Here was the original problem:
West leads K♣️. North wins trick one with A♣️.
Now what?
Solution–
The principle of vacant spaces says that the probability that an opponent holds a particular card is proportional to the number of vacant spaces remaining in their respective hands. West’s double on round two shows both unbid suits. Presumably West is 5-5 or 5-4 in the minors. In our search for the Q♠️–West has three or four “vacant spaces” and East has 12 “vacant spaces”. We should play East for the Q♠️. Start trick two with dummy’s K♠️. Here was the entire deal:
When this hand was played declarer started the ♠️s with a small ♠️ to the A♠️. Thus ensuring a trump trick for the defense. That one trick turned a 50% board into a 25% board.
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Since I play that X as a 4/4 ish in minors not sure I would play for the finesse. If however they showed a 5/5 by bidding a “Sandwich nT” I would have finessed the Q they way you say. BTW I would have bid 1nt rather than X.
Even if the double shows 4-4 in minors that’s 5 vacant spaces vs 12 vacant spaces, i.e. 12 to 5 in favor of east holding the Q.