Spade Heart Wigan Bridge Club Diamond Club
Posteriori Probabilities
Even the odds are only a probability?
Even the odds are only a probability?
I think most of us are aware of some of the basic odds and probabilities associated with cards and bridge in particular. So you will be pleased to know we will not be boring you with all those trivial facts and figures here.

But sometimes however we find that the basic odds alone do not always work out do they? Ever wondered why?

Well generally these odds and probabilities can be affected by other influences and factors.

It is these other influences and factors that give us what is known as Posteriori Probabilities.


Number of Cards Missing

Possible Distribution

Percentage Probability


1 - 1


2 - 0



2 - 1


3 - 0



2 - 2


3 - 1


4 - 0



3 - 2


4 - 1


5 - 0



3 - 3


4 - 2


5 - 1


6 - 0



4 - 3


5 - 2


6 - 1


7 - 0



4 - 4


5 - 3


6 - 2


7 - 1


8 - 0


*Percentages have been rounded to nearest whole number.




I am sure you have already seen tables like the above before or are already aware of the odds of how the missing cards in a suit may break.



This table taken in isolation alone is a good guide.





Have you ever wondered why the club experts always seem to drop your singleton trump King, when everyone else in the room is taking the finesse. Then on the next hand they take the finesse and get it right?






Ever wondered why they also make the percentage play when holding 7 cards divided 4-3 in one suit, making all 4 tricks in the suit by taking a finesse. Then on the next hand they play for the drop and again make 4 tricks from a similar holding?


Why are they so lucky and why do they always seem to guess right?



Firstly, I am pleased to inform that they do not always get it right, it just seems that way!! Secondly they are not just merely playing with the above ‘Tabled odds in isolation’.



Posteriori Probability also plays a part in their calculation of these seemingly ‘lucky guesses’



So what is Posteriori Probabilities?  A posteriori refers to knowledge derived from experience. The odds in isolation are a ‘priori probability’ but in bridge however, as we know, other factors can influence these odds. Such as the bidding (or lack of it on occasions), the opening lead or the play of the hand up to a given point, these occurrence’s will all have an effect and can alter these odds either way. This then becomes the Posteriori Probability.



A simple example for instance would be when an opponent doubles your opening bid of 1 Heart. You end up as declarer in 4 Hearts with the doubler having a singleton. You find that you also have a 4-4 spade fit and therefore have 5 cards missing in this suit. Do not think for one minute that the odds of this suit dividing 3-2 are 68% as in the above table. May I suggest with the double and the play so far, that this suit is almost definitely breaking 4-1?



So the odds have dropped dramatically from 68%. Put it another way if the choice is between placing a free bet of a £100 on this initially 68% chance or a free bet of £10 on the 28% odds my choice is the latter.  So in light of this additional information, in real terms the odds of a 4-1 break have actually risen to (without going into any complex deduction and calculation) somewhere nearer to 100% haven’t they?



Another example occurs when the opponents bid a suit pre-emptively during the auction. If you have to make a decision early in the play, about a missing honour in the trump suit for instance, do so in the knowledge that the division of the missing cards has now altered in view of the bidding.



So that is why on many hands the expert will put off the vital decision until the very last possible moment if permitted……. Not because he does not know what to do, on the contrary, he is trying to obtain as much information about the hand to enable him to make the most informed decision!!



So the next time you have a hand and the decision involves the ‘old maxim’ 8 ever 9 never, then just remember…………………



…………….. It should be - 8 ever 9 never, usually but not always, is there any other things from this hand I need to consider ‘old maxim’