There's more to it than 'eight ever, nine never'.
Most bridge players know that the mnemonic 'eight ever, nine never' refers to what to do when missing a queen in a suit where you hold eight or nine cards. With eight (or fewer) cards you should normally finesse but with nine (or more) you should probably play for the drop. But what about if you are missing a king or a jack? This article presents a simple, general strategy for all situations and modifies the 'nine never' advice.
It is well known how to calculate the relative probabilities of different distributions occuring in a random deal. (See Wikipedia - Contract bridge probabilities.) In general, this involves rather tedious calculations but for the moment, let's consider the simple case where you are missing a king and have 11 cards in the suit between your two hands. The opponents have the remaining two cards. They can be distributed 1 - 1 or 2 - 0. If they are 1-1 then playing for the drop will work but how often will this occur?
You may think that it will occur 50% of the time since there are only 4 possible distributions: Kx - 0, K - x, x - K, 0 - Kx. and two out of these four possibilities are 1 - 1. The different possible distributions aren't equally likely, however, because each hand has to have exactly 13 cards in total. If we give the king to one hand then there are 12 vacant places (Law of vacant places) in that hand that might get the second card but there are 13 vacant places in the other hand so it is more likely to get the remaining card. This is why the 1 - 1 distribution is slightly more likely than 2 - 0 and playing for the drop is a better than 50% chance. It will actually succeed 13 out of 25 (12+13) times or 52% of the time. The 'vacant places' effect is the reason why playing for the drop is better than the finesse (a 50% chance).
The good news is that you don't have to do any difficult maths. I can tell you that the actual probabilities for dropping a king, queen or jack when you have all the higher honours are as follows:
Cards Held | Missing Honour | Drop Probability | Best Action |
11 | King | 52% | Drop |
10 | King | 26% | Finesse |
9 | Queen | 53% | Drop |
8 | Queen | 33% | Finesse |
7 | Jack | 54% | Drop |
6 | Jack | 36% | Finesse |
I don't expect you to remember this table. Luckily, there is a nice simple way of deciding what to do in all cases:
Divide the missing cards as evenly as possible between the two opponents, with the missing honour in the longer hand if there is one. If the honour would drop in this situation then you should play for the drop. If not, then the finesse is the better option.
For example, suppose you have a 10 card trump suit missing the king. The opponents have Kxx. Divide these as Kx - x. The king won't drop under the ace so the finesse is the better option Or, suppose you have AKQTx opposite xx. The opponents have 6 cards including the jack. Divide these as Jxx - xxx. The jack would drop under the AKQ so play for the drop - simples eh!
I can't claim to be the originator of this strategy, I came across it as a comment by someone to a YouTube video and I thought it was an extremely simple way to remember what to do which ought to be more widely known. Unfortunately, I don't remember the name of the person who made the comment.
The strategy can also be employed when top cards have been played from a suit to analyse the remaining position. Suppose you have KQTxxx opposite xx and you lead from the short hand towards the king which loses to the A. You now have QTxxx opposite x and the opponents have 3 cards including the jack. Divide their cards Jx - x and you see the jack doesn't drop under the queen so take the finesse next time.
Note, however, the strategy does not apply when opponents have discarded cards from the suit since what they have left is not random. They have chosen what to discard.
Looking back at the probability table you will see that when the drop is the better option it is only a few percent better than taking the finesse. In these cases, you should take account of even small inferences from the bidding about the opponents' distribution. To go back to our first example, if an opponent overcalls then he probably has at least 5 cards in the suit and hence has only 8 other cards that could be the king of your suit. If his partner has 4 cards in the overcaller's suit then he has 9 cards which could be your king. The probability that the overcaller has the king is therefore 8/(8+9) or 47% and the probability that their partner has the king is 9/(8+9) or 53%. This slight difference in suit lengths outweighs the average advantage that playing for the drop has over the finesse and if the more probable position for the king makes it finessable then you should play for the finesse and not the drop.
When the finesse is the better option, however, then it is better by at least 14% and you would need much stronger indications from bidding or play to to decide to play for the drop. 'Eight ever' is strongly worded and correctly so. Unfortunately, 'Nine never' is also strongly worded but inappropriately. 'Nine possibly' is nearer the case but not so snappy!