen) is evidently best. Note that its edge over the near-equivalent Option F occurs in Case 16 when West is void — hence, you will know to change horses, whereas leading the queen allows no recovery.

It should be apparent why Options A and B are equal. Basically, it’s just a guess whether to play East for the king or jack, so the winning and losing cases balance out. Similarly, for Options C and D. In practice, however, it is usually better to choose A over B, and C over D, since there is some chance that if East has the king he will hop with it. Admittedly, the chance of such misdefense is close to zero against an expert, but it grows considerably against weaker defenders.

Top Main

An Inside Look

The previous a priori percentages are interesting and offer good general advice, but in actual play they are almost never accurate. When was the last time you needed to play a card combination before the opening lead? It’s like aiming a bow and arrow at a distant moving target; by the time the arrow gets there, the target will be gone. In this case you will always have more information before you play hearts, so the original percentages will not apply.

Even if you extract no information about clubs or diamonds, you will certainly know the exact trump division; and even if they break 2-1, one defender will have more space for hearts. In that event the changes will be slight, so Option E still rates to be best.*

*Actually, if East has the shorter spades, Options C, D and E become exactly even in theory. Given the equal choice, the edge would go to Option C because of the slight extra chance that East might misdefend and hop with the king.

Now consider a more extreme scenario, which is what actually occurred in the problem deal. While drawing trumps and ruffing diamonds, East shows out on the third diamond, pitching a club. It also happens that West shows out on the second trump, pitching a diamond. You now have extracted information in two suits. West is known to have begun with one spade and seven diamonds, and East with two spades and two diamonds. Hence, you know eight of West’s cards, leaving five unknown spaces; and four of East’s cards, leaving nine unknown spaces.

What about East’s club discard on the third diamond? Doesn’t this account for a known card in East? No! This information was volunteered and has no bearing on the 5:9 proportion of unknown cards between West and East. Essentially, the only information given by the club discard is that East is not void in clubs, but you knew that anyway. He would never pitch a heart in view of dummy, so the club pitch means nothing.

Given the 5:9 ratio of unknown cards, let’s recalculate the percentages:

Percentages for 5:9 Space Ratio
# West East Percent # West East Percent
1 KJxxx — 0.0500 9 Jxxx K 0.4496
2 KJxx x 1.3487 10 Jxx Kx 5.3946
3 KJx xx 5.3946 11 Jx Kxx 12.5874
4 KJ xxx 4.1958 12 J Kxxx 6.2937
5 Kxxx J 0.4496 13 xxx KJ 1.7982
6 Kxx Jx 5.3946 14 xx KJx 12.5874
7 Kx Jxx 12.5874 15 x KJxx 18.8811
8 K Jxxx 6.2937 16 — KJxxx 6.2937

Percentages for 5:9 Space Ratio
#	West	East	Percent	#	West	East	Percent
1	KJxxx	—	0.0500	9	Jxxx	K	0.4496
2	KJxx	x	1.3487	10	Jxx	Kx	5.3946
3	KJx	xx	5.3946	11	Jx	Kxx	12.5874
4	KJ	xxx	4.1958	12	J	Kxxx	6.2937
5	Kxxx	J	0.4496	13	xxx	KJ	1.7982
6	Kxx	Jx	5.3946	14	xx	KJx	12.5874
7	Kx	Jxx	12.5874	15	x	KJxx	18.8811
8	K	Jxxx	6.2937	16	—	KJxxx	6.2937

Option Winning Cases Percent
A 4-10, 12-16 80.6194
B 4-6, 8-16 80.6194
C 1-9, 13-16 75.7243
D 1-5, 9-16 75.7243
E 1-12, 16 66.7333
F 1-12 60.4396

Option	Winning Cases	Percent
A	4-10, 12-16	80.6194
B	4-6, 8-16	80.6194
C	1-9, 13-16	75.7243
D	1-5, 9-16	75.7243
E	1-12, 16	66.7333
F	1-12	60.4396

Wow! A complete turnaround from worst to first. Options A and B now tie for best. This is obvious if you think about it: East is likely to have longer hearts; so finessing through East is more likely to work. As suggested before, the edge would go to Option A because of the slim extra chance that East may hop with the king in Case 11.

Top Main

Looking Deeper

Since you took the effort to ruff two diamonds, why not ruff two clubs as well? This might reveal even more information. Indeed, if West showed out of clubs, this would give a complete count of the distribution.

Suppose both opponents follow suit when you ruff two clubs. The obvious deduction is that, with three more cards of each player known, the unknown space ratio becomes 2:6 instead of 5:9. Indeed, West cannot have more than two hearts, so half of the original heart distributions are now impossible. Based on the 2:6 space ratio, the percentages are calculated again below. (I kept the same case numbering as before.)

Percentages for 2:6 Space Ratio
# West East Percent # West East Percent
4 KJ xxx 3.5714 12 J Kxxx 10.7143
7 Kx Jxx 10.7143 14 xx KJx 10.7143
8 K Jxxx 10.7143 15 x KJxx 32.1429
11 Jx Kxx 10.7143 16 — KJxxx 10.7143

Percentages for 2:6 Space Ratio
#	West	East	Percent	#	West	East	Percent
4	KJ	xxx	3.5714	12	J	Kxxx	10.7143
7	Kx	Jxx	10.7143	14	xx	KJx	10.7143
8	K	Jxxx	10.7143	15	x	KJxx	32.1429
11	Jx	Kxx	10.7143	16	—	KJxxx	10.7143

Option Winning Cases Percent
A 4, 7-8, 12, 14-16 89.2857
B 4, 8, 11-12, 14-16 89.2857
C 4, 7-8, 14-16 78.5714
D 4, 11-12, 14-16 78.5714
E 4, 7-8, 11-12, 16 57.1429
F 4, 7-8, 11-12 46.4286

Option	Winning Cases	Percent
A	4, 7-8, 12, 14-16	89.2857
B	4, 8, 11-12, 14-16	89.2857
C	4, 7-8, 14-16	78.5714
D	4, 11-12, 14-16	78.5714
E	4, 7-8, 11-12, 16	57.1429
F	4, 7-8, 11-12	46.4286

Comparing this with the 5:9 table shows the options are in the same order, but the rich got richer and the poor got poorer. Option A is now close to 90 percent. Does this seem a little suspicious to you? If so, read on.

Top Main

A Compromise

I do not believe the above percentages are true because of the way the information was obtained. What actually happened for you to adopt a new percentage table? Not much. All you did was lead clubs a few times and both opponents followed suit, which was more or less expected. A show-out would have been newsworthy, but it didn’t happen. Evidently, this is just another application of the principle of extracted-vs-volunteered information.

Are the percentages from the 5:9 table still true then? No! They could hardly be right since half of the cases are now impossible. The true percentages can be found only by eliminating the impossible holdings (from the 5:9 table) and increasing the percentage of each remaining holding proportionally to obtain a 100-percent total. Since I seem to be chart-crazy, let’s look at one more set of tables with these calculations made.

Adjusted Percentages for 5:9 Space Ratio
# West East Percent # West East Percent
4 KJ xxx 5.2632 12 J Kxxx 7.8947
7 Kx Jxx 15.7895 14 xx KJx 15.7895
8 K Jxxx 7.8947 15 x KJxx 23.6842
11 Jx Kxx 15.7895 16 — KJxxx 7.8947

Adjusted Percentages for 5:9 Space Ratio
#	West	East	Percent	#	West	East	Percent
4	KJ	xxx	5.2632	12	J	Kxxx	7.8947
7	Kx	Jxx	15.7895	14	xx	KJx	15.7895
8	K	Jxxx	7.8947	15	x	KJxx	23.6842
11	Jx	Kxx	15.7895	16	—	KJxxx	7.8947

Option Winning Cases Percent
A 4, 7-8, 12, 14-16 84.2105
B 4, 8, 11-12, 14-16 84.2105
C 4, 7-8, 14-16 76.3158
D 4, 11-12, 14-16 76.3158
E 4, 7-8, 11-12, 16 60.5263
F 4, 7-8, 11-12 52.6316

Option	Winning Cases	Percent
A	4, 7-8, 12, 14-16	84.2105
B	4, 8, 11-12, 14-16	84.2105
C	4, 7-8, 14-16	76.3158
D	4, 11-12, 14-16	76.3158
E	4, 7-8, 11-12, 16	60.5263
F	4, 7-8, 11-12	52.6316

As seen above, the total percent chance of each option now falls about midway between those from the 5:9 table and those from the 2:6 table. Even if you don’t believe the ideas expressed here, you might agree that this compromise feels right.

The point to stress is that, in the absence of extracted information, percentages determined at a previous trick remain true as long as all the previous possibilities exist. Once a specific case becomes impossible, it is knocked off the chart, and the percentages of the remaining cases increase proportionally.

Top Main

A Trivial Analogy

As further evidence that an opponent following suit is volunteered information, suppose there are only two cards left when you play this simple holding:

A Q
Table
3 2

Also assume you previously learned that each opponent began with four cards in this suit, and each remains with two. Obviously, your chance of the finesse winning is 50 percent. Now suppose you lead the two and West follows low. Does this affect your chances?

One might conjecture that after West plays, he has only one unknown card; while East has two unknown cards. Therefore, the missing king is more likely to be with East by 2-to-1 odds. Surely, this is nonsense because West’s play was volunteered. The low card was his choice, not yours. Since West would always play the lower card from K-x, the information is meaningless, and your chances are still 50 percent. In other words, percentages do not change when the only new information is an opponent following suit.

Top Main