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Jack of all Trades increases all ability checks in which the bard is not proficient. Passive checks are ability checks. (PHB5e p.54 & 175, respectively)

Does that mean I can add half my proficiency to Passive Perception (Wisdom)?

I know it works on initiative, so probably on passive checks as well, right?

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Yup, you halve-and-add your proficiency bonus to even passive checks.

Jack of all Trades. ...add half your proficiency bonus, rounded down, to any ability check you make that doesn't already include your proficiency bonus. (PHB p.54, emphasis mine)

Combined with

Passive Checks. A passive check is a special kind of ability check....

Here's how to determine a character's total for a passive check:

10 + all modifiers that normally apply to the check.

(PHB p.175, emphasis mine)

So as long your Jack of All Trades modifier normally applies to your Perception (Wisdom) check, then it applies to the passive version.

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