[pescetom]

As declarer you hold A73 of trumps and dummy holds KJ9865.
You cash the A to which opps follow with 2 and 10 respectively.
You then play the 3 to which LHO follows with 4.
Which honour do you now play from dummy and why?

[Stephen Tu]

In the absence of other info you play for the drop, the K, aka the "9 never" of the "8 ever 9 never" saying. It's basically a vacant space thing, LHO has 11 spots, RHO 12; at this point it's 12/23 that RHO has it.

If you know from bidding or play of other suits that RHO is longer than his partner (e.g. he preempted) by two or more, you finesse, because that tilts the other way. One vacant space difference = tossup.

Note this is NOT a restricted choice situation because the Q and T are not equals, RHO has no good reason to ever play the Q from QT doubleton.

Also other considerations of the hand may lead one to play against the normal odds, e.g. hooking may prevent a dangerous hand from being on lead, or losing the hook guarantees the suit has split and you will be able to utilize the 3rd trump in the short hand for ruffing, or losing the hook leaves the opponent immediately endplayed.

[HardVector]

The above post has it right, in the scenario you have given, you play the king. 8 ever, 9 never refers to the odds on play for a missing queen. The odds advantage are very slight, however, and can be affected by additional information. For instance, let's say you know that the person to the right has a 7 card outside suit. Now the odds shift back to taking the finesse because there are now fewer missing spaces that could contain the Q.

[Vampyr]

It might be useful to add for some readers that vacant spaces doesn’t really apply to, say, an opening lead where leader has five and his partner has three, or two. Leader looked at his hand before leading.