Can't possibly?

[Regina]

Test No. incompl/elem-21 "Remove the Spoon", question 7

I can't ......... give up drinking tea, doctor.

(a) probably
(b) possibly
(c) likely
(d) impossibly

Test No. incompl/elem-21 "Remove the Spoon", answer 7

I can't possibly give up drinking tea, doctor.

Correct answer: (b) possibly

Your answer was: incorrect
I can't probably give up drinking tea, doctor.
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Whu here probably?

[Torsten]

It means that it is impossible for the speaker to give up drinking tea.
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Test Of English for International Communication
TOEIC Preparation & TOEIC Vocabulary

[Polska]

could you set forth the diffference between them?

[Mister Micawber]

Possibly here is a minimizer, not a disjunct, and can take the position after the modal:

I can't possibly give up drinking tea.

Probably as a disjunct cannot hold that position:

I probably can't give up drinking tea.
Probably, I can't give up drinking tea.
I can't give up drinking tea, probably.

Can't possibly does not mean it is impossible; it just minimizes the effect of the main verb.
_________________
Canadian-American native speaker
who teaches English for a living at Mister Micawber's
ESL cafe: Interview with Mister Micawber