Allow me a moment of lucidity...
The conversation these two people are having are NOT prepositions of fate.
Fate is specifically a predetermined destiny. So, if someone were to commit suicide under ANY circumstances it would be their fate to do so. Therefore, commiting suicide does NOT disprove the existance of fate.
The problem with fate has to do with people who believe in both FATE and GOD. Why? Because God gave man freewill. If fate exists, then there is no actual freewill, because everything is predetermined.
The conversation that takes place in this cartoon is bollocks and represents no philosophical quandry over fate whatsoever. Sorry.