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By Fred Gitelman
Most of the mathematics in Bridge involve adding and subtracting numbers that are smaller than 13. There are some Bridge situations, however, in which knowing something about probability theory can really make a difference. I hope that the mention of probabilities will not scare you off. This is not advanced math -- most of the calculations of probabilities in Bridge involve no more than memorizing a few numbers.
Here are some sample deals that illustrate why a basic knowledge of Bridge odds can be helpful:
Dummy
A x
x x
x x x
x x x x x x
Declarer
x x
A Q x x
A K Q x
A K Q
You play in 3NT against the opening lead of a Spade. Think about how you would play the hand as declarer.
There are eight top tricks (the Spade and Heart Aces, three Diamond tricks, and only three Club tricks as that suit is blocked). The ninth trick could come from either a successful Heart finesse or a 3-3 break in Diamonds. Unfortunately, you cannot combine your chances in both suits. If you take a Heart finesse immediately and it loses, the defense will take enough Spade tricks to beat you. Alternatively, if you try for 3-3 Diamonds at once and that suit does not break, you do not have a second dummy entry to fall back on the Heart finesse. You must choose between the Heart finesse and the 3-3 Diamond split. Which will it be?
The only way to solve a problem like this one is by using probabilities. The probability of the Heart finesse succeeding is 50% as it is equally likely the King of Hearts will be dealt to either defender. The odds of a 3-3 Diamond split is not the type of number that is easy to figure out in your head. This is one of those numbers you should memorize ? the chances of a 3-3 break are about 36% (35.53% to be more exact).
Thus, you should take an early Heart finesse on this deal. The Heart finesse is a solid 50% chance while relying on Diamonds only succeeds around 1/3 of the time.
Here is a table that tells you the approximate percentages of various suit breaks when six cards are missing:
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Split
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Percentage
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6-0
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2%
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5-1
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14%
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4-2
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48%
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3-3
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36%
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So, if you were wondering why suits don't seem to split 3-3 for you very often, you can see that you should not be surprised.
Let's change the previous play problem a little:
Dummy
A x
x x
x x x x
x x x x x
Declarer
x x
A Q x x
A K Q x
A K Q
Once again you play 3NT on a Spade lead, but this time you have an eight-card Diamond fit (as opposed to the seven-card fit you had before). Now the alternative to the Heart finesse is to rely on a 3-2 Diamond split (rather than a 3-3 Diamond split).
Here is the table for the approximate chances of various suit splits when five cards are outstanding in a suit:
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Split
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Percentage
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5-0
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4%
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4-1
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28%
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3-2
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68%
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So a 3-2 Diamond break will occur about 68% of the time. You should therefore play on Diamonds instead of relying on the 50% Heart finesse. That extra small Diamond in the dummy makes a big difference as to how the hand should be played. If the Diamonds turn out to be 4-1 or 5-0 and the Heart finesse would have worked, you can at least have the solace that you played with the odds (I know, you would rather have the points for making 3NT!).
When the defense has two cards in a suit:
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Split
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Percentage
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2-0
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48%
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1-1
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52%
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This table implies that when you have an eleven-card fit missing the King, it is correct to play for the King to drop singleton (as opposed to finessing). The odds are very close, however, and it is often the case that clues from the bidding or the defense are enough to swing the balance in favor of finessing in this situation.
When the defense has three cards in a suit:
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Split
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Percentage
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3-0
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22%
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2-1
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78%
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We can use this information to decide how to play a combination like:
Dummy
A x x x x
Declarer
Q J 10 9 8
There are three cards missing including the King. The King will be singleton about 26% of the time (1/3 of the 78% for 2-1 breaks -- each of the three possible singletons are equally likely), so playing dummy's Ace on the first round will work this often. The finesse against the King, however, will work 50% of the time and is therefore the correct play by a margin of almost 2 to 1. Even experts often misplay this suit combination, for some reason.
When the defense has four cards in the suit:
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Split
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Percentage
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4-0
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10%
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3-1
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50%
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2-2
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40%
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Some players think that this table contradicts the “9 never” part of the famous maxim “8 ever, 9 never.”“9 never” states that with nine cards between declarer and dummy missing the Queen, the declarer should play to drop the Queen (play for the 2-2 break) rather than finesse against the Queen (play for the more likely 3-1 break).
“9 never” is in fact correct as can be seen from the following argument. Suppose that declarer faces this suit combination:
Dummy
A K J 10 9
Declarer
x x x x
The Ace (or King) is cashed as both defenders follow but the Queen does not appear. You cross back to your hand in another suit and lead a small card in this suit towards the dummy. Left Hand Opponent (LHO) follows with the remaining small card. “9 never” says to go up with dummy's honor, but my table suggests that a 3-1 split is more likely than a 2-2 split (indicating you should finesse).
The reason that “9-never” is correct is that at the time of the decision, many of the splits that were originally possible are known not to exist. In particular:
- We know that the Queen is not singleton (or it would have dropped on the first round).
- We know that the suit is not 4-0 (or one of the defenders would have discarded on the first round).
- We know that LHO was dealt neither a small singleton nor a doubleton Queen (as we would have known by his play on the second round).
In fact, there are only 2 possible distributions left
The odds of LHO being originally dealt a small doubleton can be computed as follows:
The suit will break 2-2 40% of the time and the Queen will be with RHO half of that time. Therefore the odds LHO being dealt a small doubleton are about 20%.
The odds of LHO being originally dealt Q-x-x can be computed as follows:
LHO will have the 3 card length one-half of the time that the suit is 3-1 (RHO will have the three-card length the other half of the time). 3-1 is about 50% so LHO will hold three cards in the suit about 25% of the time. The Queen will be singleton one-fourth of this time (as there are four cards out and each is equally likely to be the singleton). So the original odds of LHO being dealt Q-x-x is about three-fourths of 25% (which comes out to a little more than 18%).
So even though a 3-1 break is more likely than a 2-2 break, Queen doubleton offside (that is, in RHO's hand) is marginally more likely than Queen third onside (that is, in LHO's hand). If you have not seen the terms offside and onside before, they are frequently used in Bridge literature when the success or failure of a finesse is being discussed.
If you are not willing to memorize these tables, here is a helpful maxim that you can use instead:
An odd number of cards rates to divide as evenly as possible, but an even number of cards does not.
This rule correctly predicts that 3-1 is more likely than 2-2 but that 3-2 is more likely than 4-1. The only exception to this rule is when two cards are missing. Two is an even number but the even 1-1 break is slightly more likely than the 2-0 break.
It should be mentioned that all percentages quoted in this article are a priori in nature. That means that these percentages are only true “in a vacuum” (that is, when other information about the defenders' hands is not available). It is often the case that inferences from the bidding or defensive play will be available. Calculating a posteriori odds (that is, when other information is already known about the defenders' hands) is more complicated.
For example if an opponent's preempt in the bidding suggest that a side suit is divided 7-2, “nine never” is not correct anymore. The mathematics needed to demonstrate this are beyond the scope of this article (and beyond the scope of almost all Bridge players to do in their heads). Please take my word for it -- when one opponent is known to be very long in a given suit, it is correct to finesse his partner for the Queen in another suit in which four cards are missing.
If you have any questions about this article and would like to send mail to Fred Gitelman, please e-mail Zidea.
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