Complementary Events in Probability

Definition

In probability, two events which constitute the total sample space of the experiment are called complementary events. In other words, the combined list of favorable outcomes for both the events equals the sample space of the experiment. For example, on tossing a coin, the sample space is {heads, tails}. The two events P(heads) and P(tails) constitute the sample space of the experiment. Thus they are complementary events.

Notation

If A is an event, then its complement is denoted by A' or "not A". In the above example, the complement of "heads" is "not heads".

Sum of probabilities of complementary events is 1

Since the complementary events together constitute the sample space, thus we can conclude that if one event does not occur, its complement will definitely occur. In other words, either one of the events will definitely occur. Thus the probability of occurring of either one of the events is 1.

Hence, the sum of probabilities of complementary events is 1.

`P(A) + P(A') = 1`

Thus, if we know the probability P(A) of an event A, we can calculate the probability of its complement by using the equation

`P(A') = 1 - P(A)`

For example, we know that the probability of getting 1 on a number cube is `1/6`. Then the probability of getting any of the other five numbers (2 to 6) on a number cube is a the compliment of the above event since it is the same as the probability of not getting 1. Thus

`P(2 " to " 6) =  1 - P(1) = 1 - 1/6 = 5/6`.

Examples

On tossing a coin, the probability of getting heads is

`P("heads") = 1/2`

and the probability of getting tails is

`P("tails") = 1/2`

The sum of their probabilities is

`P("heads") + P("tails") = 1`

Thus the two events "getting heads" and "getting tails" are complementary events.

Similarly, on rolling a number cube, the probability of getting 1 is

`P(1) = 1/6`

The probability of not getting 1 is

`P(1') = 1 - 1/6 = 5/6`

Thus, getting 1 and not getting1 on rolling a number cube are two complementary events.

Complementary events are always Mutually Exclusive

As we know from this post, mutually exclusive events are those events which can not occur together. Complementary events are always mutually exclusive, that is, they can not occur together. For example, on tossing a coin, the result can be either heads or tails, but not both.