Everyone knows the adage “eight ever, nine never” which refers to finessing for the queen of trumps. I recently received this question about it, which is answered below.
Why do you play for the Queen to drop when you have nine trumps, isn’t a 3-1 split more likely than a 2-2?
It is true that when you are missing four cards, a 3-1 split is more likely than a 2-2 split. However a specific 2-2 split is more likely than a specific 3-1. When you cash the Ace and lead towards the KJ and everyone follows, there are only two possible cases left, the 3-1 where the queen is finessable and the 2-2 where the queen is dropping. Read on for a fuller explanation of why the 2-2 is more likely.
Each division of the cards does not have equal probability. Once the first card of a suit is dealt to a hand the probability of various distributions has already changed. A way to think about it is imagine that you have 4 marbles which roll down a chute and then fall randomly to the right or left. Once the first marble has fallen, next time they will be 2-0 or 1-1. Then the next marble creates a 3-0 or 1-2 or 2-1, or 0-3. You can see that the possibilities for the final marble are limited by what has happened with the first three.
The actual probability of a single 2-2 (of which there are six) is 6.78% and a single 3-1 (of which there are eight) is 6.22%. Finally the two cases of 4-0 are 4.78%. And those are your 16 cases. Note that 6 times 6.78 is 40.68% which is the probability of a 2-2.
This explains why you play from the drop missing Queen fourth of a suit. Suppose you cash the Ace and lead towards the KJ and everyone follows. Now there are only two possible cases left, the 3-1 where the queen is finessable and the 2-2 where the queen is dropping. As you have already learned, a specific 2-2 is more likely than a specific 3-1, so play for the drop unless other distributional information changes the odds. For example the probabilites change if you know from a preempt that seven of the “marbles” in one hand are diamonds, there is less room in that hand for other “marbles”. Now the 3-1 (and the 4-0) are more likely with the preemptor having the shortness.
If the probabilities of bridge interest you seriously, Borel wrote a book on the mathematics of bridge which is out of print. A simpler and excellent book is