Complementary Events

Definition and Properties

In probability theory, complementary events refer to the situation where the occurrence of one event means the non-occurrence of its complement. The complement of event A (denoted as Ac, not A, or A') consists of all outcomes in the sample space that are not in A.

• Complement of an Event: If event A occurs, its complement Ac does not occur, and vice versa.

• Total Probability: The sum of the probabilities of an event and its complement is always 1.

Formula:

Example: If rolling a six-sided die, the probability of not rolling a 4 is the complement of the probability of rolling a 4.

• Probability of rolling a 4:

• Probability of not rolling a 4:

Applications and Practice Problems

Complementary events are useful for calculating probabilities when it is easier to find the probability of the event not happening.

• Example: Drawing a card from a standard deck of 52 cards, the probability of not drawing a queen is:

  • Probability of drawing a queen:

  • Probability of not drawing a queen:

• Practice: When drawing a marble at random from a jar of 6 red, 3 green, and 1 yellow marble (total 10 marbles), the probability of not drawing a green marble is:

  • Probability of drawing a green marble:

  • Probability of not drawing a green marble:

• Practice: If a weatherman states the probability that it will rain tomorrow is 10%, the probability that it will not rain is:

Summary Table: Complementary Probabilities

Additional info: Complementary events are a foundational concept in probability, often used to simplify calculations and to check the completeness of probability models.