# odds

The odds in favor of an event is the ratio of the probability that the event will occur to the probability that the event will not occur, and is typically denoted by $$a\!:\!b$$, where $$\frac{a}{a+b}$$ is the probability the event occurs. (The expression $$a\!:\!b$$ is spoken as “odds of $$a$$ to $$b$$.”)

In terms of expected value, if the odds of an event are $$a\!:\!b$$, then out of $$a+b$$ trials we would expect the event to occur $$a$$ times, and not to occur $$b$$ times. For example, if a bag contains 8 marbles and exactly 5 of them are blue, then the odds of drawing a blue marble at random from the bag is 5:3 (“5 to 3”).

It is common also to speak in terms of “odds against,” in which case we are simply stating the odds of an event not occurring. In the example above, the odds against drawing a blue marble are 3:5.