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Split Assumptions

By Ralph Welton

This is one of a series of Declarer Play articles. These articles build upon each other, so I recommend that you study them in order.

You are declarer in a no trump contract. The opening lead is made, and Dummy comes down.

It's time to count winners and plan. Better planning starts with accurate counts for both offense and defense.

Counting your own tricks is easier than counting defensive tricks.

You can readily see all your own high cards, and which suits might take extra tricks from extra length.

How a suit splits is very important. If you have eight spades, including all the top cards, the number of winners you can cash depends on how the suit splits between your own hand and the dummy.

4-4 split = 4 tricks
Q T 6 2  

A K J 8
5-3 split = 5 tricks
Q 6 2

A K J T 8  
6-2 split = 6 tricks
6 2

A K Q J T 8  
7-1 split = 7 tricks

A K Q J T 8 6  

Counting long suits for the defense is not so straightforward

You can't see how their suits split, or how many skaters they may have. But we may be able to figure it out, or make an assumption that's better than a pure guess.

That's where split statistics come in. Statistics can help us see the opponents' cards in terms of how they split between the two defenders' hands. Knowing the split leads to better planning.

You're probably concerned that I'm going to suggest we study charts of percentages for each of the ways a given suit might split.

Numbers... calculations... memorization... math!

No. None of that. Completely unnecessary.

You see, statistical percentages only tell you what happens most often when you average a huge number of random deals. "Eight cards will split 6-2 17% of the time." True, but not helpful.

You're not planning your declarer play for a huge number of random deals – just one deal. What you want to know is how the opponents' cards split on that one deal. A probability chart can't tell you that.

But we do need a place to start, so here's the one simplified (statistical) guideline you need to remember:

Suits are most likely to split equally, without hitting it exactly.

Which means...

  • 10 cards split 6-4 (not 5-5)
  • 9 cards split 5-4
  • 8 cards split 5-3 (not 4-4)
  • 7 cards split 4-3
  • 6 cards split 4-2 (not 3-3)
  • 5 cards split 3-2

That's it. It's a starting point. Not a final answer.

A Bridge Bear must also pay attention to the bidding, the opening lead, and other factors. Each time we consider another factor, it will help us confirm or reject our split assumption.

Split Assumption Practice

We begin with a two-step process for counting individual suits. Count cards to figure out how many they have. Then use the simplified formula to determine the split assumption.

example 1

T 8 2  

K 6

How many spades do they have?

They have 8 spades.

We don't yet know how they split, but what assumption should we start with?

5-3 split

example 2

7 4 2

A T 8 6  

How many hearts do they have?

They have 6 hearts.

What split assumption should we start with?

4-2 split

example 3

A 9

Q 4  

How many clubs do they have?

They have 9 clubs.

What is the split assumption?

5-4 split

example 4


A Q 4  

How many diamonds do they have?

They have 10 diamonds.

What is the split assumption?

6-4 split

example 5

6 4 2

Q T 5  

How many hearts do they have?

They have 7 hearts.

What does the Bridge Bears' statistical shortcut say about the split?

4-3 split

Little Bear: "Statistics are a lot easier than I expected!"

Me: "Just remember, our shortcut is only a starting point. We have to look at other sources of information too. Sometimes we will have to reject or adjust the starting assumption."

example 6

K J 2
T 8
A J 2
A K T 9 6  

Q T 9 6
K 6 4
K 7 4
Q J 7

Contract: 3N – you need 9 tricks.

Opening lead is the 3. You play Dummy's ten. Third hand plays the Q. Obviously you will play your king.

How many winners do you have?

5 clubs, 2 diamonds, and the K = 8 winners. You need one more to make 3N.

You have two options for more tricks.

If you force out their A, how many new spade winners would you establish?

3 new spade winners

If you finesse the J, and the Q is on-sides, you get one more winner.

It might seem like three certain tricks (spades) is better than one uncertain trick (diamonds). But spades loses the lead, while diamonds might not. So you have to count their winners before you can decide.

Let's count. They have the A and some heart winners. How many heart winners? Use the split assumption for hearts to figure out how many heart winners they have. And remember that hearts have already been played once.

They have 8 hearts, so the split assumption is 5-3. Hearts have already been played once, so they have 4 heart winners.

If they cash all their winners – the A and their 4 hearts, is your contract safe?

No! If they cash 5 tricks, there will only be 8 left for you. Down one. You cannot give them the chance to capture the lead with their A or your contract fails.

So... which is better for you – leading spades or leading diamonds?

Diamonds gives you a 50-50 chance to make the 9 tricks you need. Losing the lead by playing spades is unwise.

example 7

K J 2
T 8
A J 2
A K T 9 6  

Q T 9
K 6 5 4
K 7 4
Q J 7

Contract: 3N – you need 9 tricks.

Opening lead is the 3. You play Dummy's ten. Third hand plays the Q. Obviously you will play the king.

This hand is almost the same as example 6. Dummy's hand is unchanged. The only difference is I've moved a spot card in your hand from spades to hearts.

How many winners do you have?

Still the same as in example 6. Five clubs, 2 diamonds, and the K = 8. You need one more winner to make 3N.

You still have the same two options for developing the extra winner you need. You can force out the A, or you can finesse the J, hoping the Q is on-sides.

Playing spades is "slow but certain." Slow means you will lose the lead. Certain means you know for sure you will develop the winner(s) you need.

Taking the diamond finesse is "fast but not certain." Fast means (if it wins) you will not lose the lead. Not certain means it might both lose the lead and give an extra trick to the defense with the Q.

OK... let's count defensive winners (using the split assumption) to see if you can afford to go slowly or if you must go fast.

They have the A and some heart winners. Use the split assumption for hearts and figure out how many winners they have. And remember that hearts have already been played once. So...split assumption?

They have 7 hearts, so the split assumption is 4-3. Hearts have already been played once, so they have 3 heart winners.

If they cash all their winners – the A and their 3 hearts, is your contract safe?

Yes. If they cash 4 tricks, there will still be 9 left for you. Nine tricks is enough to make 3N.

So... which is better for you – leading spades or leading diamonds?

Spades is slow but certain. If you take the diamond finesse and it loses, the Q becomes the setting trick. Ouch! This time "slow but certain" is better.

example 8

K J 2
T 8
A J 2
A K T 9 6  

Q T 9 6
K 6 4
K 7 4
Q J 7

Contract: 2N – you need 8 tricks.

This hand is exactly the same as example 6. But I've changed the contract from 3N to 2N.

Opening lead is the 3. You play Dummy's ten. Third hand plays the Q. Of course you will play your king.

The count of your winners is the same. You have 8 winners – 5 clubs, 2 diamonds, and 1 heart.

You can establish 3 more winners with a "slow but certain" spade play, or hope for one more winner with a "fast but not certain" diamond finesse.

In example 6, we decided that you should take the "fast but not certain" diamond finesse because the defense already had enough winners to defeat 3N – A and 4 hearts (5-3 split assumption). If you gave them the lead with their A, your contract would fail.

Is the diamond finesse still a good choice when your contract is 2N?

You only need 8 tricks to make your contract, and you have them already. Risking the diamond finesse risks the contract. If it fails, the Q becomes the setting trick. They will take A, 4 hearts, and the Q.

We just decided that taking the diamond finesse was a poor plan because it risks the contract when you already have enough tricks.

Does the "slow but certain" spade play also risk the contract?

That was a tricky question. Yes, it risks the contract. The split assumption is just that – an assumption. It's definitely better than guessing, but it's not always right.

On this hand the split assumption for hearts is 5-3. But if the actual split turns out to be 6-2, playing spades gives the lead away when the extra heart skater makes enough defensive winners to defeat your contract.

You should not take chances when you already have enough tricks to make your contract. Don't play either spades or diamonds. Simply cash your tricks and make your contract.

There are always 13 tricks won in every bridge hand. No more, no less. That truth can tell you if a plan you're considering is good or bad.

Add the cashable tricks for both sides.

  • If the total is less than 13, you might be able to establish and cash more winners.
  • If the total is more than 13, somebody is going to be unhappy – unable to cash all their winners. If you lose the lead, you will be the unhappy one who doesn't get to cash all your winners. Only risk losing the lead if you need an extra trick to make your contract.
  • If the total is exactly 13, be very careful. If you give up the lead while establishing a 14th trick, you won't get to cash it. You may even wind up with fewer tricks than you had. Consider what happens if you have 9 tricks and they have 4 (9+4=13), and then you take a finesse. Someone will lose one of their "winners". If they win the finesse, they can cash 5 (five!) tricks (the 4 they had, plus the card that wins the finesse), leaving only an unhappy 8 for you. If you win the finesse, you can cash 10 tricks, leaving only an unhappy 3 for them.

So... Be careful, and don't be greedy.

Split assumptions when establishing winners

Most deals require the establishment of additional winners. But you must be careful about suits where both sides can establish extra winners in the same suit.

example 9

K J 3

Q 2

They have one cashable spade trick. You have none.

You can develop two high card winners by driving out the A.

But you must be wary of doing so because the defenders have more spades than you do. Playing your honors would establish skaters for the wrong team.

How many spades do they have?

You have 5, so they have 8.

What is the split assumption?

The split assumption is 5-3.

We have honors for three rounds of the suit, so their skaters are the fourth and fifth rounds. That's 2 skaters for them, plus their A.

Your two high-card winners will be cashable before their two skaters – which is good. Nevertheless, every time you play spades, they get closer to being able to cash their two skaters.

You should delay playing this suit until after you have established your winners in other suits... where they can't establish skaters.

The spades in this example come from a hand Little Bear played in a recent duplicate game. Let's see what went wrong when he played spades too soon...

example 9, expanded

K J 3
Q T 8
A 6 2
K T 6 5  

Q 2
K J 6 4
K Q 4
A Q J 7

Little Bear Gets No Honey

I don't want to put Little Bear on the spot, so I'm going to pretend YOU are playing the hand. You don't mind if I have you "make the mistake," do you?

Contract: 5N – Opening lead: 5

You need 11 tricks to make your contract.

You have 7 winners – 4 clubs and 3 diamonds. You need 4 more winners from hearts and spades, so you will have to drive out both major suit aces.

Little Bear started with spades, so we're going to pretend you repeat that mistake.

You remember the guideline, "Play the honor from the short hand first," so you lead the Q and play the 3 from Dummy. You're surprised when they let your Q win.

Next you lead the 2, and play Dummy's J. This time they win, playing their A. That's two rounds of spades. You have none left in your hand and only the K in the Dummy.

They now have the lead, and they lead a third round of spades, which your K wins. Now you're out of spades in both hands.

Use the split assumption to figure out how many ready-to-cash spade skaters they have after the third round of spades.

The split assumption is 5-3 and three rounds have been played. So they have 2 skaters. Even if the actual split turns out to be 4-4, they will have 1 skater, which is all they need to defeat your contract.

After driving out the A, you have your two extra spade tricks, so you turn your attention to hearts. What will they do with your first heart lead?

They will play their A to capture the lead, and cash their spade skater(s).

Your contract fails because they were able to establish the setting trick(s) in spades while they still had the A to capture the lead.

To prevent that from happening, you need to drive out the A before you play spades.

How do we figure out which suit to start? We count cards and do split assumptions. They have 8 spades but only 6 hearts, so they are far more likely to have length, with potential for a skater, in spades. So play hearts before spades.

When the defenders might have skaters...

example 10

A Q 3

5 4 2  

You have one diamond winner. They have none.

You can finesse, hoping the K is on-sides. If the finesse wins, you keep the lead and have one additional winner. That seems good...

But the finesse might lose. That gives the defense an additional winner – and the lead.

Even worse, you know they have something good (for them) they can do with the lead. They can continue diamonds, driving out your A and establishing more diamond winners for themselves.

OK, pretend you've just lost your finesse and they've driven out your A. If the split assumption is correct, diamonds can be led four times. Consider how many times they've already been played. Their remaining diamonds are winners because your A and Q are gone. How many uncashed diamond winners do they have?

The split assumption is 4-3, so they would have 2 ready-to-cash winners.

Before taking the finesse, you had the lead. If the finesse loses and they continue diamonds, you'll get the lead back with your A. But you are not "back where you started." You have lost a great deal.

  1. You have lost a trick to the K.
  2. They now have two more ready-to-cash diamond winners. That's three new winners they didn't have before you finessed.

plush toy bearLittle Bear: "I like to take finesses!"

Yes, Little Bear, beginners often like to take finesses because they have just recently learned how they work. Strong players hate the huge downside of losing a finesse in a suit where the opponents can establish skaters in the same suit. They always look for alternates that are not so risky. You should too.

When you might have skaters...

example 11

A Q 3

J 7 5 4 2  

As in example 10, you have one winner, and the defense has none. You also have the same possible finesse, where you hope the K is on-sides.

But this time taking the finesse is not a potential disaster.

What is the split assumption?

You have 8 diamonds, so they have 5. The split assumption is 3-2.

How does the play go? The first diamond trick is a finesse. Let's say it loses. After you recapture the lead in another suit, you go back to diamonds.

You win the second round of diamonds with your A, and continue with a third round, which you win with your J.

How many times have diamonds been played?

3 times

Assuming the 3-2 split assumption is correct, they will have no diamonds left after you play them three times. So even if you lose the finesse, you might make four diamond tricks – the A and the J are high card winners, and you have 2 skaters.

The main point of these two diamond examples is....

It's risky to finesse when the opponents can win and establish skaters in the same suit.

But if you are the one who can establish the skaters, it's not so risky. In fact, it's usually quite good to establish long suits.

You have to count cards, do a split assumption, and count tricks to figure out what's risky and what's not so risky.

Which Suit First? Which Suit Last?

example 12

J T 8 7  
A K 5 4
Q 3 2
6 5

Q 9 4
Q 6 3
A K 5 4

Contract: 3N. You need 9 tricks to make 3N.

Opening lead: 2. The split assumption for their 6 cards is 4-2.

How many winners do you have?

You have 6 winners – AKQ, AKQ

You need 3 more winners to make your contract.

You can drive out the A and promote 2 club winners.

You can drive out the AK and get 2 spade winners.

And, if you're lucky, you can get one extra winner from diamonds. Why do I say you have to be lucky for a diamond spot card to be a winner?

The split assumption for their 6 diamonds is 4-2, and you'd need a 3-3 split to have a skater. You'd have to be lucky for the split assumption to be wrong in exactly the way you'd wish it to be wrong.

Furthermore, if you play three rounds of diamonds, and they do split 4-2 (as expected), your opponents will have a 4th round winner (a spot card higher than your 5).

That would help them instead of helping yourself, so you should delay playing diamonds as long as possible. On this hand, play diamonds last.

Meanwhile, how many winners do they have?

3 winners – AK, A

How many more do they need to set your contract?

They need 2 more winners to set your contract.

The opening lead was a heart. If they persist in hearts, how many extra winners can they develop?

The split assumption for their 6 hearts is 4-2. You have the top 3 heart honors, so they can only develop one very slow heart winner.

If they do develop that one very slow heart winner, that brings their total to 4 winners – not enough to set your 3N contract.

So you can afford to develop your extra club and spade winners "slowly." All you have to do to make your contract is avoid helping them develop a fifth winner. That's why we won't be playing diamonds any time soon. It would be bad if diamonds don't split and the bad guys get an extra winner.

Now let's look at spades and clubs. If the split assumptions are correct for those suits, you could play out all your cards in one of them and your opponents will not get any extra skaters.

Which suit is that?

Spades. They have 6 spades, so the split assumption is 4-2. They can win with their AK, but you own all the lesser honors and a four card suit. So they get nothing more than the two honors they started with. That means spades is the first suit for you to play.

We've worked out our plan:

  1. Win the opening lead in hearts.
  2. Drive out their spade masters.
  3. Drive out their club master.
  4. When your lesser honors are established, cash all your tricks and make your contract.

This plan leaves out the details of losing tricks and recapturing the lead. The first time we lead spades, they win and lead a second heart. We win and lead a second spade. They win and lead a third heart. We win and lead a club. They win and cash their heart skater. Then we win whatever suit they lead and cash all our tricks.

With our plan, we will establish 10 winners – 2 spades, 3 hearts, 3 diamonds, and 2 clubs. Why will we only be able to cash 9 tricks?

They have already won 4 tricks – A, K, A, 1 heart skater. There are only 13 tricks in a bridge hand, so there are only 9 left for us. Two of our winners will crash. :(

plush toy bearLittle Bear: "The split assumption for their 6 hearts is 4-2. But what if the assumption is wrong and the actual split is 5-1. Wouldn't they have 2 heart skaters – enough to defeat our contract?"

Me: Ah... good question, Little Bear.

You are correct that two heart skaters would be enough to defeat 3N. But we would discover it when the player with the singleton makes a discard on the second round of hearts. If that happens we would have to change our plan.

We could develop 2 extra tricks in spades, and hope for a lucky 3-3 diamond split. If luck is with us, we would be able to cash our 9 tricks before the defense can cash their heart skaters.

There are other more advanced possibilities for coming to 9 or 10 tricks before the defenders can cash hearts, but that's beyond the scope of this page.

Anyway, it's important to figure out when split assumptions are wrong. So that's what we'll be talking about next.

plush toy bearGo to the next topic:

Listen to the Bidding

Ralph Welton with BuffyBridge Bears is run by a retired teacher and ACBL life master who has 35 years teaching experience and who's been playing bridge for over 50 years. I don't claim to be one of the top players, but I do understand how slowly beginners need to go when they are trying to learn how to play bridge.