Is a "drop-finesse" better than a simple "drop" play with a nine card suit in bridge?

Question

Regarding finesses for a missing queen in a key (e.g. trump suit), there is a proverb of "eight ever [always] nine never." Instead, one is supposed to play for a drop. (This, and most other proverbs need to be taken with grain of salt).

A "blind" finesse (without further information), is 50-50. If you have nine cards in a suit headed by the ace and king (and play them to drop the queen), your chances of success are 52.5%. That is, the opposing cards will fall 2-2. 40% of the time. Some 50% of the time, they will be split 3-1.And of these, the singleton will be the queen one-fourth of the time, or 12.5% of the total (40%+12.5%)= 52.5%.

But suppose you have one or more finesse possibilities. That is, you have AJxxx. Dummy has KTxx. Then you can lead one of your tops "blindly" and catch the queen 12.5% of the time. A finesse the other way would succeed 50% of the remaining times =.5*87.5% or 43.75%. The total chances of success would be (43.75+12.5%)=56.25%> 52.5%.

So a "drop" (top card) play followed by a finesse has an advantage over a pure drop (two top card plays), right? And wouldn't this be even more true if East or West showed strength by opening, meaning that you would finesse "around" him to one hand after playing the top honor in the other hand.

And wouldn't the exception be if you had only a "one way" finesse e.g. Axxxx opposite dummy's KTxx and East opened, meaning that the Q was probably "offside?

Answer 1

Your calculation seems off. You seem to have missed an approximately 5% chance of a 4-0 break which the play for drop will catch, but have that implicitly included in the calculations for the drop-finesse.

If you include that, the play for drop indeed comes out to be more, but by very little.

A possibly simple way to figure this out is to compare the holdings where one wins but the other loses.

Say you have AJT9x opposite K87x (in hand) and you decide to play the K first (in both lines).

Taking the finesse next wins over the drop when RHO has Qxx (assuming both follow to the first trick, otherwise it is the same for both lines).

Playing for the drop wins over the finesse when RHO has Qx.

Since RHO will have Qxx 3/4ths of 1-3 split = 3/4ths of ~25% = ~19%, while RHO will have Qx 1/2 of ~40% = ~20%, playing for the drop is better.