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Probability is the possibility of either happening or non-happening of any event. We often use probability in our daily lives; for example, we calculate the possibility of winning or losing a game. While tossing, we know there is a 50% probability of either getting heads or tails. Complementary probability includes events that are mutually exclusive and exhaustive. It means that in two events, one of the events will only occur if the other does not. Therefore, we can say that a complement of an event will be the outcome that will not happen. Moreover, the sum of these two events will always be unity, i.e., equals 1. A complementary event requires two possible equal outcomes.
Terminology Used in Probability
To better understand the complementary probability, we must know its basic terminology. Therefore, the basic terminologies used in probability are as follows:
Experiment: A probability experiment is a trial conducted to get any outcome.
Random Experiment: This has a well-defined set of outcomes but no surety of any outcome, for example, in a coin toss. The trial is the term that denotes this experiment.
Events: The event of any experiment is the total number of outcomes of that experiment.
Sample Space: The total number of possible outcomes for any experiment or event constitutes the sample space. For example, the sample space of the dice will be numbered 1 to 6.
Favourable Outcome: In the probability experiment is the desired outcome of any event. For example, a team needs heads to win a coin toss, so its favourable outcome will be heads.
Equally Likely Events: As the name suggests, are events with equal possible outcomes.
Exhaustive Events: It is when the set of all outcomes is equal to the sample space of an experiment.
Mutually Exclusive Events: These events do not co-occur, i.e., together — for example, an even and odd number simultaneously on a single dice. We can either get an even or an odd number on a dice (not both).
Properties of Complementary Events
The two basic properties of complementary events are as follows:
These events are mutually exclusive, i.e., these two events are not supposed to co-occur.
The other property is exhaustive. Being exhaustive means that the event and its complement are bound to fill up the sample space. Therefore, S = P ∪ P’.
Mathematical Expression of Complementary Events
The rules of complementary events in mathematical expression are stated below:
P (E) + P(E’) = 1.
P(E) = 1 – P(E’).
P(E’) = 1 – P(E).
Examples of Complementary Events
Example 1: While playing a dice game, you want any number greater than 4. On the other hand, your opponent wants you to get a number less than 4. We know that you will get a number greater than 4, only if the outcome does not have a number less than 4. In this case, we see that these two events are mutually exclusive and exhaustive.
Example 2: While predicting tomorrow’s weather, your friend thinks it will be cool. On the other hand, you believe that the weather will be warm. These two events are also mutually exclusive, and one will only happen if the other does not.
Solved Questions on Complementary Events
Q 1- Prove that L and M are two independent events if P (L ⋃ M) = 1 – P(L’) P(M’) using rules of complementary events.
Solution:
According to question: P(L ⋃ M) = 1- P(L’) P(M’)
The rule of complementary events states:
P(A’) = 1 – P(A).
Therefore, P(L ⋃ M) = 1 – [1 – P(L)] [1 – P(M)].
P(L ⋃ M) = 1 – [1 – P(L) – P(M) + P(L).P(M)].
P(L ⋃ M) = 1 – 1 + P(L) + P(M) – P(L).P(M).
P(L ⋃ M) = P(L) + P(M) – P(L).P(M).
Hence proved that L and M are two independent events.
Q 2- What is the probability of choosing a perfect square from numbers given from 1 to 65.
Solution:
Probability of choosing a perfect square randomly = P [say].
Number of favourable outcomes = 8 {1, 4, 9, 16, 25, 36, 49, 64}.
The total number of outcomes = 65.
P = number of favourable outcomes (perfect squares) / total number of outcomes (total numbers).
Probability of choosing a perfect square randomly = 8/65.
Conclusion
We know that probability is the possibility or chances of any event happening. There are different types of probability, and one is the complementary probability. In simpler words, any two events are mutually exclusive. One event occurs only when the other does not. These two mutually exclusive and exhaustive events will be unity, i.e., 1. A detailed study of complementary events and probability is given above.
