Well there I was lying in the French sunshine listening to
the sound of the breaking Atlantic rollers surrounded by the beautiful people
when inevitably, as it does at times such as this, my mind turned to É.bridge.

What, I thought, is the
probability that a 5-card suit headed by AKQ opposite a small doubleton in hand
would break 3-3 to supply 5 tricks. Well actually itŐs pretty simple to
discover that the suit will break 3-3 about 36% of the time, about 48% of the
time it will break 4-2 and the rest of the time it will break 5-1 or 6-0. If
both defenders follow to the first two rounds of the suit with little cards
then we can eliminate the 5-1 and 6-0 breaks so we are left with 3-3 and 4-2 in
the ratio of 35:48 or about 11 to 15. The new % figures are 42% for the 3-3
break and 58% for the 4-2 break. So the chance of the
3-3 break has improved but still offers a much lower chance than a straight
finesse in another suit which offers a clear 50% chance of success.

S 83

H Q5432

D 752

C AQ3

S A2

H AK

D AK83

C JT852

Contract 3NT, small spade led.
There are 7 top tricks and there are chances for extra tricks in hearts and
clubs. You take the Ace of Spades (no point holding up) and both defenders
follow to two rounds of hearts. Now we are at the crossroads – do we
cross to the Ace of clubs and play the heart Queen hoping for a 3-3 break or do we chance the club finesse? If we get it wrong
the defenders will take at least four spade tricks sharpish.
Well on the basis of the above theory the club finesse gives a 50% chance and
the heart break is now about 42% so the finesse is a
much better shot.

Now letŐs strengthen the North
hand a little by replacing a small heart with the ten of that suit. They still
lead a spade and they both still follow to the first two rounds of hearts. Is
it again a 42% chance that the suit will break 3-3? Not so you may be surprised
to read. In the first example all the defendersŐ cards were insignificant in
that it didnŐt matter in what order they were played, they could false card at
will. But now the defenders hold a very significant card, namely the heart Jack
and neither would willingly play this card. As neither defender did play this
card then as well as eliminating the 5-1 and 6-0 distributions we can now also
eliminate those 4-2 distributions which include a
doubleton including the heart Jack. If you list all the possible doubletons (15)
you will see that one third of them contain the Jack and so we must now
eliminate a third of the original 48% probability of a 4-2 break, which is now
reduced to 32%. The probability of the 3-3 break is
unchanged at about 35% so the ratio is now 35:32 or just over 52% in favour of
the 3-3 break. So in our example hand the heart drop is now a better bet than
the club finesse.

Here is another example:

AT982

K3

Needing tricks in this suit you
start with the King and then lead the three up to the Ace. Incidentally thatŐs
much better than finessing the ten because an honour doubleton in the East hand
is more likely than a small doubleton. You can easily work this out for
yourself by writing down all the doubletons that East can hold and counting how
many contain either the Jack or the Queen. Both defenders follow with small
cards- what is the chance of the Queen and Jack now dropping together in a 3-3
break? As before we can eliminate the 5-1 and 6-0 distributions. We can now
also eliminate the 4-2 breaks that contain a doubleton honour (18/30) and the
3-3 breaks where one defender holds both honour cards (8/20). Having eliminated
18/30 of the 4-2 breaks we are left with 12/30 so the % is now the original 48%
x 12/30 = 19%. Likewise if we eliminate 8/20 of the 3-3 breaks we are left with
12/20 and the 3-3 % is now 35% x12/20 = 21%. So the ratio of 4-2 to 3-3 breaks
is 19:21 which is again just over 52% in favour of the
hearts dropping.

There is another way of looking at
this. Taking this suit in isolation, after both opponents have played to the
King and Ace the location of all the small cards is known, the only cards
missing are the Queen and the Jack. Suppose East now
plays the Queen on your lead of the ten. East has now played three spades and
has ten vacant places in his hand. At this point West
has played only two spades and has eleven vacant spaces. The odds are therefore
11:10 or 52.4% that West holds the remaining spade honour. The same result is
reached as in our previous calculation.

The reciprocal of all this is that
if a defender does show up with an honour on the second round then the 4-2
distribution remains a big favourite.

Nobody would be expected to work all
this out in real time at the table but it is easy to understand the principle
that when both defenders play insignificant cards on the first two rounds the
suit is now more likely to break 3-3 than 4-2. ItŐs marginal I know but if you
play with the odds all of the time you will win most of the time.