I recently played a bridge hand that raised a probability question for which the bridge texts give a standard answer. It did not seem obvious to me taht this answer was correct, and the texts do not explain their answer well. After further thought, I concluded that the advice in the texts is largely correct, but the explanation often is not. The hand is one in which you are declaring, and between declarer and dummy you have 9 of a suit including the Ace-King-Jack. The question is, what course of action gives you your best chance of capturing your opponents' Queen, and winning all of that suit's tricks?
The standard advice is eight ever, nine never, meaning that if you hold just 8 of a suit then you should always try a finesse (by leading a low card from dummy, and then playing the King over the Queen, or the Jack if the Queen is not played). But if you hold 9, then you should never finesse but instead just play for the drop (lead the Ace and then the King, hoping that the Queen will drop). Some texts say that the finesse has a 50% probability of success, while playing for the drop has a 53% probability. (And some give other strange advice, like "Decide which way is best for you and do it every time. If you vacilate between positions, the odds won't work in your favor", which has absolutely no validity.)
In fact, playing for the finesse the "right" way has a 56.22% chance of success. Does that mean that you should play for the finesse? No, because playing for the drop the "right" way has a 57.92% chance of success, as I now explain.
To avoid additional complications, suppose the suit in question is trump (so there's no way to win the Queen by trumping it, nor by any other strategy other than the finesse or the drop). And, assume declarer holds the Ace and King and Jack and Ten, and you have sufficient cards and entries to lead from declarer and from dummy as often as needed. Furthermore, assume that you have no information at all about your opponents' cards based on their bidding or previous play.
To determine whichs strategy is better, we need to consider probabilities. In the absence of any other information, the opponents' 26 cards are equally likely to be divided up into any two 13-card hands. So, the probability that the opponent just right of declarer (i.e. the RHO) has "i" trump cards is given by:
Prob(RHO has i trump cards) = (4 choose i) * (22 choose 13-i) / (26 choose 13) , for i=0,1,2,3,4.(Here "n choose k" means the total possible number of ways of selecting k cards out of n cards total.) This gives the following probabilities for the number "i" of trump cards held by the RHO: Prob(0) = 4.78%, Prob(1) = 24.87%, Prob(2) = 40.70%, Prob(3) = 24.87%, Prob(4) = 4.78%. This allows us to compute the probability of success with various strategies:
In summary, the probabilities of success are: 50% for the simple finesse, 53.14% for the simple drop, 56.22 for the smarter finesse, and 57.92% for the smart drop. So, it is indeed true that with a 9-card suit, and plenty of entries, and no information at all about your opponents, it is best to play (smartly) for the drop. But just barely, and only because you can switch to the finesse strategy if LHO is void.