This is one of the many derogatory phrases that emerged in the latter half of the 20th century which allude to someone ‘having a bit missing’, ‘not all there’ etc. The implication being that the victim of the jocular insult lacked a portion of their brain. In this case the deck is of course a deck of cards. Anyone ‘not playing with a full deck’ wouldn’t be expected to make much of a fist of card playing. The first example that I can find of the phrase is in a George Carlin comedy routine about a member of the Ku Klux Klan, on the Merv Griffin show in 1965.
Other common ‘missing’ phrases are:
– Doesn’t have both oars in the water.
– One sandwich short of a picnic.
– The light’s on but no one’s at home.
– A few bricks short of a full load.