Learning to Play Bridge - one advantage of bad bidding is that you can practice playing atrocious contracts

Bridge Club's Cheat Sheet

What you see above is a standard SAYC (or Standard American Yellow Card) bidding 'cheat sheet' for what many people know in Europe as a version of the 5-card major system.

A '
cheat sheet' is a type of 'crib sheet', 'reference card' or System Card for the prior listing of a partnership's understandings. These cards can be consulted by the opponents in order to better understand the explanation provided by an opponents after a particular call (bid) or play is questioned. 'Cheat sheets' have nothing to do with cheating at bridge. And for your information cheating at bridge is not in itself difficult, Dirty Hands (2016) talks about cheating in professional bridge. Here is a Wikipedia article on the latest situation concerning Fantoni and Nunes. More prosaically we must note that cheating is about a pre-mediated act of illegally obtaining hand or card information from a partner without being detected. It is not the same as doing something which might be considered 'unethical', such as making an inference from a partner's hesitation. This type of thing can be sanctioned by a Tournament Director without establishing intent.

On this webpage we are going to comment on the basic elements of the card game called
bridge, using this 'cheat sheet' as a guide. We will first look at how simple bidding systems evolved and what is the statistical basis for the way cards are distributed to players. We will then start in the long trip to understanding the meaning of each 'declaration' on the 'cheat sheet'.

There will be other webpages dealing with the play of the cards, interventions, defence, ...

A Bit of History

bidding systems were designed to permit a player to open one of a suit when they held a better than average hand, and to open at the two level with an even better hand, and at the three level with a very strong hand. This was the basis of the so-called strong-two bids. Today weak-two bids are common practice.

Again in the old days a simple reply at the two level could be made with 7-8 points, so a partner would have to bid at the three level with a good suit and 11-12 points or more. Then bidding systems introduced the idea that a two level reply was
forcing to 2 No Trumps (NT), or even game forcing. This creates more bidding space because a 2-over-1 is forcing (possible to game), and therefore there is a wider variety of bidding sequences (hence greater accuracy) to move to game or a slam. A hand that is worth a bid over 1, but is not worth a game forcing bid 2-over-1, must be bid 1NT.

Four card major suits were once the standard opening one-level bid, and the search for a 4-4 fit was the cornerstone of many bidding systems. Now many of the popular bidding systems use an
opening 5-card major, and search for an early 5-3 fit.

Modern day bidding systems are now focussed on what is more successful when 1,000's of hands are analysed. And bidding systems evolve as new forms of defence are found to counter the advantage of a bidding innovation. Initially the weak NT was popular, but new styles of defence were introduced making the
strong NT a better alternative today. All NT bids should define no more than a 3-point range (better still a 2-point range) and a very specific set of distributions, and above all should inform the partner as to whether to continue bidding or not.

Opening bids once required 13 points with a 4-card suit, 12 points with a 5-card suit, and only 11 points with a 6-card suit. Todays bridge players want to enter the bidding as quickly and as often as possible. More than 50% of computer dealt hands have partnership point counts between 17-23. Aggressive players are willing to bid game with a 10-card suit and a 20-20 point split. But if one pair is holding a 10-card suit, then it is likely the other pair also holds a 10-card suit, and therefore there is a potential for a double game swing. All the more reason to get in quick.

opening weak-two bid should be about 8 points (range 6-10) with a 6-card suit with two of the top three honours, or three of the top five honours. If the remainder of the cards are dealt out, the partner should receive about 11 points and 2 cards in the opening suit, precisely the combined points and cards to justify a 2 or 2 contract. The same logic applies with the same number of combined points and a distribution 5-3, or 4-4 with the possibility to ruff. Computer simulations show that 2 or 2 will make or go down with a profit. So in all cases it is desirable to get to a two level contract as quickly as possible. Take a good hand with a 6-card major in front of a partner opening a weak 2 in the other major. If the respective fits are 6-1 in both majors computer simulations show that it is better to play the contract in the 6-card major of the weaker hand. Another good reason to allow the opening weak-two bid. Opening weak-two bids also allows partner to make a T.N.T. (Total Number of Tricks) raise based upon the number of cards they are holding in that suit. A final reason to support the opening weak-two bid is that the weak hand could otherwise be forced to reply with 1NT despite holding a 6-card suit (remember many partnerships play the 2-over-1 forcing game).

Transfers and relays are now common in most bidding systems, simply because they significantly extend the range of possible bids open to the partnership. For example, a 2NT following a transfer bid over an opening 1NT can have additional meaning based upon the partnership agreement.

Variations on the
Roman Blackwood are now common and provide a powerful way to quickly determine the number of 'key' cards held by partner, and even if partner is also holding the Queen of trumps. Given that 3NT+1 is always better than 5 or 5, some partnerships are using 4 or 4 as a Roman Blackwood request for a slam in a minor suit.

The above 'little history' is just to reminder the reader about some of the key principles that they will be expected to apply in adopting a bidding system and conventions. A very good starting point is to think of all bids (including Pass) as having an approved range of points, and as being defined as either sign-off, encouraging, invitational, forcing, or forcing to game (or eventually slam).

Silly contracts are always linked to one partner forgetting one or more of the basic rules of their agreed bidding system.

Milton Work Point Count

You need some simple and practical way to start evaluating a bridge hand. What to take into consideration - high cards, suit distribution, quality of suits, controlling cards, fit with partner, general 'shape', …? The most basic evaluation method (initially conceived for a 'balanced' hand) is to assign numerical values to the top four honour cards.

Point Count:
   Ace = 4 HCP     King = 3 HCP     Queen = 2 HCP     Jack = 1 HCP  

There are 40 total
high card points (honour point count) in the deck, and the addition of the points in a particular hand is known as the high card point (HCP) count. An 'average' hand would be expected to hold 10 HCP in the form of 1 Ace, 1 King, 1 Queen, and 1 Jack.

Simply put, you need a better than average hand (say 12 HCP) to open the bidding.

There are several ways to refine the way you look at your HCP. Statistical analysis of hands show that this system of point counting undervalues Aces and 10’s. There are several ways to “adjust” your HCP, the ones I like are (i) add a HCP if you hold all 4 Aces, (ii) deduct a HCP if you hold no Aces, (ii) add a ½ HCP for each 10 in your hand.

Distributional Points

You should supplement the HCP with
distributional points, either suit length or suit shortness. When playing in a suit I prefer to count ruffing potential, e.g. suit shortness. Initially I add 3 points for a void, 2 for a singleton, and 1 for a doubleton. However once a 'fit' (usually meaning 8 cards in a suit) has been found I will tend to give 5 points for a void, 3 for a singleton, and keep 1 point for a doubleton. This type of distributional point count works well for 'three-suit' hands, but some players prefer to re-evaluate the distributional strength of their hand only when they are going to be dummy and only when they have the necessary trump support (avoiding both partners counting extra distributional points for the same suit).

With 'two-suit' and 'one-suit' hands it is often better just to count playing tricks rather than distributional points. Counting playing tricks assumes a reasonable distribution of cards and HCP around the table, including in partners hand. It does not mean that one hand is able to win a defined number of tricks without any help from the partner and against the worst-case distribution of cards and points with the opponents.

Some players will add additional distributional points for long suits. e.g. 1 point for any 6-card major suit and 2 points for any 7-card major suit. With these declarations it is sensible to only add the points if the long suit also holds at least two of the top three honours, or three of the top five honours.

The addition of 3 points for a void, etc., and the negative point adjustment for unprotected honours, and the extra points for long suits, the 4 Aces and for 10's are all together called the
Pavlicek Point Count. An important point here is that 13 Pavicek (total) point count becomes the threshold for making an opening bid. Pavlicek also added a couple of additions. Firstly he adds a point for any suit in dummy that is Qxxxx or better. Equally he adds 1 point for a 5-card suit if partner raised the suit without knowing it was a 5-card suit. He adds 2 points for every trump card over five if they had not been 'shown' to partner. And finally he adds an additional point for each card over three in a side suit. Adding everything together he sets game firmly at 26 points.


Sometimes it is difficult to evaluate the game or slam potential of a hand, e.g. should you pass or bid game or even a slam? One way is to look at your '
controls', e.g. the Aces and protected Kings you would expect to hold for your particular HCP. If you have 5 HCP I would expect you to hold an Ace or King, if not you hand is 'weaker' than 'average'. And so on, 7-8 HCP expect 2 controls, 10 HCP expect 3 controls, 12-13 HCP you should have 4 controls, 15 HCP expect 5 controls, 17-18 HCP expect 6 controls, and with 20 HCP you should expect to have 7 controls.

Another equally useful approach for these '
boarder line' hands is to look at the presence or absence of specific negative features that would or would not reduce the value of you HCP count. These are:
Honour doubletons KQ, KJ, QJ. Qx, Jx unless in partners suit - reduce by 1 HCP.
Honour singletons (exempt the singleton Ace).
Honours in opponents' suit when deciding to support your partner's suit (these points could be useless to your partner), and in particular honours in suits shown by your left hand opponent (they are sitting 'over' you).

There are also specific distributions that
enhance you hand and should make you consider a more 'aggressive' bid, namely:

Two or three honours in long suits, or better still honour sequences in long suits - add 1 HCP.
Honours in partner's suit when deciding to support it.
Honours in own suit when deciding to overcall.
Two or three intermediate cards in a suit (8, 9, 10) especially if headed by honours.
Honours in suits shown by your right hand opponent (you are sitting 'over' them).

And many players like to open the
suit to make overcalling more difficult.

Assessing a 'boarder line' hand is also about the
defensive strength of you hand. Good defensive values include:

Honours in shortish side suits, e.g. Kxx.
Honours and/or length in opponents suit.
Lack of honours in your own suit.

You might find it better to play a contract rather than defend if you have an abundance of honours and length in your own suit, and a lack of defensive values. Again your decision will also be affected by your vulnerability and that of your opponents.

There are some 'quick fixes' to assess the opening value of you hand, e.g. the 'rule of 20' where you open if your HCP plus the number of cards in your two longest suits add to 20 or more. Another way is to add the number of cards in your 5+ length and the number of honour cards in the same suit (Ace down to and including the 10). With 7 open at the 1-level, with 8 open at the 2-level, and with 9 open at the 3-level.

Losing Trick Count

A technique which I personally like to use to assess '
boarder line' hands (and this can be when deciding to bid, over-bid, to support partner, to pass, ... as well as to bid game or even slams) is the losing-trick count. This is used when you have found a trump fit. The question is often, do you raise, do you go to game, do you stop, do you invite, ....? The idea is to weigh shape and fit rather than just point counts. You look only at the top 3 cards in each suit. You then evaluate losers in each. If you have a void, you have 0 losing tricks. A singleton (other than an Ace) is 1 losing trick. An Ace and King in a doubleton is 0 losing trick, whereas Ax, Kx or KQ is 1 losing trick, and naturally something like Qx, Jx or xx is 2 losing tricks. So clearly with 3 cards such as AQx or KQx you have 1 losing trick. This gives your losing trick count for the hand. If you open you are presumed to have 7 losing tricks, so this helps you decide to open or pass with certain boarder line hands. If you partner also has 7 losing tricks he knowns that 7+7=14, subtract from 18 leaves +4, so you can bid up to level 4 or game in a major. If your partner has 8 losers they would jump to the level of 3, e.g. 18-(7+8)=3. and with 9 losers you simply raise to 2, e.g. 18-(7+9)=2. And of course if you only have 5 losers then 18-(7+5)=6, a slam is certainly possible. Some people count to 18 and others to 24 but the result is the same, e.g. 24-(7+7)=10 total tricks.
Clearly if your partner supports your suit you can then 'count' the number of losing tricks, and you can invite or even raise to game as required.
There are more complex loser counting systems, but I find this one is simple and provides help in deciding about boarder line hands. Some experts think that this count system undervalues Aces and overvalues Qx or singleton Kings.

Total Number of Tricks

In situations were you have a fit but are
competing with your opponents it is sometime difficult to decide to make an additional bid or pass. A simple rule is that you will make (more or less) the number of tricks corresponding to your combined holdings in trumps (this is the so-called T.N.T. or Total Number of Tricks). This is particularly useful when you partner has made a weak-two or weak-three opening bid or overcall, indicating 6, 7 or even 8 cards in a particular suit. If you have a weak defensive hand you can bid to the level of the combined holdings in your suit. Clearly you need to be careful about the vulnerability at the table.

Some Basic Probabilities

Wikipedia puts it quite succinctly, you need to have some understanding of probabilities in order to decide which strategy to adopt in bidding and playing bridge hands. Success is dependent upon the distribution of opponent's cards, and in the absence of other information (evidence) players need to identify the most probable route to success.

Below we have a short and simple discussion about what are
a priori probabilities, such as how cards and HCP might be assumed to be distributed before any additional evidence is taken into account. Naturally bidding and the actual playing of cards will change a players assumptions about how the cards and HCP are actually distributed in any specific hand.

Warning - Discussing probabilities is not the same as discussing what might or might not be actually declared and played in competitions, nor is it the same as what might best be played either by humans or by computer programs. In addition, I must stress that we have below what are called
a priori odds, or what we would expect to be the distribution of points, suits or face-value cards in a 'fair', 'random', 'unbiased' world with 'all things being equal'.

And we must also remember that the distribution of points, suits and face-value cards is not the only kind of uncertainty in bridge. Bidding systems are not perfect, and along with the strategies of the opponents (bidding and card play), a different form of uncertainty is introduced which is more dependent upon human psychology than on statistics. It must be said that studies on optimal play almost always focus on understanding the best line of play and defence when all four hands are visible to all participants (i.e. so-called double dummy).

Point Count

It is possible to calculate the probability to
hold any specific number of HCP in any specific individual hand. For example you have the highest probability (9.41%) to hold exactly 10 HCP, and a slightly lower probability (9.36%) to hold exactly 9 HCP, but a still lower probability (8.94%) to hold exactly 11 HCP. The probability to hold 10 HCP or less is 56%.

Being dealt
7 to 12 HCP accounts for just over 50% of all hands. This breaks down to one hand in every four will have 6-8 HCP, two hands in every seven will have 9-11 HCP, and 1 hand in every five will have 12-14 HCP.

Being dealt 15 to 17 HCP accounts for 10.1% of all individual hands (but only about 4% of all hands have the strong No Trump distribution).
Being dealt 16 to 18 HCP only accounts for 5.7% of all individual hands. Moving from a 1NT point count of 16-18 HCP to 15-17 HCP meant moving from 1 hand in 49 to 1 hand in 25, making an opening 1NT twice a likely with 15-17 HCP as with 16-18 HCP.

Being dealt 20+ HCP accounts for only 1.45% of all hands and thus will occur around once in every 70 hands, and being dealt 24+ HCP accounts for only 0.1% (once in every 1,000 hands).
Dropping the strong 2 Diamond bid and keeping a strong 2 Clubs to indicate 20+ HCP is a no-brainer, despite the fact that many experts question the efficiency of the weak 2 Diamond opening.

A nice way to look at the HCP distribution in a hand is to consider an evening's bridge playing 24 hands. You would expect to find no hands with 0-2 HCP, 3 hands with 3-5 HCP, 5 hands with 6-8 HCP, 7 hands with 9-11 HCP, 5 hands with 12-14 HCP, 3 hands with 15-17 HCP, and no hands with more than 17 HCP. Players spend far too much time learning bidding options and conventions for hands with lots of points and don't focus enough on part-score bidding, competitive interventions, and card play. And yes, I know it adds to 23 and not 24, but that just because of the rounding error. You chance of getting a hand with 18+ HCP is still only 4% or 4 times in 100 hands. So maybe you will get one of those hands next week! Good luck, but remember that with at least two of those wonderful 18+ HCP hands, you partner will Pass. Also for the occasional 1NT opening (15-17 HCP), partner will pass half the time (46%) unless you use the Jacoby Transfer Convention (responder has about a 30% chance of having a 5-card major in front of an opening 1 NT).
So given that more than 50% of your 24 hands will only have 5-11 HCP, there is all the more reason to get to know well the
weak-two bids even if a hand with a 6+ card suit will probably only occur once for you in an evenings bridge.
Playing with a partner you must add you HCP together. In nearly half (46%) of the 24 hands you will have together 18-23 HCP, so you had better be good at bidding and playing part-scores. Maybe 4-5 hands (23%) will have 24+ HCP and could be played in game. You and your friends, playing 24 hands a week, will have to wait 3 months before getting a small slam with 33+ HCP.


The most probable distributions in any individual hand are 4-4-3-2 (21.6%), 5-3-3-2 (15.5%), 5-4-3-1 (12.9%), 5-4-2-2 (10.6%) and 4-3-3-3 (10.5%). These 5 distributions represent more than 70% of all individual bridge hands.

You have a 44.34% probability of holding a hand with one or more 5-card suits, but no 6-card suits.

You have a 35.08% probability of holding a hand with one or more 4-card suits, but no 5-card suits or longer (only the 4-4-4-1 distribution which occurs with a probability of 2.99% is not also included in the NT type distribution).

You have a 39.85% probability to hold an NT type distribution with 4-4-3-2, 4-3-3-3, or 5-3-3-2 with a 5-card minor suit (54% of NT type hands will have the distribution 4-4-3-2, and an NT type distribution with a 5-card minor will occur only 19% of the time).

You have a 16.55% probability of holding a hand with one or more 6-card suits, but no 7-card suits or longer.

You have a 3.53% probability of holding a hand with one 7-card suit, but no 8-card suit or longer.

You have a 0.47% probability of holding a hand with one 8-card suit, but no 9-card suit or longer.

You have a 5.10% probability of holding a void in one or more suits (one in every 20 hands).

You have a 30.6% probability of holding one or more singletons (every 3 or 4 hands).

You have a 53.8% probability of holding one or more doubletons (every second hand).

Points and Distribution

You have a 6.51% probability of holding 12-14 HCP and a 4-3-3-3 or 4-4-3-2 distribution (and an addition 0.54% probability of holding one of those distributions but with exactly 18 HCP). You have a 0.97% probability of holding 12-18 HCP and a distribution 4-4-4-1. You have a 1.59% probability of holding 12-14 HCP and a distribution 5-3-3-2 with a 5-card minor suit. You have a 4% probability of holding 12-18 HCP and a distribution 5-4-2-2 or 5-4-3-1 or 5-4-4-0 with a 5-card minor.
So you have around a 13% probability of holding a hand with between 12-18 HCP and a distribution suited to an opening bid of a '
better minor' (so excluding the strong NT distributions with a 5-card minor suit).

You have a 6.50% probability of holding 12-18 HCP and at least one 5-card major suit.

You have a 4.08% probability of holding 15-17 HCP and an NT distribution (one in every 25 hands).

You have a 2.18% probability of holding a weak 6-card major (6-3-2-2 or 6-3-3-1) with 5-10 HCP, and an additional 1.09% probability of holding a weak 6-card
suit with 5-10 HCP.

Point Count Adjusted for Distribution

Above we saw HCP and card distribution probabilities for a 13-card bridge hand. But what happens if you take that purely statistical distribution of HCP and add additional distributional points based upon the Pavlicek Point Count? Remember this point count system adds 3 points for a void, 2 points for a singleton and 1 point for a doubleton, adjusts for unprotected honours, and adds extras point for long suits, as well as a point for holding all 4 Aces and ½ point for the 10's.

What happens is that the probabilities of holding hands with a specific number of total points changes, sometimes quite dramatically.

Based purely on HCP the hand with the highest probability was one holding exactly 10 HCP (9.41%). However adjusted for distributional points for all types of hands holding exactly 10 HCP, the probability of the occurrence of that type of hand drops to 9.02%, and the hand with the highest probability becomes the one with 11 total points (9.16%). Based purely on HCP the probability to hold 10 HCP or less was 56%, now the probability to hold 10 total points or less is just under 44%.

Being dealt 7 to 12 HCP accounted for 51.2% of all hands, but now adjusted for distributional points the range 7-12 total points (inclusive) accounts for only 45% of all hands.

Being dealt 15 to 17 HCP accounted for 10.1% of all individual hands, but now adjusted for distributional points the range 15-17 total points (inclusive) accounts for 18% of all hands.
When looking at the range 15-17 HCP we found that about 4% of hands were No Trump ones. Traditionally we do not count distributional points when evaluating NT hands. However many experts do add ½ points for useful 10's, and some experts down-grade hands with exposed honour cards (e.g. QJ doubleton). The increase from 10% to 18% is largely based upon hands with less that 15 HCP but which have strong distributional characteristics (e.g. singletons, or solid long suits) and thus move into the 15-17 total points range. There are also a few hands that have 15-17 HCP but are re-evaluated at 18+ when distributional points are added.

Being dealt 20+ HCP accounted for only 1.45% of all hands, but now adjusted for distributional points the 20+ total points accounts for 3.51% of all hands.

Partnership Combined Points (2 hands)

The previous probabilities were for individual hands, but it is also possible to calculate combined probabilities for the HCP and card distributions in 2 hands.

Based purely on HCP, a partnership can expect to hold a combined 25+ HCP 17.5% of the time (one in every 6 hands). This excludes additional points added for distribution, vulnerability, etc. And a partnership can expect to hold 30+ points about 2.2% of the time.
Expert players will try to push to game when holding a combined 23-24 HCP, particularly if vulnerable. A partnership can expect to hold a combined 24+ HCP 23.4% of the time and a combined 23+ HCP 30.3% of the time. This means that experts will try for game on up to 50% of their contracts. This equally true for expert opponents, so it can be that up to 50% of all contracts played at a table could be game attempts.

If one partner holds exactly 12 HCP there is a 28.1% probability that the other partner will hold 12 HPC or more. If one partner holds exactly 15 points there is a 36.5% probability that the other partner will hold 10 HPC or more, and a 19.1% probability that they hold 12 HPC or more. If one partner has exactly 10 HPC there is only an 8.5% probability that the other partner will hold 16 HPC or more. The probability distributions are not symmetrical.

If a partnership has 25 HCP how will the opponents HCP be distributed? The opponents HCP will be distributed between 5-10 and 10-5 a total of 67% of the time. If the partnership has 26 HCP the opponents HCP will be distributed between 5-9 and 9-5 a total of nearly 60% of the time.

Partnership Combined Distribution (2 hands)

By far the most common distribution across two hands is 8-7-6-5 which occurs with a probability of 23.6% (about every 4th hand). The next most likely distribution for a partnership is 7-7-6-6 with a 10.5% probability. The third most likely distribution is 9-7-6-4 with a probability of 7.3%, and then both 9-6-6-5 and 8-7-7-4 have a probability each of 6.6%. The fact is that if one partnership has an 8-card suit, their adversaries will also have an 8-card suit. Or put another way
if your adversaries have bid an 8-card suit, you and your partner also have an 8-card suit. Of course the distribution of the 8 cards can range from 8-0 through 4-4 to 0-8.

The only distributions that do not include an 8-card suit are 7-7-6-6 and 7-7-7-5, which total only 15.7% of all combined hands. Or put another way,
84% of all distributions across two hands include at least one suit of eight or more cards. This means that in more than 4 in every 5 games both teams will have at least one 8-card suit.

The definition of a 'good fit' is often used when a partnership holds an 8-card 'primary' suit, and another 7-card or 8-card 'secondary' suit (this will occur respectively about 30% and 10.3% of time).

One problem that partnership's face is when their two hand have the same shape. It can occur that partnerships play in a suit contract with few ruffing opportunities, whereas they should be playing in NT. There is a 40% probability that if one partner holds 4-3-3-3, then the other partner will hold the same exact distribution. It is nearly the same for two hands having the same distribution 4-4-3-2 (39.6%). The only other significant probability is for two hands with a 5-3-3-2 distribution (12%).

Interestingly if one partner holds a hand with a singleton there is only about a 2.5% probability that the other partner also has a singleton. If partner has a void, then there is only about a 0.03% probability of the other partner also holding a hand with a void.

If one partner holds exactly 5 cards in a suit, the probability of the other partner holding exactly 3 cards in the same suit is about 31% (29% for only 2 cards and only 17% for 4 cards). The probability that the other partner has less than 3 cards in the same suit is 45.6%, so the probability that the partner holds 3 or more cards in the same suit is 54.4%, meaning that
a partnership has a 1 in 2 chance of finding an 8-card fit.

If one partner has exactly 4 cards in a suit, the probability of the other partner holding exactly 4 cards in the same suit is only 22.2% (or 33.7% for holding at least 4 card support).

If one partner holds exactly 6 cards in a suit, the probability of the other partner holding exactly 2 cards in the same suit is about 33% (28% for 3 cards and only 19.5% for 1 card). The probability that the other partner has less than 2 cards in the same suit is only 23.8%, so there is a 76.2% probability that partner is holding at least 2 cards in the same suit.

Warning - These probabilities are not symmetrical. For example if one partner holds exactly 3 cards in a suit, the probability of the other partner holding 5 cards in the same suit is not 31%, but is only 9.1%. The only symmetrical case is when one partner holds exactly 4 cards in a suit, then the probability of the other partner holding exactly 4 cards in the same suit is always 22.2% whichever way you look at it.

When discussing probabilities players must be careful to understand the exact significance of what is written. If one partner is holding exact 4 cards in a suit there is a 22.2% probability that the other partner is holding exactly 4 cards in the same suit. However there is a 33.7% probability that the other partner is holding 4 or more cards in the same suit. And naturally there is a 44.1% probability that the other partner is holding 3 or less cards in the same suit. You can quickly see the justification for looking for fits of the type 5-3 because with one partner holding 5 cards in a suit there is a 54.4% probability that the other partner is hold 3 or more cards in the same suit.

If one partner has two 4-card suits then there is a 63% probability that they will find an 8-card or better fit in one of the two suits. This rises to 83.5% when one partner has two 5-card suits (for at least one 5-3 fit).

Understanding these probabilities affects directly how partnerships should bid their hands. Imagine a partner with a 6-4 distribution. Is it better to repeat the 6-card suit (and hide the existence of the 4-card suit) or to introduce a new 4-card suit (and hid the existence of the 6th card)? With a 6-card suit there is a 76.3% probability to find an 8-card or better suit. Alternatively showing only a 5-4 distribution there is a 74.9% probability of finding an 8 card or better fit in one of the two suits. So it is marginally better to repeat the 6-card suit. All the more true if the 6-card suit is a major. This is based purely on card distribution and not on game values, etc.

Distribution and HCP for NT Hands

3NT hands can be bid with 24, 25 or 26 HCP, but much depends upon how those points are split between the two hands. With only 24 HCP computer simulations show that a partnership has a 41% probability to make 3NT if the points are distributed 12-12, 13-11, or 14-10, but the probability to make 3NT drops significantly beyond the 18-6 point split (36%).
Another reason for the move to bidding 1NT with 15-17 HCP.

With a combined 25 HCP the partnership has a 59% of making 3NT with a points distribution of 13-12, 14-11, 15-10, and 16-9, and the probability to make 3 NT drops below 50% only for point distributions 21-4 or worse.

With a combined 26 HCP the partnership has a 75% of making 3 NT with a points distribution of 14-12, 15-11, 16-10, and the probability to make 3NT drops below 50% only for distributions 25-1 or 25-0. For all distributions of 20-6 points or better the probability of making 3NT still exceeds 70%.

Even or close-to-even HCP splits have the best change of making 3NT. Unbalanced HPC splits down to about 7 HCP in one hand have a 4% lower chance of making 3NT. The logic is simple. Making most 3NT hands will depend upon one or two directed leads (for example to exploit a finesse), but weak dummies provide few or no entires (or useful card combinations in general).

Naturally this discussion depends upon the opponents have their points more or less evenly distributed. NT hands are more difficult to play if the split of points in the hands of the opponents is not more or less even. This is all the more true if the stronger hand of the opponents is sitting behind the stronger playing hand.

We can begin to see why 25 HCP and 15-10, 16-9, and 17-8 card distributions are used as the target for making 3NT (59% of the time), but with just 24 HCP 3NT has only a 41% probability of success. Text books often say that partnerships should always play 3NT with 25 HCP, but 3NT with only 24 HCP is a break-even proposition. The books spend a lot of time looking at "either 8 tricks or 9 tricks" in NT. The problem is that playing 3 NT and making 8 tricks can be a valid risk-reward strategy, but making only 7 tricks can be a 'zero'. How to decide to 'up-grade' a hand and arrive in 3NT with only 24 HCP, and end up only making 7 tricks, or to 'down-grade' a hand and just play 1NT? We know that with 24 HCP, split 14-10, there is a 41% probability of making a 3NT game, a 35% probability of taking only 8 tricks (1-down), and an 18% probability of taking only 7 tricks (2-down). But the reality is that we do not know partners HCP to within 1 point, so often the discussion is a bit academic. Maybe the only sensible way to look at things is to 'down-grade' a hand when non-vulnerable, and to 'up-grade' when vulnerable.

Perhaps a more realistic questions is about already being in 2NT with 24 HCP, and trying to decide to accept or not accept a partners invitation, so bidding 3NT or Passing. Vulnerable it is right to continue from 2NT to 3NT with a better than 32% probability of making game. So with 24 HCP bidding a vulnerable 3NT (over 2NT) is always the best option unless the point split is 20-4 or worse. But with 24 HPC bidding a non-vulnerable 3NT (over 2NT) is profitable only when the point split is 15-9 or less. The conclusion is that with the points split more or less evenly (15-9 or better) always accept an invitation and bid 3NT. This solves the "1 NT - 2 NT - ?" question. The same computer simulations show that it is not profitable to routinely bid 3NT with only 23 HCP, vulnerable to not.

Probabilities look a little academic and dry, but they have a direct impact on both bidding and card play. A common sequence of bids might be 1 Diamond - 1 Spade - 2NT, showing that the opener has a balanced 18-19 points. If responder replied with an honest 6 HCP they should always bid a vulnerable 3NT (i.e. they have a greater than 32% probability of making a vulnerable game). With an honest 6 HPC but a non-vulnerable game they should Pass (in this situation partner needs at least 7 HCP to make a game bid worthwhile). A partner that replied with a shoddy 5 HPC has pushed opener into a 2NT bid with only a 36% probability of making just 2NT.

In the above example just 1 HCP can make a difference. In fact in part-score and game contracts 1 HCP adds ½ a trick to the probability of success (this advantage disappears in slams). For example KQJ is worth 2 tricks, whereas AQJ is worth 2½ tricks if entries exist for the finesse.

Based purely upon probabilities it would be wise to pass a 20-21 point 2NT bid when holding 4 points or less. However distribution is not the only criteria. Access to the weak hand could be a determining factor in being able to make a finesse or to developing a long suit. So a partner might pass with Qxx, Jxx, Jxx, xxxx, but bid 3NT with Axx, xxx, xxx, xxxx or QJxx, xxx, Jxx, xxx.

A word of warning, computer simulations are about simulating the different distribution of cards (and thus points) but they assume that the games are played in the best way possible against the best defence possible. So you better know how to best play the hands. Playing for 3NT with only 24 HCP could be a waste of time if you do not know how best to play the contract. However playing 3NT with only 24 HPC could be easily if the opponents are not good players and make the wrong lead. You decide.

But before you decide, consider the results of this study of alternative bidding and play options on a very large set of double dummy hands. What they found was that there was virtually no chance of 'going down' in the game contracts that were actually bid and where the partnerships held 26+ HCP and a least 8 trumps. What was surprising was that 77.1% of the contracts that could have been game declared and 'made' were in fact not bid. Clearly real world bridge players tend to prefer a system that keeps the 'going down' rate low, and are less concerned about missing game contracts. The problem is to find the bidding system that actually predicts success better than using a simple HCP model. Some experts have suggested that real world players do not properly integrate distributional points into their HCP bidding models.

Probability of Holding Specific Cards

A player has a 51.8% probability of holding 2 Aces when they have 16 or more HCP.
A player has a 55% probability that a strong NT hand is holding 2 Aces (1.6% holding 0 Aces, 26% holding only 1 Ace, 17% holding 3 Aces, and less than 1% holding 4 Aces).
A player has a 50% probability that a 2 NT hand (20-21 HCP) is holding 3 Aces (7% probability that the hand had all 4 Aces).

Missing Card Splits (in opponents hands)

2 missing cards will split
1-1 52% of the time and 2-0 48% of the time

3 missing cards will split
2-1 78% of the time and 3-0 22% of the time

4 missing cards will split
3-1 49.7% of the time, 2-2 40.7% of the time and 4-0 9.6% of the time
One of the 'classical' card playing challenges is when declarer holds 9 cards in a suit with the two top honours, but is missing the Queen. The question is to finesse or to play Ace and King to drop Q-x? A simplistic reply might be that a 2-2 split only occurs about 40% of the time, so a player must finesse. We must remember that a 3-1 split occurs with a probability of 49.74% and a 4-0 split occurs with a probability of 9.56%. Often the finesse can be taken only in one direction, but occasional it can be taken in either direction. In the first case the exposed honour should be played. There is a 12.4% probability that the Q is a singleton in a 3-1 split. There is also a 4.78% probability that the Q is still on the right side (for declarer) in a 4-0 split. So the total probability of dropping a Q by playing a top honour and adapting to the outcome (play a successful finesse or play to drop) is in fact 58%. Naturally this is based purely on the probabilistic distribution of cards, and often more information can be inferred from the bidding and play of the cards.

5 missing cards will split
3-2 67.8% of the time, 4-1 28.3% of the time and 5-0 3.9% of the time

6 missing cards will split
4-2 48.4% of the time, 3-3 35.5% of the time, 5-1 14.5% of the time, and 6-0 only 1.5% of the time.

Playing Tricks

The expression 'playing tricks' often pops up in bridge manuals and is particularly useful when looking at 6+ card suits for weak two and pre-emptive bids. You count 1 point for each winning card (e.g. A-K-Q would be 3 points), and 1 point for the 4th, 5th, 6th, cards, etc. You also count ½ points to represent potential winning honours (e.g. singleton K would be ½ point, K-J-10 would be 1½ points, A-Q-10 would be 2 points, and A-Q-J would be 2½ points).

The most probable are hands with 5 playing tricks (12.6%). Around 45% of all hands hold between 4 and 6 playing tricks.

Losing Tricks

You will remember that the losing trick count is useful for evaluating hands that could be played in a suit. Each suit can only have 3 losers. A singleton is just 1 loser, an A-K-x-x would also be just 1 loser, and a J-10-x-x-x would be just 3 losers. Generally an opening bid is assumed to have 7 losers, and a reply at the one level is presumed to represent 9 losers. Subtracting the number of losers for the partnership from 18 tells us the level that the hands can be played at, e.g. 18 - 7 (opening bid) - 9 (one level reply) = 2, suggesting that 8 tricks can be won and a 2-level contract would be appropriate.

The most probable number of losers in a hand is 8 (24.5% of all hands hold 8 losers). Holding 7 losers has a probability of 23%, or about 1 hand in 4.

Holding 10 losers or more happens 10.4% of the time, or 1 in 10 hands should be automatically Passed unless you are forced to speak.

Holding exactly 9 losers happens 17.6% of the time, or around 1 in every 6 hands.

Holding only 5 losers happens only 6.9% of the time, and holding 4 or less losers in one hand happens just 2.8% of the time.

Quick Tricks

Quick tricks are also often called 'defensive tricks' and represent tricks that might be won taken from just the two top cards in each suit, and assuming that the shortest suit in the hand is trumps. A-K-J-10 would be just 2 quick tricks, A-Q-J would be 1½ quick tricks, K-Q-J-10 would be just 1 quick trick, and even a singleton K would be a ½ quick trick.

About 50% of all hands have 1, 1½ or 2 quick tricks. Nearly 70% of all hands have 2 or less quick tricks, and only 20% of hands have 2½ or 3 quick tricks in them.

Some players expect an opening hand to hold at least 2½ quick tricks, and that game is possible with a combined 5 quick tricks. Just as with losing tricks, quick tricks can be used to evaluate 'boarder line' hands. You must/will always open the bidding with 14 HCP, but do you open with 12 HCP or 13 HCP and only 1½ or 2 quick tricks? Using quick tricks can stop players adding too many 'soft values' to their hand, and can certainly help in deciding to compete, to double, or to Pass. Milton Work attributed 4 points for an Ace and 3 points for a King to stress how important these cards are in controlling the timing needed to make game. Opponents who hold Aces and Kings can gain time to create addition winners, and defeat a contract. Some players can go too far in the bidding when they try to offset the lack of quick tricks with 'soft values', and this is also true for those players that double a game contract based on a soft HCP count rather than on quick winners. Example, holding Q-J-x, J-x, A-x-x, Q-J-10-x-x, looks nice as a defensive hand, but actually only holds 1 quick trick. Keeping the same number of points, this hand K-Q-x, x-x, A-Q-x, x-x-x-x-x has 2½ quick tricks, and could be even stronger if sitting behind the opener.
Some players judge their competitive overcalls and defensive bids based upon quick tricks.

It is very important to only use a 'take-out' Double in a competitive auction when holding good quick tricks. This kind of Double in a competitive auction informs partner about the nature of your HCP holding, e.g. it could mean 13 HCP and with at least 2 quick tricks. This helps a partnership understand when to Pass or Double the final contract. If partner is holding 13 HCP in 'softer' values an overcall might be a better option than making a defensive oriented Double.

Many expert players have said that quick tricks and controls are the key to make a good assessment of a bridge hand.


We saw that Aces and Kings are controls, and controls can be easily counted if you give Aces 2 points and Kings 1 point. Hands are most likely to hold 3 controls (20.9%) or 2 controls (20.5%). More than 60% of all hands contain 3 or less controls, and nearly 80% have no more than 4 controls.

Vacant Places Method

What has been described above are the
a priori probabilities of how cards, suits, and face-value cards would be expected to be distributed in an ideal or totally random world. Bidding systems were developed based on these probability distributions and on hypotheses about what would or would not be both successful (in the widest sense of the word) for a partnership. It must be said that I have not yet found any websites that focus on the way to update the a priori probability distributions of opponents's cards based upon evidence as it emerges during the bidding and play. This problem is considered too complex for players to manage during a game, so they look for simplifications such as the Vacant Places Method.

The Vacant Places Method is an aid or technique for playing the cards. I have included it here because it is based upon the underlying probabilistic distribution of cards. It is a simplification that tries to represent the way probability distributions of points, suits and face-values change as a function of 'evidence', i.e. the bidding and play of the cards.
The idea is simple. Imagine at the beginning the declarer knows that in partnership with dummy they hold 9 out of 13 cards in a particular suit, so they start by thinking that the remaining 4 cards might be distributed 2-2 (40.7% probability). However, declarer also does not have the Queen, so they will need to determine which opponent is more likely to hold that particular card. As the different tricks are played out the exact distribution of some suits and the exact location of some cards and face values becomes known. This information (or evidence) can be used to better to attribute the still unidentified cards (distribution, suit and face values) to a number of so-called 'vacant places'.
The bidding may help declarer determine if that hypothesis is likely or not. The lead also may help declarer confirm or disprove that idea.
Let us continue with the idea that declarer is playing a contract, say, holding 9
, but missing the Queen. Does declarer finesse the Queen? There is about a 60% chance that the Queen is sitting Q singleton in front of x-x-x, or Q-x-x in front of a small card singleton, or Q-x-x-x in front of a void. By playing either the Ace or King (and keeping open the finesse option) declarer can eliminate the Q singleton and the Q-x-x-x options. So now declarer knows that the possible distributions are 3-1 or 2-2, but he still does not know where the Queen is. Playing a small card to his finesse, the right-hand opponent follows suit. Does declarer finesse the right-hand opponent because they hold Q-x-x, or does declarer play the other top honour intending to drop Q-x in their left-hand opponents hand? We already know that declarer had a 58% probability of finding the Queen by playing intelligently his top honours and keeping open the finesse option. But what does 'vacant places' tell us? Let's assume that declarer knows nothing about the underlying distributions in the opponents hands. But what declarer does know is that the right-hand opponent still has 11 card, and the left-hand opponent still has twelve cards. The outstanding Queen can be any one of 11 cards with the right-hand opponent, or any one of 12 cards still with the left-hand opponent. With no other information there is a slightly higher probability of the Queen occupying one of 12 'vacant places' out of a total of 23 'vacant places', thus about a 2.2% advantage to dropping the Queen with the left-hand opponent. This logic is dependent upon the fact that all the Hearts but one are accounted for. So in this case the logic of 'vacant places' comforts the decision to intelligently play off the two top honours, hoping to drop the Q-x.
But let us imagine that during the auction the left-hand opponent opened a weak-two in
. Declarer can see that their partnership hold only 5 cards in . Declarer thus can infer that the left-hand opponent holds 6, and that the right-hand opponent holds only 2 cards. Turning to , the left-hand opponent now only has only 6 'vacant places', and the right-hand opponent has 9 'vacant places'. There is now a 9/15 (60%) probability that the right-hand opponent is holding the Queen, so the finesse becomes the preferred option (remember declarer had an a priori probability of 58% to drop the Q by intelligently playing out their two top honours). It is of course possible that the left-hand opponent mis-bid a 7-card suit, but this will not change the overall conclusion. In this case the inference is still valid despite the fact that declarer does not have complete knowledge of the distribution of the suit.
There is quite a lot written about the 'vacant places' method applied to bridge, some of it looks a bit over-simplified. The essence is that if we have 'complete' evidence or knowledge about one or more suits, the "the probability that an opponents holds a particular card in any other suit is proportional to the number of vacant spaces remaining in that hand". We saw that the weak-two bid allowed the declarer to know the distribution of that suit in the opponents hands. In a different situation this complete knowledge of a suit could be provided by (say) the honest lead of the 4th card in a players longest suit.

More information

If you want to delve into the finer details of these probabilities check out the
Occasional Enthusiast and TaigaBridge. You may have to hunt around a bit but all the above data can be found 'somewhere' on the web, or you can also calculate them yourselves. Here are a few more webpages that might be of interest, Hand Evaluation Stats, Durango Bill, Many of the statistical tables on these different websites are the same, but not all, so it's important to read carefully the exact description of the calculations being made.

Contract Requirements

Partnerships need (in your two hands) to make these contracts:

Suit part-score (bid of 1, 2 or 3 in any suit) - 18-24 HCP and at least an 8-card trump fit.
A partnership has a 57% probability of holding between 18-24 HCP, and a 84.3% probability of holding at least one 8-card suit

No Trump part-score (bid of 1NT or 2NT) - 20-24 HCP and no 8-card major suit fit.
A partnership has a 37% probability of holding between 20-24 HCP, and a 51% probability of holding an NT distribution but not holding at least one 8-card major suit

Major-suit game (4H, 4S) - 25+ HCP and at least an 8-card major suit fit.
A partnership has a 12.7% probability of holding 25+ HCP, and a 42.2% probability of holding at least one 8-card major suit

Minor-suit game (5C, 5D) - 28+ HCP and at least an 8-card minor suit fit (and no 8-card major fit).
A partnership has a 5.8% probability of holding 28+ HCP

No Trump game (3NT) - 25+ HCP and no 8-card major suit fit.
A partnership has a 12.7% probability of holding 25+ HCP, and a 51% probability of not holding at least one 8-card major suit

Small slam (any bid of 6) - 33+ HCP (ensuring at least 3 Aces).
A partnership has a 0.35% probability of holding 33 or more HCP

Grand slam (any bid of 7) - 36+ HCP and all four aces.
A partnership has a 0.03% probability of holding 36 or more HCP

These contract guidelines apply when you have relatively balanced hands. If you have a strong trump fit, long side suits and/or unusual distribution, you'll need fewer points to make these contracts.


Below I have outlined a very simple set of opening bids and replies in order to get the reader to start to think about guidelines for bidding. These bidding 'rules' are generic and could be slightly different depending upon a detailed partnership agreement.

Warning: Bidding systems involve inevitable ambiguities. The systems are a set of rules that cover different situations (or distribution of cards), but rules can overlap. Based upon the cards and the bidding, different rules can be considered which can produce different bids. Experts talk of two types of games, some are like chess where all the information is available to all parties, and some are like bridge and poker where the players do not have access to all the information (cards are hidden). Increasingly the difference between the best bridge players is in bidding to game, given that their card playing skills are comparable. Reducing ambiguity in bidding systems is a desirable target, but at the same time systems must be simple enough to memorise. There are literally hundreds of different bidding systems, and
Wikipedia just lists a few of the more popular, but there are multitude of different systems ranging from the 'natural' to the 'unusual' and through to the bizarre.

'Natural' bidding systems are said to be easy to learn and use, but they employ opening bids that often have a wide range of points and are therefore imprecise and sometimes difficult to control. However since they are often used, they are well developed, well taught, and can be played almost anywhere in the world.

Acol, originally a British bidding system, is supposed to be natural and simple, and employs light opening bids and systematic limit bids. Many Acol players use a weak No Trump of 12-14 HCP, and strong-two bids. However, there are many variations that can include a Multi 2 Diamonds.

Majeure Cinquième (5-card major), originally a French system, employs opening bids with 5-cards majors, a 15-18 HCP No Trump, weak-two's in majors, an opening 2 with 18+ HCP and a 5+ suit, and an opening 2 forcing game and any shape.

Standard American is similar to the Bridge World Standard system, and employs opening bids with 5-cards majors, a 15-17 HCP No Trump, weak-two's in majors and in , and an opening 2 with 22+ points and any shape.

Many natural systems, and the Majeure Cinquième and the Standard American use
Stayman, Jacoby transfer, and a form of Blackwood. Many of the earlier versions have or are migrating to more or less the same basic set of rules.

In addition there are a number of '
Strong Club' bidding systems (sometime also called 'Precision' systems) which reserve an opening artificial 1 to mean 16+ HCP and any distribution. Thus all other opening bids are limited to less than 16 HCP.

There are literally hundreds of bidding systems being used around the world, so the key is to pick one that corresponds your needs, learn it, and above all use it often with the same partner. Most problems with the main bidding systems are the result of one partner not following the bidding rules and not respecting or understanding the principles underlying those rules.

Some Rules for Opener

Opener is the first player to make a bid other than Pass

First or 'Opening' Bid

You should
always open the bidding at the 1-level if you have 13-20 HCP.
Remember you have a 19% probability of holding between 12 and 20 HCP with any distribution.

You can open a hand with 11-12 HCP, if you have a long suit, distributional strength and/or good quick tricks.
Dropping to a lower limit of 12 HCP increases the probability of holding such hands from 19% to 26%, and dropping the lower limit to 11 HCP increases the probability of holding a hand in the point range 11-20 HCP to 34%. The key here is suit length, you have a 46% probability of holding a 5-card suit, but only a 17% probability of holding a 6-card suit. Also there is a 36% probability of holding a singleton or void. Some players feel that it is better to 'get in quick' so they can properly inform their partner about any unusual distributional features. Other players might consider downgrading their 6-card suit and employ a weak-two opening bid.

Counting losing tricks helps you decide about 'boarder line' hands. Some players will open if they have 2½ quick tricks and a good suit. Sometimes it might be better to pass if the likely response of your partner will make your next bid awkward, e.g. your partner bids your singleton and you have no place to go. Some 'boarder line' hands are better described by not opening and waiting to make an overcall later in the bidding. Sometimes it is safer to open a vulnerable 1-bid than to make an overcall later in the bidding, e.g. you can compete better for a partial score and also pass if needed. If you have an easy rebid you should always try to open.

Many plays will open 'weak' in the 3rd position (10-11 HCP) if they have a good suit or if they can pass on any reply of their partner. In the 4th position it is better to only open if 'strong' (13-14 HCP).

meaning of opening bids:

1 No Trump means exactly 15-17 HCP and balanced distribution (4-3-3-3, 4-4-3-2, or 5-3-3-2 with a 5-card minor). Distribution points are not counted. Some players will open 1NT with a good 14 HCP.
Remember you have 4.1% probability of holding a 1NT opening hand. This can be compared with the old 16-18 HCP range used in the past, which had a probability of occurrence of only 3.55%.

1 or 1 means 13-21 HCP and at least 5 cards in your suit.

1 or 1 means 13-21 HCP and at least 3 cards in your suit (and no 5-card major in hand). A 1 or 1 opening is often called a 'better minor', it tells partner you have opening points, but you do not have the requirements for 1, 1 or 1NT. Choose your longer minor, not your stronger minor. If you have two 3-card minors, always open 1 to keep the bidding low. Some players use a 1 for at least 4 cards, but this means that they might be forced to open 1 with only 2 cards. The best idea is to only open 1 with at least 4 cards, except when you have 4, 4, 3 and only 2, then you also open 1. The better minor is the best approach since a 1 opening with only 3-cards (and no alternative bid) will happen only 4% of the time. You have a 43% chance of having 4 cards in , a 34% chance of having a 5-card suit, and a 19% chance of having 6 or more . Remember opening 1 with only 3 cards will happen only 16% of the time. You have a 38% chance of having 4 cards, a 29% chance of having a 5-card suit, and a 17% chance of having 6 or more . With 5-4 or 5-5 in minors most players will try to open 1 and rebid 2, even when the maybe 5-cards and are 4-cards. A hand with 5 and 4 and just an opening point count is a difficult hand to bid. After opening a minor try if at all possible to get into 1NT, showing 12-14 HCP. Remember if you bid 1 then rebid 2, you are showing 4-5, 4, and 16+ HCP.

2 No Trumps means exactly 20-22 HCP and balanced distribution (same as for 1NT but some players will open 2NT even with a 5-card major).
Remember that you have a 1.2% probability of holding any type of hand with 20-22 HCP (and you only have a 1.5% probability of holding 20+ HCP)

2, 2, 2 means a weak hand (5-11 HCP) with a good 6-card suit. No more than one Ace or King outside your suit. Many players will open very weak when not vulnerable, and if vulnerable you can expect a stronger suit and more playing strength.
Remember that you have a 2.2% probability of holding 5-10 HCP with a 6-3-2-2 or 6-3-3-1 distribution.

2 means strong, artificial and forcing (21+ HCP) normally unbalanced with a good 5-card suit (6-card minor), quick controls in other suits. You may use this bid with fewer points if you have a long and strong suit and you have at least 8 playing tricks in a major and 9 playing tricks in a minor (or you have only 4 losing tricks).
Remember that you have only a 0.8% probability of holding 21+ HCP with any distribution.

3 of a suit means a weak hand (5-9 HCP) with a long, strong suit (7 or more cards with at least 2 of the top 3 honours, or 3 of the top 5 honours). Some players will bid 4 of a major with 8 or more cards and solid honours, a so-called 'shut-out', but be careful of vulnerability. A preempt also show poor defensive values, e.g. no Aces or Kings in side suits and no 4-card majors. A preempt usually means you have a max. of 1½ defensive tricks, and you should assume one of the opponents will have a singleton in your suit. You are not expected to bid again unless specifically asked to by your partner, e.g. bidding a new suit, cue-bidding opponents suit, or asking for aces.

3 No Trumps means 25+ HCP.
Remember that you have only a 0.05% probability of holding 25+ HCP with any distribution.

If you have two 5-card suits, open the higher-ranking suit, then rebid the lower-ranking suit. If you have opening hand and 5
and 5, open 1 and bid at your next turn, allowing partner to choose between your two suits without raising the level of the bidding.

Second Bid of Opener

Let us have a look rapidly at your
second bid options (assuming your partner has “responded” and your opponents have passed):
Pass means you are minimum (max. 15 HCP), you do not have a 5-card suit worth rebidding, your partner initially passed and only bid when 'forced' to by your bid or that of your opponents, you are not unhappy with your partners suit and you can see no future in an alternative contract.
A new suit at the 1-level means at least 4 cards in the suit (13+ HCP).
Simple rebid of your suit means minimum opener (max. 14 HCP) might suggest extra length (usually a 6+ cards), but you may have no option but to re-bid your 5-card suit. It is a good idea to think about your rebid before opening and avoid opening 'light' if you might end up having to rebid a poor 5-card suit. Try to avoid re-bidding a 5-card minor suit, and prefer 1NT rebid if at all possible.
Single raise in partner's suit means 4 cards in the suit partner responded and a minimum opener (max 14-15 HCP).
No Trump means a balanced minimum opener (12-14 HCP) WITHOUT 4 cards in partner's suit and without a new 4-card suit you could bid at the 1-level. To show a stronger balanced hand, you would have opened 1NT (15-17 HCP).
Bid at the 2-level a new lower ranking suit than your first suit means at least 4 cards in the suit (showing at least 15 HCP).
Bid at the 2-level a new higher ranking suit than your first suit means at least 4 cards in the suit (showing at least ……).
Jump-rebid your suit shows a strong opening hand with a long suit (…..).
Jump-raise in partner's suit shows a strong hand with a fit for partner (…..).
2NT shows a very strong balanced hand (18-19 HCP), open a suit bid and then jump in No Trump.
Raise to game in partner's suit means excellent distributional hand (19 HCP) and good support for responders suit (you should have only 5 losing tricks).
Raise to game in your own suit means excellent distributional hand (19 HCP), your opening suit is long and solid, and you have values in responders suit (you should have only 5 losing tricks).

Rules for Responder

After your partner opens the bidding.

Respond to partner's opening bid if you have 6 HCP or more.
If partner opens 1
or 1 and you have 3-card support, always raise to confirm the 8-card trump fit.
If partner opens 1
or 1 and you have a 4+ card major, always respond 1 of your major. If you have two 4-card majors, respond the cheaper major to keep the bidding low.
If you have fewer than 10 HCP, DON'T go to the 2-level unless you're raising partner's suit (to confirm a trump fit) or rebidding your own extra-long suit. With some unbalanced hands, you'll have to respond 1 NT to keep the bidding low.

Your first response (basic options)
Pass means less than 6 HCP, and/or more than 9 losing tricks. You can pass even an opening 1
1 of a new suit means at least 4 cards in your suit and at least 6 HCP (e.g. the usual 1
over 1 , but can also be 1 over 1 but you are also saying to your partner that you do not have 3-cards in ). This also means no more than 9 losing tricks.
2 of a new minor (non-jump) means a 5+ card suit and at least 10-11 HCP (e.g. 2
over 1).
2 of a new major (non-jump) means a 5+-card suit and at least 10-11 HCP (e.g. 2
over an opening 1).
Single raise of opener's major means at least 3 cards in opener's suit and 6-10 HCP.
Single raise of opener's minor means at least 4-5 cards in opener's suit, 6-10 HCP and no 4-card major suit.
1 No Trump = a weak hand (6-10 HCP) with no support for opener and no 4-card major. If you have 4-card hearts and 6-10 HCP, and your partner opens 1
, you bid 1NT.
Jump-raise in partners suit shows a good hand with support (11-12 HCP) and usually with 8 losing tricks (invitational to bid game with a better than minimum hand).
Jump-shift in a new suit means a very strong hand (17-19 HCP) and a long, strong suit, clearly interested in a slam.
Jump to 2NT shows a stronger balanced hand (11-12 HCP) and usually with 8 losing tricks (invitational to bid 3NT with a better than minimum hand).
Jump to 3NT shows a stronger balanced hand (13 min. HCP) and more importantly only 7 losing tricks.
Jump to game in partners suit shows a good hand with support (13 min. HCP) and more importantly only 7 losing tricks.

What do you do if your opponents intervene? We will deal with this separately, but as a general rule if you can make your normal or “natural” response, bid it.

Sources of Information on Bidding Systems and Card Play

Bridge Guys is a horrendously designed website, but is home to webpages on Conventions and much else, but you will have to hunt around to find stuff.
A number of Bridge Club's hold introductory and training courses, and some put the documentation up on the web. For example the
Willow Valley Bridge Academy has put on the web a variety of training material that looks well organised and up-to-date.
The Oxford Bridge Club has
Ian's Bridge Clinic, which is just one example of how bridge clubs can document and discuss interesting hands and more generally run club tournaments, etc.

Fancy Playing Bridge on the Web?

These five packages are certainly worth a look
Jack Bridge, WBridge5, Shark Bridge, Ginsberg's Intelligent Bridge Player (on Bridge Base) and Bridge Baron.
And there are also
Micro Bridge, Q-Plus Bridge (from 2016), Blue Chip Bridge, Bridge Buff, BridgeComposer, Dealmaster, Deal, and DDS that might also be worth a look. GoTo Bridge allows players to select different bidding systems from SAYC, ACOL, 5-card major, Polish Club, and 2/1.

There is an annual
World Computer-Bridge Championship which in 2018 will be in its 22nd year, and WBridge5 won it in 2017.

But '
buyer beware' concerning any and all computer-bridge software.