No, it's wrong. The article applies to the entire noun phrase. "A" is correct
If you say "an privacy" it'll sound like English isn't your first language.
I stand corrected, and was actually returning to edit my original post when I saw this. I was always under the impression that the form a or an takes is dependent on that which it determines. Apologies.
Also, it's not even the noun phrase according to the sources I'm looking at (namely Merriam-Webster) which state that it is always dependent on the word directly following the article, not even the entire phrase.
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