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For example, if the odds of a football team winning a match are 1/2, it means the bookmaker considers it more likely that they will win than not. On the other hand, if the odds against a team winning are 2/1, it means the bookmaker considers it less likely that they will win than not.
A probability of 70% means that when you observe the event, the prediction should be borne out (in the long run) 7 times out of 10. When you're talking about a non-repeatable event, then the best interpretation is as a level of confidence in the prediction.