1 how likely it is that something will or will not happen :
probability
the odds
the odds of
You can narrow the odds of a nasty accident happening in your home by being more safety-conscious.
You can narrow the odds of a nasty accident happening in your home by being more safety-conscious.2 difficulties which make a good result seem very unlikely COLLOCATIONS 
against (all) the odds (=although there are great difficulties) overcome the odds (=succeed although there are great difficulties) battle against the odds the odds are stacked against somebody (=there are great difficulties) impossible odds enormous/heavy/overwhelming odds
Their job was to hold on despite impossible odds, in order to give the rest of the army time to strike.
difficulties

against (all) the odds (=although there are great difficulties) overcome the odds (=succeed although there are great difficulties) battle against the odds the odds are stacked against somebody (=there are great difficulties) impossible odds enormous/heavy/overwhelming odds
Their job was to hold on despite impossible odds, in order to give the rest of the army time to strike.3
be at odds
a) to disagree
Briggs found himself at odds with his colleagues.
Briggs found himself at odds with his colleagues.b) if two statements, descriptions, actions etc are at odds with each other, they are different although they should be the same
be at odds with
Mark's account of what happened is at odds with Dan's.
She gave him a sweet smile, totally at odds with the look of dislike in her eyes.
Mark's account of what happened is at odds with Dan's.
She gave him a sweet smile, totally at odds with the look of dislike in her eyes.4 the numbers that show how much money you will win if you bet on the winner of a horse race or other competition :
The odds are 6-1.
horse racing etc
DGGDSH
The odds are 6-1. (at) long/short odds (=high or low numbers, that show a high or low risk of losing)
Everyone was surprised when Desert Zone won the race, at very long odds.
Everyone was surprised when Desert Zone won the race, at very long odds. lay/offer (somebody) odds British English
They are laying odds of 8-1 that the Conservatives will win the next election.
They are laying odds of 8-1 that the Conservatives will win the next election.5 used to say that what someone does or what happens is not important :
Pay me now or later - it makes no odds.
it makes no odds
British English spoken
Pay me now or later - it makes no odds.6 to pay a higher price than is usual or reasonable :
Most residents live in tiny apartments and pay over the odds for them too.
pay over the odds
British English informal
Most residents live in tiny apartments and pay over the odds for them too.