I donÕt think there is any doubt whether Pythagoras was a bridge player. ItÕs all right going on about the square on the thingy being equal to the sum of the squares on the other two whatsits but what earthly use is that to someone who has to make six spades after his partnerÕs over optimistic view of his hand? For the same reason I doubt if Newton had time after spending his waking hours thinking about his Laws of Motion to give much thought to the correct way of asking for Queens in Roman Key Card Blackwood.
However this cannot be said about the Reverend Robert Bayes who was a bit of a brain box in the eighteenth century and came up with a theorum surprisingly called Bayes Theorum to do with Probability Theory which of course marked him down as an enthusiastic bridge player keen to work out whether to play for the drop or to take the finesse.
Now I will keep this simple. The key idea is that the probability of event A given event B depends not only on the relationship between A and B but also on the absolute probability (occurrence) of A not concerning B and the absolute probability of B not concerning A .
Is that clear then?
I donÕt mind admitting it is all Greek to me as is ArchimedesÕ principle – another guy who didnÕt play bridge. However apparently it has an application to bridge and is known as the theory of restricted choice. Roughly stated, the theory is that when a defender follows suit or wins a trick with one of two equivalent cards the probability of his having both cards is halved. DonÕt ask me why – get in touch with the Reverend Bayes for that information – but with a few examples we can see the thoughts of the Bayes boffin in action.
Take this suit:
You play a small one and your finesse of the ten loses to either the King or Queen. (My finesses always lose too!). Everyone knows it is now correct to finesse again and some people wisely say Ôrestricted choice you knowÕ.
The theory is that if East above held both the Queen and King then he could play either with impunity and that because he played one rather than the other reduces the probability of his holding both by 50%. If he held both he could play either 50% of the time but holding only one he has to play it 100% of the time. This is the bit I donÕt understand. ItÕs the guy who holds both honours who has a choice so itÕs more a theory of freedom of choice rather than a restriction. I still donÕt really understand the theory but I do know that it works!
I look at the above problem in another way. The chance of the two honours being split is 52% and the probability that they are both held by the same player is 48% or 24% each.
Going back to
when East wins the first trick with an honour then the chance of West holding both honours must be eliminated and we are left with a ratio of 52%:24% or 68% in favour of the second round finesse.
You play low to dummyÕs Jack and EastÕs King wins the trick. Later on you lead low again – should you play the Queen or finesse the nine? The restricted choice merchants tell us that if East held both honours he could have won the trick with either the King or the Ace and the likelihood of his holding both honours is therefore halved and again we have a 68% chance if we play to the Queen rather than the 50% chance of the finesse. I reach the same result by saying that West cannot have both honours if East wins the first trick and so this 24% has to be eliminated and we are left with the 52:24 ratio that we had before.
Yet one more example for you to consider.
You can lose only one trick so you play a low card to the Ace, West plays the ten and East the five. You then lead the 4 from dummy and East plays the 8. Should you play the Queen or the 9?
If the ten was a singleton you have no chance so you need to discover whether West played the ten from KT or JT. Again, restricted choice theory says that given the first holding West had no reasonable alternative to playing the ten but given the second holding West would have played the ten half of the time so only half of the probability of this holding can be counted so the play of the 9 is twice as good as the play of the Queen. This sounds logical to me. The chances of West holding KT or JT are exactly the same but with KT he has only one logical play whereas with JT he has two.
The lead of the King fetches the 5 from West and the Queen from East. On the next round West plays the six. Finesse or drop? Restricted choice says that because East could have played either the Queen or the Jack on the first round the probability of this holding must be halved and the finesses is nearly twice as good as playing for the drop. You can reach the same conclusion in another way by comparing the initial likelihood of East holding Q or J singleton with QJ doubleton. The answer is roughly 12:6 in favour of the singleton honour and therefore of the finesse.
Can we draw a conclusion from all this? I think we are on safe ground if we state that when an opponent drops one of two touching honours on the first round there is a strong presumption that his partner holds the other honour and you will not go far wrong if you always finesse in these situations.