Lesson 4B   Main

Going with the Odds

  by Richard Pavlicek

The title of this lesson often frightens people as it may seem to be a study in mathematics. Not really; you don’t need an electronic calculator to be a good bridge player. All you need to know are a few “easy numbers” and what to expect about suit breaks. The rest is mostly common sense.

Finessing Odds

Most finessing plays require that a particular card be located favorably, i.e., held by a specific opponent. Assuming the enemy hands are unknown, this is clearly a 50-50 chance — half the time it will succeed; half the time it will fail.

Sometimes a line of play will depend on two or more finesses so it is important to know how to figure the chances. If one finesse is a 50-percent chance, then the chance of two finesses both working (or both failing) is simply 50% × 50%.

Two finesses will both succeed only 25 percent of the time.

At least one of two finesses will succeed 75 percent of the time.

1. S A 10 9
H 5 4
D A Q 6 5 4
C J 5 3
S K 6 5 4
H Q J 8 2
D J 8
C 10 8 7
Table S Q 8 7 3
H K 10 6 3
D K 10 9
C 4 2
3 NT South
Lead: H 2
S J 2
H A 9 7
D 7 3 2
C A K Q 9 6

Declarer has eight top tricks. The lead indicates a 4-4 heart division so declarer can afford to give up the lead once. The diamond finesse is a 50-percent chance while the double finesse in spades offers a 75-percent chance — all you need is for West to have either spade honor.

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Suit Breaks

As declarer you must know what to expect regarding the division of the outstanding cards in a suit. The following table lists the most likely and next most likely breaks of 2-8 missing cards:

CardsMost likelyNext likely
21-1 52%2-0 48%
32-1 78%3-0 22%
43-1 50%2-2 41%
53-2 68%4-1 28%
64-2 48%3-3 36%
74-3 62%5-2 31%
85-3 47%4-4 33%

It is not so important to memorize the percentages, but it is crucial to know which breaks are most likely. Here is a neat memory aid:

An odd number of cards usually break as evenly as possible.

An even number of cards usually do not break evenly, except in the close case of two cards.

2. S K 3 2
H A K 7 3 2
D A K 4
C A 3
S 9 7 5
H Q 10 9 5
D Q 10 5 3
C J 6
Table S Q J 10 8
H J 8 4
D J 9
C Q 10 9 4
3 NT South
Lead: D 3
S A 6 4
H 6
D 8 7 6 2
C K 8 7 5 2

Declarer must decide whether to establish the hearts or the clubs. The club suit requires a 3-3 break (note the lack of entries to succeed against a 4-2 break); the heart suit requires a 4-3 break. Win the first trick and lead the H 2. Later you will set up the long heart.

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Finesse or Suit Break

A common situation for declarer is having to choose between two plays — one involving a finesse, and the other involving a suit break. In most cases all you need to know is whether the required suit break is a favorite. If so, then it has to be better than a 50% finesse; you don’t really care how much better.

I’ll have you know I memorized all the percentages.

Great. Now we can lose with a little dignity!

3. S A Q 8 6 4
H J 9 4 2
D A 6 5
C 4
S 10 7 2
H 5
D Q J 10 2
C K J 9 7 6
Table S K J 9 5
H 7 3
D K 8 4
C Q 10 5 3
6 H South
Lead: D Q
S 3
H A K Q 10 8 6
D 9 7 3
C A 8 2

After winning the D A and leading a heart to the ace, declarer can be sure of 11 tricks — six hearts, three side aces and two club ruffs. The 12th trick might come from the spade finesse or by attempting to establish the long spade. Which play is better?

The spade finesse is clearly a 50-percent chance. Establishing the long spade requires a 4-3 spade break (62 percent) and four entries to dummy (three to ruff spades and one to reach the last winner). Are the entries available? Yes, declarer can use the S A, H J and two club ruffs in that order.

The recommended play is actually even better. Declarer will also succeed against 5-2 breaks when the king is doubleton. For math buffs this extra chance can be calculated as 31 percent (the probability of a 5-2 break) times 2/7 (the ratio for the 5-to-2 odds against the king doubleton) or about 9 percent. This raises the total chance of establishing a spade trick to 71 percent.

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Multiple Chances

A particular line of play will often have more than one chance of success. To estimate the total chances, you need to understand the method of combining percentages. No, you won’t need a calculator. What you need is a logical mind.

Here is an example to illustrate how to choose the better play when multiple chances are involved.

4. S 4 3
H K J 10 5 2
D K J 5
C A K 5
S 9 2
H A Q 7 4
D 10 9 7 2
C J 10 9
Table S 10 5
H 9 8 6 3
D Q 8
C Q 8 7 6 3
6 S South
Lead: C J
S A K Q J 8 7 6
D A 6 4 3
C 4 2

Declarer has 11 top tricks, and the 12th must come from either hearts or diamonds. Dummy’s heart honors offer a straight 75-percent play by taking two ruffing finesses — if East has either the H A or the H Q, you will succeed. The question is whether this is better than playing on diamonds.

Looking at the diamond suit in isolation, you can make an extra trick any time the finesse works (50 percent). If the finesse loses, you still succeed if diamonds are 3-3 (36 percent × 50 percent) which comes to 68 percent. This appears to fall short of the 75-percent play in hearts.

But wait! If the diamond finesse loses and diamonds do not split 3-3, you still have a chance that the person with the long diamonds also has the H A — he will be squeezed. Estimating this to be about 45 percent of the remaining 32 percent, adds another 14 percent, which brings the total to 82 percent. Hence, playing on diamonds is the better play.

Win the C K and lead the H J (just in case you get a friendly cover) but ruff it and draw trumps. Next lead a diamond to the jack and queen. Win the club return, ruff a heart and lead all your trumps. West is squeezed in the red suits.

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Enemy Information

The basic odds and percentages assume that nothing is known about the enemy hands. In many cases there will be enemy bidding to reveal suit lengths and indicate the likely location of high cards. This information takes priority over all the normal percentages.

If one opponent has doubled or bid (excluding weak bids), he is more likely to have any missing ace or king.

The probabilities of suit breaks in one suit are affected by the layout of another suit (if known).

If an opponent is shorter in one suit, he is likely to be longer in another; and vice versa.

The opponent with the greater length in a suit is more likely to hold a specific card in that suit.

5. S A
H A 9 7 3
D J 8 7 5
C A K J 2
3 S
4 H
S K Q 9 8 7 6 4
H 2
D Q 10 6
C 4 3
Table S 5 3 2
H Q 10 4
D A K 9
C Q 10 9 5
4 H South
Lead: S K
S J 10
H K J 8 6 5
D 4 3 2
C 8 7 6

The normal play with nine cards missing the queen is to play for the drop, but declarer takes exception because of West’s preempt. West is known to have extreme length in spades so the chance of West having short hearts is increased. The correct play in trumps is to win the ace then finesse against East. Next declarer should lead diamonds because if that suit breaks 3-3, he won’t need the club finesse.

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Restricted Choice

A principle in probability theory states that, if a card played by an opponent may have been played by choice or necessity, it is more likely to have been played by necessity. That is, the opponent’s choice of plays is more likely to be restricted than a free selection.

Are you confused? Most people are, but the principle has proved to be sound and it is easy to apply at the bridge table. Here is how:

If you are missing two touching cards and an opponent drops one of them when you lead the suit, the odds are he does not have the other card.

6. S A 4
H K 8 3
D A Q 6 3
C A Q 7 4
S K 10
H Q J 10 6 4
D 10 8
C 8 6 5 2
Table S J 5 3
H A 9 7
D J 9 5 4
C K 10 3
4 S South
Lead: H Q
S Q 9 8 7 6 2
H 5 2
D K 7 2
C J 9

You ruff the third round of hearts and play a spade to North’s ace. You intended to lead a spade toward the queen, but the fall of West’s 10 creates the option to finesse the nine. Which play is better?

You were missing touching cards (jack and 10) so when West plays one of them the rule of restricted choice suggests that he is less likely to have the other. Therefore, it is better to play East for the jack and finesse the nine.

This hand has another interesting point. Suppose West wins the S K and shifts to a club. Should you take the finesse? Or go up with the ace and hope for a 3-3 diamond break? The finesse (50%) seems better, but this overlooks an extra chance. Refusing the finesse also gains when the player with long diamonds has the C K — a squeeze play that works as the cards lie.

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© 1990 Richard Pavlicek