The art of being lucky

(probability in bridge)
Matthew Kidd, 2009

“Dans les champs

de l'observation le
hasard ne favorise
que les esprits
préparés.”
- Louis Pasteur

“It’s better to be “Chance favors the

lucky than good.” prepared mind.”
 Hoping for a 3-2 break

♠432 ♠432

♠J87 ♠T9 ♠J987 ♠T

♠AKQ65 ♠AKQ65

(3-2 break) (4-1 break)

 What does a “3-2” break mean?
• Most of the time a 3-2 break means either
3-2 (LHO has 3, RHO has 2) or
2-3 (LHO has 2, RHO has 3).
• Sometimes it means exactly the case where LHO has 3
cards and RHO has 2 cards.
• Usually it is clear which, but not always (ask if confused).

CHO
Center
Handed
LHO Opponent RHO
Left Right
Handed Handed
Opponent Opponent
You
 The Wrong Way*
N choose K

Possible card combinations for one opponent Count

- (void) 1 = (5,0) Æ 3.125 %
J, T, 9, 8, 7 5 = (5,1) Æ 15.625 %
JT, J9, J8, J7, T9, T8, T7, 98, 97, 87 10 = (5,2) Æ 31.250 %
JT9, JT8, JT7, J98, J97, J87, T98, T97, T87, 987 10 = (5,3) Æ 31.250 %
JT98, JT97, JT87, J987, T987 5 = (5,4) Æ 15.625 %
JT987 1 = (5,5) Æ 3.125 %
32

Odds of 3-2 break would seem 1 1

to be 2 x 31.25% = 62.5% 1 2 1 Remember
1 3 3 1 Pascal’s
1 4 6 4 1 Triangle?
1 5 10 10 5 1
*but not horribly wrong
 What’s wrong?
• We are not merely flipping coins!
• There are cards in the other
suits, “spectator cards”.

Probability of holding a specific 2 or 3 card combination (e.g. JT or T87) >

Probability of holding a specific 1 or 4 card combination (e.g. T987 or 8) >
Probability of holding 0 or all 5 cards.
 The Correct Way
• The opponents hold 5 trump and 21 other
cards (2 x 13 – 5).
• Total number of LHO/RHO layouts is (26,13)

P(LHO has 3 trump) = (5,3) * (21,10) / (26,13)

= 33.91 %

Lesson: Odds of 3-2 break are actually 2 x 33.91 = 67.8% (5.3% higher)
 Comparison of methods
35

30

25
Percent

20

15

10
LHO: 5
5 RHO: 0

0
5-0 4-1 3-2 2-3 1-4 0-5

Lesson: The Bridge Gods smile more often than they frown.
Split probabilities for 2-7 outstanding cards
60 52.0
40 39.0 39.0
Percent

40
24.0 24.0
20 20
11.0 11.0

0 0
2-0 1-1 0-2 3-0 2-1 1-2 0-3

40.7 40 33.9 33.9

40
24.9 24.9
20 20 14.1 14.1
4.8 4.8 2.0 2.0
0 0
4-0 3-1 2-2 1-3 0-4 5-0 4-1 3-2 2-3 1-4 0-5
40
40 35.5 31.1 31.1
24.2 24.2
20 20 15.3 15.3
7.3 7.3
0.7 0.7 3.4 3.4
0.3 0.3
0 0
6-0 5-1 4-2 3-3 2-4 1-5 0-6 7-0 6-1 5-2 4-3 3-4 2-5 1-6 0-7

For even number of outstanding For odd number of outstanding

cards, second most favorable split is cards, most favorable split is most
most likely (except for 2 cards). likely.
 The finesse
(a 50-50 proposition)
♥AQ ♥AQ

♥K… ♥… ♥… ♥K…

♥xx ♥xx

(onside K) (offside K)

Vocabulary: declarer finesses the queen, finessing against the king.

 The double finesse
♥AQT ♥AQT
3 tricks – 25%
♥KJ… ♥… ♥K… ♥J…
2 tricks – 50%
♥xxx ♥xxx
1 trick – 25%
(both onside) (J offside)

♥AQT ♥AQT

♥… ♥KJ… ♥J… ♥K…

♥xxx ♥xxx

(both offside) (K offside)

Improve your chances with an endplay!
e d
nc
v a ic!
Ad Top

Lead this!
 Finesse or drop? (9-card fit)
Play the ♠A and then either:
♠KJ32
1) Lead to the ♠J
(playing for the finesse)

♠? ♠? 2) Lead to the ♠K
(playing for the drop)

Which line is best?

♠A7654

or
 Drop works when suit is 2-2 or Q is singleton
♠KJ32 ♠KJ32
♠KJ32
♠T98 ♠Q ♠Q ♠T98

♠T8 ♠Q9 ♠AJ654 ♠AJ654

24.9% x 1/4 24.9% x 1/4

♠A7654

40.7%

Total: 40.7% + 24.9% x ¼ + 24.9% x ¼ = 53.1%

 Split probabilities for 2-7 outstanding cards
60 52.0
40 39.0 39.0
Percent

40
24.0 24.0
20 20
11.0 11.0

0 0
2-0 1-1 0-2 3-0 2-1 1-2 0-3

40.7 40 33.9 33.9

40
24.9 24.9
20 20 14.1 14.1
4.8 4.8 2.0 2.0
0 0
4-0 3-1 2-2 1-3 0-4 5-0 4-1 3-2 2-3 1-4 0-5
40
40 35.5 31.1 31.1
24.2 24.2
20 20 15.3 15.3
7.3 7.3
0.7 0.7 3.4 3.4
0.3 0.3
0 0
6-0 5-1 4-2 3-3 2-4 1-5 0-6 7-0 6-1 5-2 4-3 3-4 2-5 1-6 0-7

For even number of outstanding For odd number of outstanding

cards, second most favorable split is cards, most favorable split is most
most likely (except for 2 cards). likely.
Finesse works when Q is onside & suit not 4-0
♠KJ32 ♠KJ32 ♠KJ32 ♠KJ32

♠Q ♠T98 ♠Qx ♠xx ♠Qxx ♠x ♠T98 ♠Q

♠A7654 ♠A7654 ♠A7654 ♠A7654

24.9% x 1/4 40.7% x 1/2 24.9% x 3/4 24.9% x 1/4

Total: 40.7% + 24.9% x ¼ + 24.9% x ¼ = 51.5%

With a 9-card fit, the drop (53.1%) slightly

beats the finesse (51.5%).
 Finesse or drop? (8-card fit)
♠KJ2 Play the ♠A and then either:
1) Lead to the ♠K
(playing for the drop)
♠? ♠? 2) Lead to the ♠J
(playing for the finesse)

♠A6543 Which line is best?

Drop works when Q is doubleton or Q is singleton

♠KJ2 ♠KJ2 ♠KJ2 ♠KJ2

♠xxx ♠Qx ♠Qx ♠xxx ♠Q ♠T987 ♠T987 ♠Q

♠A6543 ♠A6543 ♠A6543 ♠A6543

33.9% x 2/5 33.9% x 2/5 14.1% x 1/5 14.1% x 1/5

Total: 32.8%
 Finesse works when Q is onside (3-2 split) or Q singleton

♠KJ2 ♠KJ2 ♠KJ2 ♠KJ2

♠Qxx ♠xx ♠Qx ♠xxx ♠Q ♠T987 ♠T987 ♠Q

♠A6543 ♠A6543 ♠A6543 ♠A6543

33.9% x 3/5 33.9% x 2/5 14.1% x 1/5 14.1% x 1/5

Total: 41.0%

With an 8-card fit, the finesse (41.0%)

significantly beats the drop (32.8%).
 “8-Ever, 9-Never”

With an 8-card fit, the finesse (41.0%)

significantly beats the drop (32.8%).

With a 9-card fit, the drop (53.1%) slightly

beats the finesse (51.5%).

Q: Should you finesse? “Eight ever; nine never.”

“A peek is worth 2 finesses”
Conditional Probability
(or never say “never”)

Q: How to play clubs:

drop or finesse?

“Weak Two” overcall: 6

hearts and a poor hand
(5-11 hcp)
 Without the 2♥ preempt
• The opponents hold 4 trump and 22 other cards.
• Total number of LHO/RHO layouts is (26,13)
P(LHO has 2 trump) = (4,2) * (22,11) / (26,13) = 40.7 %

With the 2♥ preempt

• The opponents hold 4 trump, 9 hearts split 6-3, and 13 other
cards.
• Total number of LHO layouts is (9,6) * (17,7) =
(9,3) * (17,10) = Total number of RHO layouts
Remember: (9,6) = 9! / 6! (9-6)! = 9! / 6! 3! = 9! / (9-3)! 3! = (9,3); symmetric!

P(LHO has 2 trump | 6 hearts) =

(4,2) * (9,6) * (13,5) / (9,6) * (17,7) =
(4,2) * (13,5) / (17,7) = 39.7 %
 Shifting Probability
A priori probability After 2 ♥ bid
50 50

40.7 39.7
40 40
35.3

30 30
Percent

Percent
24.9 24.9

20 20
14.7

10 10 8.8
4.8 4.8
1.5
0 0
4-0 3-1 2-2 1-3 0-4 4-0 3-1 2-2 1-3 0-4

Note: 2-2 split is still quite likely. Preempts do not reduce the chance of a
favorable (trump) split nearly as much as intuition might suggest. This is
generally true. Do not live in fear!
 Recomputing the probability
♠A764 ♠A764 ♠A764 ♠A764
Finesse
total:
♠T98 ♠Q ♠xx ♠Qx ♠x ♠Qxx ♠Q ♠T98 58.8%

♠KJ532 ♠KJ532 ♠KJ532 ♠KJ532

14.7% x 1/4 39.7% x 1/2 35.3% x 3/4 35.3% x 1/4

♠A764 ♠A764 ♠A764

Drop
♠T8 ♠Q9 ♠Q ♠T98 ♠T98 ♠Q total:
52.2%

♠KJ532 ♠KJ532 ♠KJ532

39.7% 35.3% x 1/4 14.7% x 1/4

Different result! Now finesse has a 6.6% edge.

 Joint Probability
Say you need the following to make your contract:
1. Favorable 3-2 trump break.
2. At least two tricks from an AQT double finesse.

P(success) = 67.8% (3-2 break) * 75% (double finesse) = 50.9%

Warning: The above calculation is fairly accurate because #1 and #2 are nearly
independent probabilities. In some cases there are significant correlations, e.g. a
favorable break in one suit makes a favorable break in another more likely. In such
cases, a more careful calculation must be made if accuracy is desired.
 Do bridge players care about the odds?
• Some don’t; the better ones do.
• In most case it suffices to know the
best line of play, not exactly how
much better it is than the
alternatives.
• It is handy to know a few numbers
(e.g. chances of 2-3 and 3-3
breaks).
• Experience counts; good players
have an instinctive feel based on
encountering common situations
many times.
• Intuition is not always right (worried
about flying? Try driving if you
really want to risk death).
• Proving partner or teammates
wrong can be satisfying, even if it
requires several hours of
computation.
 More on bridge probability
• Bridge Odds for Practical Players (Hugh Kelsey)
• Dictionary of Suit Combinations, J.M. Roudinesco
• SuitPlay program
 Selected Bridge Websites
• American Contract Bridge League
http://www.acbl.org
• La Jolla Unit
http://lajollabridge.com
• Soledad Club (Mon Aft, Thu Eve games)
http://www.soledadclub.com/soledad-
bridge-club.htm
• Adventures in Bridge (games every day)
http://www.adventuresinbridge.com/
Example Matchpoint Result (Top = 17)
6N – best NS result
6♣/♦ – 2nd best NS result

3N+3
3N+2
5♣/♦+1

5♣/♦

4♣/♦+2