Complementary Events in Probability

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:

• $P(A) + P(A^c) = 1$

• $P(A^c) = 1 - P(A)$

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: $P(4) = \frac{1}{6}$

• Probability of not rolling a 4: $P(\text{not 4}) = 1 - \frac{1}{6} = \frac{5}{6}$

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: $P(\text{queen}) = \frac{4}{52}$

  • Probability of not drawing a queen: $P(\text{not queen}) = 1 - \frac{4}{52} = \frac{48}{52}$

• 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: $P(\text{green}) = \frac{3}{10}$

  • Probability of not drawing a green marble: $P(\text{not green}) = 1 - \frac{3}{10} = \frac{7}{10}$

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

  • $P(\text{not rain}) = 1 - 0.10 = 0.90$

Summary Table: Complementary Probabilities

Event	Probability of Event	Probability of Complement
Rolling a 4 on a die	$\frac{1}{6}$	$\frac{5}{6}$
Drawing a queen from a deck	$\frac{4}{52}$	$\frac{48}{52}$
Rain tomorrow (probability 10%)	$0.10$	$0.90$

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