Two events are said to be complementary when one event occurs if and only if the other does not. The probabilities of two complimentary events add up to 1.
For example, rolling a 5 or greater and rolling a 4 or less on a die are complementary events, because a roll is 5 or greater if and only if it is not 4 or less. The probability of rolling a 5 or greater is = , and the probability of rolling a 4 or less is = . Thus, the total of their probabilities is + = = 1.
Example: If the probability of an event is , what is the probability of its complement?
The probability of its complement is 1 - = - = .
The odds of an event is the ratio of the probability of an event to the probability of its complement. In other words, it is the ratio of favorable outcomes to unfavorable outcomes. We say the odds are "3 to 2," which means 3 favorable outcomes to every 2 unfavorable outcomes, and we write 3 : 2. For example, the odds of rolling a 5 or greater are 2 : 4, which reduces to 1 : 2.
Example 1: If we flip a coin two times, what are the odds for it landing heads at least once?
Favorable outcomes: 3 -- HH, HT, TH.
Unfavorable outcomes: 1 -- TT.
Thus, the odds for it landing heads at least once are 3 to 1, or 3 : 1.
Example 2: If the probability of an event happening is , what are the odds for that event?
Since the probability of the event is , the probability of its complement is 1 - = - = . Thus, the odds for that event are : , which is equivalent to 2 : 5.