It's just a fancier way of saying Good Luck
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It means that the chances you have of winning the games are in your favor it's like some capitol's good luck ;)
Effie Trinket says that in "The Hunger Games"
and it is actually "May the odds be ever in your favor"
She meant may all the luck be with you, because they would need it otherwise they would get killed.
"May the odds be ever in your favor."
She doesn't. As far as I've read she isn't in Mockingjay.
Effie Trinket said, "May the odds be ever in your favor!!" after she said, "Happy Hunger Games"
odds are 1:6
"Against All Odds (Take a Look at Me Now)" by Phil Collins