Sick of Probabilities?

So many percentages and probabilities, it's impossible to remember them all, so you end not remembering any of them. The reality is that better players know the odds, but often the importance is not the exact percentages, but to know which is the better of two alternatives. So what is worth remembering?

Here I've tried to collect together what people think are the most important percentages.

Lets start by picking up a bridge hand.

Hand Evaluation and Bidding

Counting your High Card Points (HCP)

You are most likely to hold exactly 10 HCP (9.4%), and in fact over 50% of all hands have 7 to 12 HCP. 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.
About 10% of all hands will have
15 to 17 HCP, but less than half of them will be strong No Trump hands (the others will have at least one 5-card major). If you play 16-18 HCP for an opening 1NT this will happen only once in every 50 hands, but if you play 15-17 HCP for an opening 1NT this will happen once in every 25 hands. With the occasional 1NT opening (15-17 HCP), partner will pass half the time (46%) unless you use the Jacoby Transfer Convention (you partner has about a 30% chance of having a 5-card major in front of an opening 1 NT).

Being dealt 20+ HCP accounts for only about 1.5% of all hands (once in every 70 hands), and being dealt 24+ HCP will happen once in every 1,000 hands. This is why many players use a strong 2
♣ for 20+ HCP and now use a weak 2.

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, 7-8 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.
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 in an evenings bridge.
Playing with a partner you must add your 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.

What about distribution?

Once you have counted your HCP you are going to want to add distributional points. I prefer to count ruffing potential, e.g. suit shortness.
The reality is that you will usually have a more or less
'flat' hand such as 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 types of hand represent more than 70% of all bridge hands. If you look carefully you have nearly a 45% chance to hold one or more 5-card suits (but no 6-card suits). You are less likely (35%) to have a hand with one or more 4-card suits in your hand and no 5-card suits.
You have nearly a 17% chance of holding a hand with at least one
6-card suit (one in every 6 hands), but only a 3.5% chance of holding a hand with one 7-card suit (about one hand in every 30).
You have about a 31% chance of having a
singleton (every 3 or 4 hands), but only about a 5% chance to have a void (one in every 20 hands). Doubletons occur quite often (54%), or every second hand.
You have a 40% chance to hold an NT type hand 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). This does not mean you have the points for an NT bid.

You should add 3 points for a void, 2 for a singleton, and 1 for a doubleton. However once you have a 'fit' (usually meaning 8 cards in a suit) you should consider giving 5 points for a void, 3 for a singleton, but keep just 1 point for a doubleton.
You also might want to consider subtracting points for unprotected honours, and adding extra points for long suits, the 4 Aces and for 10's.

What to do about 'border line' hands?

There are always hands where you are not sure what to do. Pass or open, go to game or stop, support or bid your own colour?
There are
negative features that reduce the value of your hand:
Honour doubletons K-Q, K-J, Q-J. Q-x, J-x unless in partners suit - reduce you hand by 1 HCP.
Honour singletons (exempt the singleton Ace).
Honours in opponents' suit when deciding to support partner's suit (these points could be useless), and in particular honours in suits shown by the left hand opponent (they are sitting 'over' you).
Hands with 5 HCP and only 1 control (Ace or protected King), and hands with 7-8 HCP and only 2 control are 'poor' hands. Hands with 10 HCP should have 3 controls, hands with 12-13 HCP should have 4 controls, 15 HCP expect 5 controls, 17-18 HCP expect 6 controls, and with 20 HCP hands should have 7 controls. Anything less is a warning sign.

There are
positive features that enhance your hand:
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 your 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 the right hand opponent (you are sitting 'over' them).

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

You can also think about the
defensive strength of your hand:

Honours in shortish side suits, e.g. K-x-x.
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 (they might be worthless if the opponents have a void or singleton), and a lack of defensive values. Remember to think about the vulnerability.

There are some 'quick fixes' to assess the opening value of a hand, e.g. the 'rule of 20' or "open if the HCP plus the number of cards in the two longest suits add to 20 or more".
Another way is to add the number of cards in the 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.

I often use the
losing-trick count to assess 'border line' hands. It's not perfect, but I've found it helps to quickly forge a go/no-go decision.

Don't forget to consider using the so-called T.N.T. or
Total Number of Tricks when supporting a partner who has made a weak-two or weak-three opening bid or overcall.

Putting HCP and distribution together

Adding distribution points to HCP means that the most probable hand now becomes one with 11 total points (9.2%), and being dealt 7 to 12 total points represents now only 45% of all hands (7-12 HCP represented just over 50% of hands).

More importantly
being dealt 15-17 HCP accounted for 10% of all hands, but now 15-17 total points accounts for 18% of all hands. This change is mostly driven by re-valuing hands with strong distributional characteristics (e.g. singletons, or solid long suits). It does not change much the strong NT type hands.

Being dealt 20+ HCP represented only about 1.5% of all hands, but now adjusted for distributional points the 20+ total points accounts for 3.5% of all hands (from one in every 70 hands to one in every 30 hands).

How does your hand fit with your partners?

Back to HCP, you can expect to hold a combined 25+ HCP 17-18% of the time (one in every 6 hands).
A partnership can expect to hold a combined 24+ HCP about 23% of the time or a combined 23+ HCP 30% of the time.
And a partnership can expect to hold 30+ HCP about 2% of the time.
Expert players will try to push to game when holding a combined 23-24 HCP, particularly if vulnerable, and this can represent something approaching 50% of the hand they play. Experts consider that they have a fighting chance of game when holding 10 cards in a suit and with a 20-20 HCP split.

If you have exactly 12 HCP there is a 28% chance that your partner will hold 12 HPC or more. If you have exactly 15 HCP, 37% of the time your partner will hold 10 HPC or more, and there is a 19% chance they have 12 HPC or more. This is not symmetrical, so if you have exactly 10 HPC there is only an 8-9% chance that your partner holds 16 HPC or more.

What about the points held by your opponents? If you have 25-26 HCP with your partner your opponents will have their HCP split 5-10 or 5-9 or better (for you) more than 60% of the time.

By far the most common distribution across two hands is 8-7-6-5 which occurs in about every 4th hand (24%). The next most likely distribution for a partnership is 7-7-6-6 which will occur in every 10th hand (10-11%).
The only distributions that don't have an 8-card suit are 7-7-6-6 and 7-7-7-5, which represent only 16% 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 hands both teams will have at least one 8-card suit.
The fact is that if you have with your partner an 8-card suit, your opponents 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.

About 40% of the time if you have a distribution 4-3-3-3 or 4-4-3-2, your partner will have the same shape.

If you have a void or singleton, do
not expect your partner to have one as well.

If you hold exactly 5 cards in a suit the chance that your partner holds 3 or more cards in the same suit is nearly 55%, meaning you have
a 1 in 2 chance of finding an 8-card fit.

If you have exactly
4 cards in a suit, you partner has a 34% chance of also holding 4 or more cards in the same suit (and that means a 66% chance of not finding a 4-4 fit). You can see why many modern bidding systems focus first on finding 5-3 fits rather than 4-4 fits.

If you havew
two 4-card suits then there is a 63% chance that you will find an 8-card or better fit in one of the two suits. This rises to 84% when you have two 5-card suits (for finding at least one 5-3 fit).

If you hold exactly
6 cards in a suit, the chance that your partner has at least 2 cards in the same suit is about 75%.

Understanding these probabilities affects directly how partnerships should bid their hands. Imagine you have 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% probability to find an 8-card or better suit. Alternatively showing only a 5-4 distribution there is a 75% 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 HCP or game values, etc.

Warning - These probabilities are not symmetrical. For example if one partner holds exactly 3 cards in a suit, the chance that partner holds 5 cards in the same suit is only 9%. The only symmetrical case is when one partner holds exactly 4 cards in a suit, then the chance that the other partner holds exactly 4 cards in the same suit is always 22% whichever way you look at it.

Worried about bidding NT hands?

Many players appear to have more problems understanding when to bid game in No Trumps (NT) than in suit contracts. And many players find No Trump (NT) contracts harder to play.

You can bid 3NT hands with 24, 25 or 26 HCP, but success depends in part on how those points are split between the two hands.

With only 24 HCP you have about a 40% chance to make 3NT if your points are split 12-12, 13-11, or 14-10, and the chance to make 3NT starts to drop significantly when your point split is 18-6 or worse (36%).

With 25 HCP you have nearly a 60% of making 3NT with a points split of 13-12, 14-11, 15-10, and 16-9, but your chance to make 3NT drops below 50% for point splits 21-4 or worse.

With 26 HCP you have a 75% of making 3NT with points split 14-12, 15-11, 16-10, and the chance to make 3NT only drops below 50% for HCP splits of 25-1 or 26-0. 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. The logic is simple. Making most 3NT contracts 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 all depends upon the opponents having 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 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 HCP bidding a non-vulnerable 3NT (over 2NT) is profitable only when the point split is 15-9 or better. 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 the partner 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 club players do not properly integrate distributional points into their HCP bidding models.