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In my first Finland game (historical focuses off) he was shot but lived, in my second (historical on) he didn't appear at all (I lost the Winter War too fast maybe?)
Looking at AAT_Finland.txt it seems the chance is based on the choice the FIN_white_death_target_country makes during their White Death event:
Ignore the snipers: 30% chance for Häyhä to be wounded
Hunt the snipers: 50% chance to be wounded, 10% to die
Base AI picking weight for both choices is 50, but the hunt option gets an extra 100 weight if historical focuses are on OR the invader has at least 1500 infantry equipment in stock