In spite of seeming very unlikely, American gets great chances (7,3,4)
I believe the answer is:
against all odds
'in spite of seeming very unlikely' is the definition.
(against all probability)
'american gets great chances' is the wordplay.
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'gets' becomes 'gains' (gaining is a kind of getting).
'great' becomes 'tall' (both can mean large).
'chances' becomes 'odds' (I've seen this before).
'a'+'gains'+'tall'+'odds'='AGAINST ALL ODDS'
(Other definitions for against all odds that I've seen before include "favouring 2,4,6,8, as you might suppose?" , "Most improbably" .)