The probability of an event is written as a fraction:
This probability tells us how likely an event is to happen.
Odds are another way of conveying the same information, or another way of saying how likely an event is to happen. Instead of comparing the number of favorable outcomes to the total number of outcomes, we compare the number of favorable and unfavorable outcomes. An unfavorable outcome is any outcome not in the event we're looking at. Try to keep this straight from an unflavorable outcome, which is one that's bland and tasteless.
Here's how we write the odds in favor of an event:
The odds against an event are:
Sample Problem
What are the odds in favor of rolling a 4 with a fair die?
There's 1 favorable outcome (rolling a 4) and there are 5 unfavorable outcomes (rolling anything else). The odds in favor of rolling 4 are 1:5.
Not that we want you to encourage you to gamble, but we wouldn't lay even money on that proposition if we were you.
Since an event must either happen or not happen, if we add up the number of favorable and unfavorable outcomes, we get the total number of outcomes. Therefore, we can go from odds to probability, or from probability to odds. If we'll be doing both, it only makes sense to purchase a round-trip ticket ahead of time.
Sample Problem
If an event has probability 
, the number of favorable outcomes is 3 and the number of total outcomes is 4. There's only 1 outcome left to be unfavorable. The odds in favor of the event are 3:1, and the odds against the event are 1:3. Oh, and by the way, there's no such thing as luck. Your odds of losing a coin flip are not higher than someone else's because the world hates you. Sorry to disappoint.
Sample Problems
If the odds in favor of an event are 1:2, there's 1 favorable outcome and 2 unfavorable outcomes, meaning there are 3 total outcomes, so the probability of the event is
.
If the odds against an event are 4:5, there are 5 favorable outcomes and 4 unfavorable outcomes, for a total of 9 possible outcomes. The probability of the event is
.
By the way, a set of odds can be reduced just like a fraction. If the odds of acing your math midterm are 15:20, you can simplify that to 3:4. Good luck.