In Bridge Play, What is Meant by “Combining Your Chances?”

Question

I was in a NT contract. After the initial rounds of play, the last few (visible) cards were something like these in two suits (call them spades and hearts):

dummy: spades KJx Hearts xxx

Me: Spades Ax Hearts AKQx

I had five "top" tricks, but needed six to make game. I could play out AKQ and hope that the hearts were split 3-3, a 36% chance. Or I could take the spade finesse, a 50% chance.

I took the spade finesse and lost. A passerby commented that I would have been "odds on" (better than 50-50), if I had "combined my chances." Apparently, I needed to start with hearts, though. Then he left.

What did he mean by "combining my chances"? And why would I start with the lesser (36%) chance?

Answer 1

Great question!

Normally, you have a few chances to make your contract (like in your example, hearts 3-3 or spade finesse). Combining chances mean timing the play in such an order such that the failure of any chance to materialize still allows you to take the next chance. Even though you might start with a lower chance, it allows you to take another shot rather than ending there.

If you do the math, this comes out to 1 - (1-c1)(1-c2)...(1-cn) where the chances of success of each chance are c1, c2, ..., cn and assuming the chances are independent. This is larger than any single chance (or a strict subset of chances).

Example 1

You are in 3NT. LHO leads a spade.

xxx
AKQx
Jxx
JTx


A
xxx
AKxxx
AKxx

You have to win this (top hand is dummy, bottom is declarer and suit order is spades, hearts, diamonds, clubs from top to bottom).

You have 8 top tricks and need just one more without losing the lead.

Now you have three major chances:

• hearts 3-3
• diamond Q dropping in two rounds (diamond finesse is a practice finesse).
• club finesse

and a minor club Q dropping singleton.

You test for club Q singlton, no luck.

Now the correct play is to test diamonds first, then test hearts and then take club finesse.

Notice that if diamond Q does not drop in 2 rounds, you can still test hearts. If the hearts don't break 3-3, you can still take the club finesse: thus combining all your chances.

Imagine if you had taken the club finesse and it loses. Now you are down, without getting to test the diamonds or hearts.

Note that you have to test diamonds first. If you test hearts first, and then diamonds, then you are in the wrong hand and cannot take the club finesse!

---

Here is another example, where you have to take a finesse first.

Example 2

You are in 6S. LHO leads a trump.

KQxxx
AQ
AQ
AKxx



AJTxxxx
xx
x
xxx

You have 11 winners and need one more. Now your major chances are

• heart finesse
• diamond finesse
• clubs 3-3

The correct order of play is to try the diamond finesse first.

If that loses, you can now discard a club from hand on the diamond A, play AK clubs and ruff a club, thus testing for clubs 3-3. If that fails, you fall back on the heart finesse.

Note that you cannot time the play in any other order. If you test clubs first, and it fails, you are now committed to guessing which finesse to take.

If you take a heart finesse first, then you cannot test for clubs 3-3 and are forced to fall back on diamond finesse.

Hope that helps.

Answer 2

You have two ways of making your sixth trick given this ending. You can find the outstanding hearts divided 3-3, or you can find the spade Queen onside.

The beauty is that you can take both chances. First, play the hearts. If they break 3-3 (or somebody has done some lousy discarding in the build up to this end position), you're set for your sixth trick with your baby heart. If they do not break, you can then play the Ace of Spades (in case somebody has the singleton Queen), and finesse the Jack.

Your chance of success now are the 36% chance the hearts split, and the 50% of the time that the Spade Queen is onside in the case that the hearts do not break, or:

36% + (50%) * (64%) = 68%

Note that you have to start with the Hearts. Testing the hearts does not result in the loss of a trick, it only establishes a trick for the opponents if the suit does not break. Testing the Spade finesse will lose a trick if it is not correct.