Understanding Complementary Events in Probability: A Comprehensive Explanation and Examples

complementary events

Complementary events are events that cannot occur at the same time and encompass all possible outcomes of an experiment

Complementary events are events that cannot occur at the same time and encompass all possible outcomes of an experiment. In other words, if two events are complementary, the occurrence of one event means the other event cannot occur and vice versa.

The probability of an event and its complement always adds up to 1. This means that if the probability of event A occurring is P(A), then the probability of event A not occurring (complement of A) is 1 – P(A).

Let’s consider an example to illustrate this concept. Suppose you have a fair coin and you want to find the probability of either getting a heads or a tails when you flip it. In this case, the complement of getting heads would be getting tails, and vice versa. Since these are the only two possible outcomes of flipping a coin, they are complementary events.

The probability of getting heads can be denoted as P(H), and the probability of getting tails can be denoted as P(T). Since the coin is fair, the probability of getting heads is 1/2 (0.5) and the probability of getting tails is also 1/2 (0.5). Therefore, P(H) = 0.5 and P(T) = 0.5.

Using the concept of complementarity, we can find the probability of getting tails by subtracting the probability of getting heads from 1. So, P(T) = 1 – P(H) = 1 – 0.5 = 0.5.

In this example, the complement of getting heads is getting tails, and the complement of getting tails is getting heads. Since these two events include all possible outcomes (heads or tails), they are complementary events.

Remember, for any two complementary events with probabilities P(A) and P(A’), the sum of their probabilities always equals 1. This means that P(A) + P(A’) = 1.

Complementary events are frequently used in probability calculations and can help simplify and solve various problems.

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