Content Page Introduction and Rules of Contract Bridge --------- P. 1-6 Odds about Card Distribution ------------------------- P. 7-10 Strategies in bidding ------------------------------------- P. 11-18 Basics of bidding ------------------------------------- P. 11 Bidding systems -------------------------------------- P. 12-14 Conventions ------------------------------------------- P. 15-18 Strategies in defending --------------------------------- P.19-26 Finesse vs. Cashing -------------------------------------- P. 27-43 Cashing winners ------------------------------------- P. 28-32 Finesse ------------------------------------------------- P. 33-35 Finesse vs. cashing winners ----------------------- P. 36-41 Conclusion ------------------------------------------------- P. 42 Reference -------------------------------------------------- P. 43
Introduction and Rules of Contract Bridge Contract bridge is a kind of trick-taking game using 52 playing cards. Contract bridge is mainly composed of FOUR main parts: 1) dealing 2) bidding for the contract 3) playing the hand 4) scoring the results Dealing is the process of card distribution and determination of where the player will be in. In bridge, players will be divided into East, South, West and North. There will be a board to determine that. The board will also indicate that the players are vulnerable or not. Vulnerable players will score more when they can fulfil their contract, but they will lose more if they can t fulfil their contract set. Vice versa, non- vulnerable players will score less and lose less. In bridge competitions, the hands are shuffled and dealt only once since the hands will be transferred to another tables and other competitors will play the same hand so that every competitor can play the same hand once. 1
After the four players gather their own hands, the bidding starts. The dealer, which has been stated on the board in competitions, or can be chosen randomly if the game is played casually, will bid first. After that, the other players will bid in clockwise order. The simplest meaning of each bid can be that a player thinks how many tricks he can win with his and his partner's hands, though there can be other meanings behind it, which will be discussed later. There are 38 different bids in contract bridge, which are s t a t e d b e l o w : 1 1 1 1 1NT ( N o T r u m p ) 2 2 2 2 2NT 7 7 7 7 7NT P a s s, Double and Redouble A bid of level N indicates that the bidder promises to take (N+6) tricks out of 13. Most of the bids are ranked in order, with 1 the lowest, followed by 1, 1, 1, 1NT, 2 and so on. The bids pass, double and redouble are not ranked in order. A player cannot make a bid lower than the last bid. If a player passes, that means the player doesn't want to bid a higher bid. 2
If a player thinks that his opponents will not be able to win the number of tricks they have bid, he could double their bid. If a player or his partner's bid has been doubled, but he is very certain that he and his partner will be able to make the contract, he can redouble the bid. You gain more if you make the contract as declarer/dummy, or you defeat the contract as defender, but you lose more if you fail to do so. After 3 players pass consecutively, the bidding ends. The final bid will become the contract. The trump suit will be determined by the contract. Among the pair who bid the contract, the player who first bid the trump suit in the contract during bidding will become the declarer. Followed by bidding is playing the hand. The play consists of 13 tricks; each trick consists of 4 cards played from 4 different hands. A is the highest and 2 is the lowest. The leader, the left of the declarer, will play the first card. Then the play proceeds in clockwise order. After the opening lead is played, the dummy (partner of the declarer) lays his hand face up on the table. The declarer is responsible for selecting cards to play from the dummy's hand and from his own hand in turn. 3
The defenders each choose the cards to play from their own hands. Dummy is allowed to prevent declarer from infringing the rules, but otherwise must not interfere with the play. Any card can be chosen from the hand as the lead, but the remaining hands must follow the suit, meaning that they need to play a card of the same suit as the lead, unless the hand has no more cards of that suit. The hand that plays the highest card in the suit of the play will win that trick, unless any of the played cards are of the trump suit, in which case the hand that plays the highest trump card wins the trick. The hand that wins the trick leads the next trick, until all the cards have been played. Success in making the contract is rewarded by points in the scoring phase for the declarer's side. If the declarer fails to make the contract, the defenders are said to have defeated the contract, and are awarded points for doing so. When the contract is made, the level of contract will be the primary factor that affects the scoring, rather than the number of tricks taken in the play. This encourages the player to bid a higher contract in order to score more, and ensures the competitiveness in the bidding process. 4
When the declarer makes the contract, his side receives points for: every trick bid and made (20 for minor suit contracts, 30 for major suit and no-trump ones, with an additional 10 points for the first trick in no-trump), overtricks (tricks taken over the contract level, again with 20 for minor suits, 30 for majors and no-trump), and some other kinds of bonuses. If the contract is doubled, the declarer s pair will get more points, but will lose more points if he fails to make the contract. If the points of the contract alone is less than 100, a bonus of 50 points will be awarded. If the points of the contract alone is more than or equal to 100, a bonus called game bonus will be awarded. If the declarer s side is vulnerable, the game bonus obtained by his side will be 500 points. If the declarer s side is non-vulnerable, the game bonus obtained by his side will be 300 points. For example, a just-made 4S contract will obtain 420 points if the declarer s side is non-vulnerable or 620 points if the declarer s side is vulnerable. 5
There is also a bonus called the slam bonus, which is awarded when the contract made requires more than or equal to twelve tricks to make. If the contract requires twelve tricks to make, it is called a small slam and a slam bonus of 750 points, if the declarer s side is vulnerable, or 500 points, if the declarer s side is non-vulnerable will be awarded. If the contract requires all thirteen tricks to make, it is called a grand slam and a slam bonus of 1500 points, if the declarer s side is vulnerable, or 1000 points, if the declarer s side is nonvulnerable will be awarded. The slam bonus will be added together with the game bonus and the points of the contract alone to obtain the final score. For example, the points obtained by making a 6D contract will be 6*20+300+500=920 if the declarer s side is non-vulnerable, or 6*20+500+750=1320 if the declarer s side is vulnerable. If the contract goes down, the defender will obtain 100 points for each trick down, if the declarer s side is vulnerable, or 50 points, if the declarer s side is non-vulnerable. After scoring, the game is ended. We may shuffle the cards and play another game when we are playing casually or we will need to wait for the cards of another table to play when in competitions. 6
Odds about Card Distribution When playing bridge, we will have different hand distribution for every game. Having more cards of the same suit will surely help us in winning, but what is the probability of getting different hand distributions? That is what we are going to investigate. When we got 13 cards in our hand, they are mainly included of 4 different suits: clubs ( ), diamonds ( ), hearts ( ), spades ( ). We usually name our hand in the way of counting the number of cards each suit got and use the number in descending order, like 4-3-3-3(evenly distribution), 6-5-2-0, e t c. In this part, we are going to find out some of the probabilities of getting different kinds of hand distributions. 7
Number of ways of getting random 13 cards =( 52 C 13 ) =635,013,559,600 Number of ways of getting 6-3-2-2 =( 13 C 6 )*( 13 C 3 )*( 13 C 2 )*( 13 C 2 )*4!/2! =35,830,574,208 Probability of getting 6-3-2-2 =35,830,574,208/635,013,559,600 =5.64% (correct to 3 significant figures) Number of ways of getting 5-4-3-1 =( 13 C 5 )*( 13 C 4 )* ( 13 C 3 )* ( 13 C 1 )*4! =82,111,732,560 Probability of getting 5-4-3-1 =82,111,732,560/635,013,559,600 =12.9% (correct to 3 significant figures) 8
Number of ways of getting 5-4-2-2 =( 13 C 5 )*( 13 C 4 )* ( 13 C 2 )* ( 13 C 2 )*4!/2! =67,182,326,640 Probability of getting 5-4-2-2 =67,182,326,640/635,013,559,600 =10.6% (correct to 3 significant figures) Number of ways of getting 5-3-3-2 =( 13 C 5 )*( 13 C 3 )* ( 13 C 3 )* ( 13 C 2 )*4!/2! =98,534,079,072 Probability of getting 5-3-3-2 =98,534,079,072/635,013,559,600 =15.5% (correct to 3 significant figures) 9
Number of ways of getting 4-4-3-2 =( 13 C 4 )*( 13 C 4 )* ( 13 C 3 )* ( 13 C 2 )*4!/2! =136,852,887,600 Probability of getting 4-4-3-2 =136,852,887,600/635,013,559,600 =21.6% (correct to 3 significant figures) Number of ways of getting 4-3-3-3 =( 13 C 4 )*( 13 C 3 )*( 13 C 3 )*( 13 C 3 )*4!/3! =66,905,856,160 Probability of getting 4-3-3-3 =66,905,856,160/635,013,559,600 =10.5% (correct to 3 significant figures) After calculating some of the probabilities about hand distribution, we found out that: 4-4-3-2 (21.6%) is the most probable hand distribution. Followed by 5-3-3-2 (15.5%) and 5-4-3-1 (12.9%) 10
Strategies in bidding Basics of bidding In contract bridge, there are two parts: bidding and playing. We are going to introduce some basic information and strategies in bidding. After the four players gather their own hands, the bidding starts. The dealer, which has been stated on the board in competitions, or can be chosen randomly if the game is played casually, will bid first. After that, the other players will bid in clockwise order. The simplest meaning of each bid can be that a player thinks how many tricks he can win with his and his partner's hands, though there are other meanings behind it, which will be discussed later. There are 38 different bids in contract bridge, which are s t a t e d b e l o w : 1 1 1 1 1 NT ( N o T r u m p ) 2 2 2 2 2 NT 3 3 3 3 3 NT 4 4 4 4 4 NT 5 5 5 5 5 NT 6 6 6 6 6 NT 7 7 7 7 7 NT P a s s, Double and Redouble 11
Bidding systems The bidding process is very simple, but the strategies behind it can be very complicated. When a pair wants to bid a good contract, they need to know information about each other's hand. This leads to the invention of bidding systems and some ways of bids with complicated meanings behind which called conventions. A pair can know about the strength and structure of each other's hand using bidding systems. In normal bidding systems, the concept of "High Card Point" (HCP) is used. "Ace" is counted as 4 HCP King" is counted as 3 HCP Queen" is counted as 2 HCP Jack" is counted as 1 HCP Having other cards does not lead to an increase in HCP We will introduce two basic Bidding systems, S A Y C and A C O L. 12
"Standard American Yellow Card" system(s A Y C ) is widely used by beginners. The most important features of SAYC are : "5 Card Majors", "Strong 1NT", "Strong 2 ", "Weak 2s" and pre-emptive open bids. The meanings of opening bids from 1 to 4 in simple S A Y C a r e listed below: Bid Meaning 1 12-21 HCP, at least 3 clubs 1 12-21 HCP, at least 3 diamonds 1 12-21 HCP, at least 5 hearts 1 12-21 HCP, at least 5 spades 1NT 15-17 HCP, balanced hand (4-3-3-3 or 4-4-3-2) 2 22+HCP 2 5-11 HCP, at least 6 diamonds 2 5-11 HCP, at least 6 hearts 2 5-11 HCP, at least 6 spades 2NT 20-21 HCP, balanced hand 3 6-10 HCP, at least 7 clubs(pre-emptive) 3 pre-emptive, at least 7 diamonds 3 pre-emptive, at least 7 hearts 3 pre-emptive, at least 7 spades 3NT 25-27 HCP, balanced hand 4 pre-emptive, at least 8 clubs 4 pre-emptive, at least 8 diamonds 4 pre-emptive, at least 8 hearts 4 pre-emptive, at least 8 spades S A Y C also includes overcalling and defending bids, but we are not going to introduce these kind of bids. 13
A C O L is a bidding system widely used in Britain. It has different forms, such as "Basic ACOL", "Benji ACOL", "Multi-2 A C O L ", e t c. We are going to introduce Basic ACOL. The meanings of opening bids from 1 to 4 in Basic ACOL are listed below: Bid 1 1 1 1 1NT 2 2 2 2 2NT 3 3 3 3 3NT 4 4 4 4 Meaning 12-19 HCP, at least 4 clubs 12-19 HCP, at least 4 diamonds 12-19 HCP, at least 4 hearts 12-19 HCP, at least 4 spades 12-14 HCP, balanced hand Game forcing or 23+HCP, balanced hand 8 or more playing tricks, 2 or more outside defensive tricks 8 or more playing tricks, 2 or more outside defensive tricks 8 or more playing tricks, 2 or more outside defensive tricks 20-22 HCP, balanced hand 6-10 HCP, at least 7 clubs(pre-emptive) pre-emptive, at least 7 diamonds pre-emptive, at least 7 hearts pre-emptive, at least 7 spades pre-emptive, with a long and solid minor suit pre-emptive, at least 8 clubs pre-emptive, at least 8 diamonds pre-emptive, at least 8 hearts pre-emptive, at least 8 spades Same as SAYC, ACOL also includes overcalling and defending bids, but we are not going to introduce them as well. 14
Conventions Bidding systems are ways of bidding that help partners to know each other's hand by simple bidding. Most of the bids are natural bids. However, in conventions, bids are usually not natural bids. A good advantage of using conventions is to save bidding spaces. If the bid is too high, the opponents may be able to defeat the contract. Stayman is one of the conventions used to save bidding spaces. For example, two partners, "North" and "South", have the following 2 hands: North: South: S: A J 3 2 S: K Q 8 5 H: K 8 7 H: A J 4 2 D: A 8 7 D: 9 6 2 C: Q J 5 C: 4 3 Let North be the opener and there is no intervention from the opponents. Using S A Y C, he will bid 1NT. South has two good suits, spades and hearts, but he doesn't know which suit in North's hand is better. If he tries to find out information of North's hand by natural bids, he may bid up to 3S, which will be very dangerous due to the weakness of the minors. 15
North: South: S: A J 3 2 S: K Q 8 5 H: K 8 7 H: A J 4 2 D: A 87 D: 9 6 2 C: Q J 5 C: 4 3 Stayman is used to respond NTs openings. It is used to ask the opener whether he has a major suit with 4 cards. If the opener bids 1NT, his partner can use stayman by bidding 2. If the opener has no major suits with 4 cards, he will bid 2, otherwise he will bid the major suit which he has 4 cards. He may bid either suit bid his stronger suit if he has four cards in both major suits. In this example, South will bid 2, using stayman. North will bid 2, since he has 4 cards in Spades. South will be satisfied with the bid, since his strength is not enough to go for game and he finds a 4-4 combination of spades with his partner. He will pass and the contract will be 2. In this example, 2 will be a good contract. A 2 contract would help the pair to obtain the most points in this hand. 16
Another commonly used convention is called Blackwood. It can be used to help a player to find out the location of aces and kings, and even the queen of the trump suit. There are many different kinds of Blackwood, such as the simplest one, "Baby Blackwood", "Key Card Blackwood", "Exclusion Blackwood", etc. We will introduce the simplest kind of Blackwood. For example, two partners, "North" and "South", have the following 2 hands: North: South: S: A 875 3 S: K Q 9 42 H: A K H: 8 3 D: A 9 5 3 D: K Q J 2 C: Q J C: A K Let North be the opener and there is no intervention from the opponents. Using S A Y C, North will bid 1. Having 5 Spades and 18 HCPs, South will be happy to see a 5-5 combination and a good chance to go for a slam. He can find out the number of aces and kings North holds using Blackwood. Using Blackwood, South will bid 4NT. If North has 0 or 4 aces, he will bid 5. If he has 1 ace, he will bid 5. He will bid 5 and 5 if he has 2 and 3 aces respectively. In the example, North will respond by bidding 5, since he has 3 aces. South will be happy to see that there will be one losing trick at most, which is in hearts. 17
After that, South will try to find out whether a grand slam is possible, so he bids 5NT to find out whether North has the king of Hearts, so that the other Heart will be covered. Similar to the response to 4NT, North will bid 6 if he has 0 or 4 kings, 6 if he has 1, 6 if he has 2 and 6 if he has 3. In the example, North will bid 6, since he has one king. South knows that North holds the king of Hearts, since he himself holds all of the other kings. He will see that all of the losers in his suit will be covered by the cards in his partner's hand, so he will be happy to go for a grand slam and bid 7NT, since they do not rely on voids and ruffing to bid the contract. The 7NT contract will be easily made during the playing part if South does not make any terrible mistakes. There are many other different kinds of conventions used in bidding, including those used to bid a suitable contract and even defensive ones. However, we are not going to introduce them. There are also conventions used in the playing part. Some of these conventions and other playing techniques will be introduced in the following chapters. 18
Strategies in defending After the contract is decided, the one sitting on the left hand side of the declarer will lead a card and start the playing part. In some contract, declarer is confident about making the contract, but the lead could destroy his whole plan. Sometimes it gives precious information to the player s partner; sometimes it disables dummy from trumping declarer s loser (the card that will lose a trick). Therefore, the lead is the start of play, and perhaps is the most important part of the play. It is very important to lead well. There are some general rules about the lead. Although they may not be always correct, it still helps a lot in most cases. 19
1. Lead the suit that your partner has bid (for natural bids) This is perhaps the most logical, easy-to-understand rule. Why your partner bids a suit? It s because he has enough strength in this suit and he thinks he can gain some tricks with it. Remember: always trust your partner. Unless you are void in that suit, you should lead your partner s suit instead of yours. It would be best if you have a singleton in suit contract, because you can always expect your partner to play a return card to you and give you a chance toruff. If, very unfortunately, your partner has not shown any suit, it is up to you to decide which card to lead. Usually, the suit that the dummy has bid is not preferred, but a trump lead may be helpful. 2. Lead top of sequence After deciding which suit to lead, it s time to choose a card from that suit. In case you have a sequence in that suit, you should lead the top card of that sequence. Y o u m a y a s k, W h a t i s a sequence? A c t u a l l y, a sequence is a group of high cards connecting together, for example, A K Q, or Q J 10. Even if one card is missing, we still consider it as sequence, for example, A Q J, or K Q 10. Usually this lead can give you one or more tricks in that suit. 20
3. Lead the fourth highest card If you decide to lead your long suit (i.e. the suit in your hand with the most number of cards), and that your suit does not contain any sequence, you may choose the fourth highest card from the suit. Note that according to pigeon-hole principle, 13 cards in four suits implies that there is at least one suit with at least four cards, so there must be a fourth card in the long suit. It is particularly useful in no-trump contract for you to win tricks in the long suit. It also leads to the application of the rule of 11. You can count the number of cards higher than your card by subtracting it from 11. For example, if your fourth highest is 6, there will be (11-6) = 5 cards higher than 6 in the remaining three hands. This is also useful for your partner to find out positions of the missing high cards, as you promise three cards higher than the one you play. 21
We have already introduced some defending techniques. Now we are going to introduce some conventions used in defending. Same as conventions used in bidding, which have been introduced in the previous chapter, conventions used in the playing part can help the defending pair to know more about each other s hand, thus enabling better decisionmaking during the play. Most of the conventions used in playing are signals, which help the defending pair to know more about each other s hand, such as the suits of winners and the strength of a suit. We are going to introduce two conventions used in playing. 22
The simplest signal is called HELD, which stands for High Encourage, Low Discourage. It is widely used by beginners because it is easy to use and understand. It can help defending pairs to know about the strength of a suit of each other s hand. Let North and South be partners, and they are against a 3NT contract. It is North s turn to play. The distribution of the cards in the game is as follows: North: S: A 842 H: 9 D: 76 5 4 C: Q J 10 4 West: East: S: K J 5 S: Q 7 6 H: A Q 4 3 H: K J 8 D: K Q 2 D: A J 8 3 C: 7 5 C: K 6 3 South: S: 10 9 3 H: 10 7 65 D: 10 9 C: A 9 8 2 23
North S: A 842 H: 9 D: 76 5 4 C: Q J 10 4 West: East: S: K J 5 S: Q 7 6 H: A Q 4 3 H: K J 8 D: K Q 2 D: A J 8 3 C: 7 5 C: K 6 3 South: S: 10 9 3 H: 10 7 65 D: 10 9 C: A 9 8 2 North will lead queen of Clubs. South knows that North has jack of Clubs as well, so he plays 9 of Clubs, which is a relatively high card, showing his preference for North to continue playing clubs. After North continues to play the clubs, North and South will eventually win four tricks in clubs and one trick in spades, making the contract goes down by one trick. If North switches to hearts or diamonds, West, the declarer, will cash the hearts and diamonds winners and make the contract. 24
Another famous convention used in playing is the Smith Signals. It is used to express the user s attitude to the suit in the lead to his partner. Let North and South be partners, and they are against a 3NT contract. It is North s turn to play. The distribution of the cards in the game is as follows: North: S: A 10 87 3 H: 9 8 3 D: 7 C: A 5 4 2 West: East: S: K 6 5 S: 4 2 H: A K 7 6 H: Q J 10 D: K J 4 D: A Q 9 8 C: Q 63 C: K J 10 9 South: S: Q J 9 H: 5 4 2 D: 106 53 2 C: 8 7 25
North leads 7 of Spades, and South plays jack of Spades. West plays king of Spades and win the trick. After that, he will cash the winners in hearts and diamonds. During the playing of diamonds, South plays 10 of Diamonds first, followed by the smaller diamonds. This shows he wants North to play the suit in the lead, which is spades. This suggests that he has queen of Spades. After West cashes all the winners in hearts and diamonds, he has to play clubs. As a result, North will win the trick with ace of Clubs. North plays a small spade to South s queen and South returns with 9 of Spades. North continues to cash its spades and downs the contract by one trick. There are many other kinds of signals used in contract bridge, but we are not going to introduce those. In the next chapter, we will introduce how probability can be used in the playing part of contract bridge to make better decisions. 26
Finesse vs. Cashing In contract bridge, sometimes you face dilemma of how to play out a suit as declarer. If you get all the high cards, it is simple: play them out and clear the suit. However, in most of the cases, you would miss some high cards. It is a good idea to play that suit at last and gather information about High Card Points and card distribution of opponents, but it is not always a successful plan. Sometimes defenders hide information, sometimes you forget them. Therefore, you would have to play the suit in a better way that you can get the desired tricks. Usually, there are two main types of suit handling: cashing winners and hope the high cards would drop during the process, or finesse (guess the position of high cards, would be explained soon). 27
1. Cashing winners The success rate of this method depends on the card distribution of that suit. It would be better for the declarer if the suit split evenly. Consider this example: North: K 3 West:??? East:??? South: A Q 6 5 4 2 You need four more tricks from this suit in a no-trump contract (we try not to discuss the problem of trumping here). There are five cards remaining, including J, 10, 9, 8 and 7 of that suit. If West holds three cards and East holds two cards, or West holds two cards and East holds three cards (we call it a 3-2 distribution), you can cash A, K and Q of the suit and the remaining cards in South to make the contract. However, with a 4-1 or 5-0 distribution, you cannot simply cash A, K and Q of the suit, otherwise you would lose the crucial fourth trick and the contract may be defeated. 28
Total number of combinations for West player= ( 26 C 13 ) Explanation: Among the 26 cards from your opponents, West holds 13 of them. Number of combinations for West player to have two cards of that suit= ( 5 C 2 ) *( 21 C 11 ) Explanation: ( 5 C 2 ) is the total number of combination of West s two cards of that suit, ( 21 C 11 ) is total number of combination of West s other cards, which must not belong to that suit. Number of combinations for West player to have three cards of that suit= ( 5 C 3 ) *( 21 C 10 ) Probability that the contract can be made = [( 5 C 2 ) *( 21 C 11 ) + ( 5 C 3 ) * ( 21 C 10 )] / ( 26 C 13 ) = 0.678 (corrected to 3 significant figures) There is quite a good chance to make this contract. 29
Here is another example: North: K 8 3 West:??? East:??? South: A 76 5 4 2 Declarer needs all six tricks in the trump suit. This time four cards are missing. The South-North pair holds two winners. Therefore, a 2-2 distribution is necessary to gain all six tricks, using this cashing winners strategy. Total number of combinations for West player = ( 26 C 13 ) Total number of combinations for West player to have two cards of that suit = ( 4 C 2 ) * ( 22 C 11 ) Probability that the contract can be made = ( 4 C 2 ) *( 22 C 11 ) / ( 26 C 13 ) = 0.407 (corrected to 3 significant figures) The chance to make this contract is not high. Note that although ( 4 C 2 ) is bigger than ( 4 C 0 ), ( 4 C 1 ) and ( 4 C 3 ), the chance of making the contract is less than one half. 30
Of course, cashing high cards isn t that simple. Sometimes it can drop high cards surprisingly. Here is an example again: North: K Q 3 West:??? East:??? South: A 10 4 2 The South player, as declarer, needs all four tricks in the trump suit for the contract. The suit has not been played before. Let s calculate the chance that he can make the contract! Total number of combination for a 3-3 distribution= ( 6 C 3 ) * ( 20 C 10 ) Is this the end? NO! The dangerous Jack of the suit may drop during the process! 31
Total number of combination for a 4-2 distribution = ( 6 C 2 )*( 20 C 11 ) + ( 6 C 4 )*( 20 C 9 ) Consider the probability of the Jack to be on the side with two cards. Since every card has equal chance to be at different positions, probability of the Jack to be on the side with two cards = 2/6 = 1/3 Total number of combination for a 4-2 distribution AND the Jack to be on the side with two cards of that suit = [( 6 C 2 )*( 20 C 11 ) + ( 6 C 4 )*( 20 C 9 )]/ 3 Probability to make the contract = {[( 6 C 2 )*( 20 C 11 ) + ( 6 C 4 )*( 20 C 9 )] / 3 + ( 6 C 3 ) * ( 20 C 10 )} / ( 26 C 13 ) = 0.517 (corrected to 3 significant figures) The contract has a good chance to be made, although it seems the other way. Sometimes it seems a stupid choice to simply play out the suit. However, if you are out of ideas, this will be a good way to make the contract with a fair chance. 32
2. Finesse The success rate of this method is based on whether the high cards lie at favorable positions. Consider this example: North: A 7 6 3 West:??? East:??? South: K J 5 4 2 There are four cards missing: Q, 10, 9, 8 of that suit. South, as declarer, needs five tricks from this suit. Now the lead is in dummy. If he uses the method of finesse, he would lead a low card from dummy, in the hope that the crucial Queen belongs to East. Let s see what happens if the Queen is with East. If he plays the Queen, declarer covers with King; if he doesn t, declarer plays the Jack. As the Queen is with East, West doesn t have a card that is strong enough to cover the Jack. The Jack is then successfully established as a winner. The probability that the Queen belongs to East = 0.5 (1/2) 33
Finessing a card has quite a high success rate! you may think. However, it may shrink to a slight chance as the number of missing cards increases North: A 3 2 West:??? East:??? South: K 10 9 4 There are six cards missing. South, as declarer, cannot afford to lose a trick in this suit in a no-trump contract. Now the lead is with dummy. What is the chance for declarer to make the contract, using finesse only? In fact, the contract is almost impossible to be made using finesse! If West holds at least one of Q and J, South is doomed; even if East holds all two of them, he would be clever enough to realize that the Ace lies in a favorable position, and would put up his Queen or Jack in the first round! In this case, the possibility that South can make the contract is exactly 0 if South tries to finesse only. 34
Y o u m a y a s k : Is there a possible way to make the contract? In fact, yes. And we will mention this scenario again in the next page. This is just to prove that finesse does NOT always have half a chance to succeed. N o w, y o u m a y a s k : What is the probability to succeed if North-South pair aims at a down-one result? If this is the case, South can afford to fail one try of finesse out of two; unless both Q and J are in West s hand, the target can be achieved. Therefore, the probability of declarer getting three tricks in this suit would be 0.75 (3/4). Finesse may sometimes be a good alternative of simply cashing all the winners. Use it wisely, and you will have a better chance to make the contract. Note that losing a trick to the dangerous opponent can be d e a d l y. 35
3. Finesse vs. cashing winners Now, you should know something about the two major methods that suits are play. However, which one is better? Let s compare them with some cases below and try to find out the better plan: cashing winners, finesse, or the two combined? Case 1: North: A 3 2 West:??? East:??? South: K 10 9 4 Well do you remember this one? It s the scenario that appeared in the last page. We have discussed about using finesse in this case, and it is impossible to take all four tricks. What about cashing winners? There are two dangerous high cards: Q and J. We need to drop them in two rounds, with A and K. There must be a 4-2 distribution, with Q, J on the side with two cards. The total number of possible combinations so that the contract can be made = [( 5 C 2 )*(2/6)*( 20 C 11 )*2] / ( 26 C 13 ) = 0.0323 (corrected to 3 significant figures) Playing high cards is slightly better here, although both have little chance to succeed. And this should be the best plan of play, other suits not taken into consideration. 36
Case 2: North: A J 3 2 West:??? East:??? South: K 10 9 4 South player is the declarer again. He needs all four tricks in this suit in order to make this no-trump contract. Now dummy has the lead. Declarer doesn t know the strength of his two opponents hands. Assume that there are side entries for the declarer to enter dummy s hand at any time. If he simply cashes A, K of the suit, the Queen must not be p r o t e c t e d b y m o r e t h a n o n e c a r d. Such number of combinations in a 5-0 distribution= 0 Such number of combinations in a 4-1 distribution = ( 5 C 1 )*(1/5)*( 21 C 12 )*2 (Explanation: the queen must be in the hand with 1 card in order to succeed; the answer is multiplied by 2 because 4-1 means both 4-1 and 1-4 ) Such number of combinations in a 3-2 distribution = ( 5 C 2 )*(2/5)*( 21 C 11 )*2 37
Probability that the contract will be made if the strategy cashing winners is used = [( 5 C 1 )*(1/5)*( 21 C 12 )*2+( 5 C 2 )*(2/5)*( 21 C 11 )*2] / ( 26 C 13 ) = 0.328 (corrected to 3 significant figures) Apparently, playing for the queen to drop is not a plan that is likely to succeed. Let s consider the chance of success if South uses finesse. In whatever position South finesses, there is half a chance to succeed. The probability of making the contract if South plays finesse=0.5 It has a higher chance to make the contract. However, there is an even better plan. This slightly increases the chance to make the contract, and will do less harm comparing to the finesse plan. First, play the Ace from dummy. This can prevent the singleton Queen from getting a trick. Then enter dummy again with another suit. Lead a low card and finesse the Queen of that suit. This covers the case that West holding a singleton Queen, and the total number of possible combinations is increased by ( 21 C 12 ). The probability of making the contract = [ 21 C 12 + ( 26 C 13 )/2] / ( 26 C 13 ) = 0.528 (corrected to 3 significant figures) This increases the chance of success to more than one half. Therefore, in some occasions mixing the play of finesse and cashing high cards can yield a better chance to success. (Of course, it would be better to gather more information from the defenders before playing this suit) 38
Case 3: North: A J 3 2 West:??? East:??? South: K 9 8 7 6 4 South player, as declarer, cannot afford to lose a trick here. South now has the lead. What is his best plan of play, and what is his maximum chance to make the contract? A) Cashing winners B) Finesse C) Mixing the two methods 39
A) Cashing winners If South plays the Ace and King of the suit, only a 3-0 distribution can stop him from winning all six tricks. Probability for South to win all six tricks = 1-[( 3 C 3 )*( 23 C 10 )*2] / ( 26 C 13 ) = 0.78 The chance is really high. Therefore, South player would consider playing A, K of the suit. B) Finesse If South plays low from his hand and finesses with the Jack, probability that South can win all six tricks is equal to 0.5 (1/2). The chance of success is much lower than playing A and K. 40
C) Mixing the two methods If South plays the King in his hand first, he has a better chance to succeed. If both East and West follow suit, South can then simply play the Ace from dummy; If East shows up (that is, he is unable to follow suit), South can place all three cards with West and perform a finesse against his holdings in the next round. The tricks cannot be taken only if East holds all three cards. Probability for South to win all six tricks = 1-[( 3 C 3 )*( 23 C 10 )] / ( 26 C 13 ) = 0.89 It is even better than Plan A, and is very likely to succeed. Of course, there are other methods that can force East player to give up the tricks, but we are not going to talk about this here. 41
Conclusion Cashing high cards and finessing are two very common practices in suit play. They are used in different occasions for a better chance to gain extra tricks. Mathematically there is always as better plan, however, sometimes it may be against the odds. Don t be surprised by the outcome. If we can make use of Probability and Logic when playing bridge, it will increase the chance of winning. 42
Reference Durango Bill's Bridge Probabilities and Combinatorics http://www.durangobill.com/brsplithowto.html Bridge Club Live http://bridgeclublive.com/a/information/bridge/convention_ca r d s / S E A c o l. a s p http://bridgeclublive.com/a/information/bridge/convention_ca rds/simplesayc.asp Mastering Bridge http://masteringbridge.com/ Wikipedia http://en.wikipedia.org/wiki/contract_bridge BridgeHands http://www.bridgehands.com/p/probability_hand_distribution. htm Easy Bridge Online http://easybridgeonline.com/ 43
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