Probability is a branch of mathematics that deals with numerical representations of how likely an event or statement is to occur. In probability, complementary occurrences occur when two events are exhaustive and mutually exclusive. As a result, when one event occurs, the other cannot occur.

Complementary events

Complementary events are two occurrences that occur only if and only if the other does not occur. Two events must be mutually exclusive and exhaustive in order to be characterised as complimentary. The sum of complimentary occurrences’ probabilities must equal one. Only when there are exactly two outcomes can complementary events occur. If one occurrence can only happen if the other does not, the two events are said to be complimentary. A complement of an event can also be described as the set of outcomes that it does not produce. Allow A to be an occasion. A’ or Ac stands for the complement of A. The events such as A and A’ are mutually exclusive here.

When only two possibilities are feasible, complementary events occur in probability. Take, for example, passing or failing a test. An experiment’s set of outcomes is referred to as an event. As a result, the sample space will always be a subset of events.

Complementary Events Properties:

To be considered complementary events, two events must share specific characteristics. The following are the details:

• Complementary events are incompatible. This indicates that two complementing occurrences cannot happen at the same moment. Complementary events, in other words, are disjointed.
• Complementary activities are numerous. This means that an event must entirely fill the sample area, as well as its complement. Thus, S = A ∪ A’.

Rule Of Complementary Events:

The rule of complementary events asserts that the sum of an event’s probability of occurrence and its complement’s probability of occurrence is always 1. Let A represent an occurrence, and P(A) represent the probability of A occurring. As a result, P(A’) reflects the likelihood that A will not occur. This rule can then be stated numerically as follows.

P(A) + P(A’) = 1

P(A) = 1 – P(A’)

P(A’) = 1 – P(A)

These three mathematical statements are interchangeable.

Point to Remember:

• Complementary occurrences occur when one event occurs only if and only if the other does not.
• Events that are complementary are mutually exclusive and exhaustive.
• The sample space is made up of an event and its complement.
• P(A) + P(A’) = 1 is the complementary events rule.
• Events that are complimentary are always mutually exclusive. This indicates that no results will be shared between an event and its complement.
• All complementary events are always exhaustive. This indicates that the sample space is made up of the event’s outcomes and its complement.

Examples:

1. A bag contains ten balls, two of which are black, three of which are red, one of which is blue, three of which are pink, and one of which is purple. Let X represent the occurrence of choosing the main colour. Determine P(X’).

        Solution: X = 3 red and 1 blue

        10 total balls

        P(X) = 4 / 10 number of good outcomes

        P(A’) = 1 – P(A) P(X’) = 1 – (4/ 10) = 6 / 10 using the complementary events rule

        P(X’) = 6 / 10 is the correct answer.

1. Using the complementary events rule, demonstrate that M and N are independent events if P(M N) = 1 – P(M’) P(N’).

           Solution: P(M’) P(N’) = 1- P(M’) P(N’)

          P(A’) = 1 – P according to the complementary events rule (A)

          P(M U N) = 1 – [1 – P(M)] [P(N) – 1]

          P(M U N) = 1 – [1 – P(M) – P(N) + P(M) + P(M) + P(M) + P(M) + P (M). P(N)]

          P(M U N) = 1 – P(M) + P(N) – P (M). P(N)

          P(M U N) = P(M) + P(N) – P(M N) (M).  P(N)

          As a result, proven.

1. A number between 1 and 40 is picked at random. Determine the likelihood of not selecting a perfect square.

Solution: Consider Z’ to represent the occurrence of selecting a perfect square. The following is the sample space:

     Z’ = 1, 4,,  9, 16, 25, and 36, 

   Total number of results = 40

   Positive outcomes = 6 P(Z’) = 6 / 50.

   P(Z) = 1 − (6/50) = 44/50

  P(Z) = 44 / 50 is the answer.

Conclusion

• When one event occurs if and only if the other does not, such events are referred to as complementary events.
• Events that are complementary are mutually exclusive and exhaustive.
• The sample space is made up of an event and its complement.
• The complementary event rule is P(A) + P(A’) = 1.