# Complement of an Event

Experiment 1: A spinner has 4 equal sectors colored yellow, blue, green and red. What is the probability of landing on a sector that is not red after spinning this spinner? Sample Space: {yellow, blue, green, red}

Probability: The probability of each outcome in this experiment is one fourth. The probability of landing on a sector that is not red is the same as the probability of landing on all the other colors except red.

 P(not red) = 1 + 1 + 1 = 3 4 4 4 4

In Experiment 1, landing on a sector that is not red is the complement of landing on a sector that is red.

Definition: The complement of an event A is the set of all outcomes in the sample space that are not included in the outcomes of event A. The complement of event A is represented by (read as A bar).

Rule: Given the probability of an event, the probability of its complement can be found by subtracting the given probability from 1.

 P( ) = 1 - P(A)

You may be wondering how this rule came about. In the last lesson, we learned that the sum of the probabilities of the distinct outcomes within a sample space is 1. For example, the probability of each of the 4 outcomes in the sample space above is one fourth, yielding a sum of 1. Thus, the probability that an outcome does not occur is exactly 1 minus the probability that it does. Let's look at Experiment 1 again, using this subtraction principle. Experiment 1: A spinner has 4 equal sectors colored yellow, blue, green and red. What is the probability of landing on a sector that is not red after spinning this spinner?

Sample Space: {yellow, blue, green, red}

Probability:

 P(not red) = 1 - P(red)
 = 1 - 1 4
 = 3 4 Experiment 2: A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a card that is not a king?

Probability:

 P(not king) = 1 - P(king)
 = 1 - 4 52
 = 48 52
 = 12 13

Experiment 3: A single 6-sided die is rolled. What is the probability of rolling a number that is not 4?  Probability:

 P(not 4) = 1 - P(4)
 = 1 - 1 6
 = 5 6 Experiment 4: A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a card that is not a club?

Probability:

 P(not club) = 1 - P(club)
 = 1 - 13 52
 = 39 52
 = 3 4 Experiment 5: A glass jar contains 20 red marbles. If a marble is chosen at random from the jar, what is the probability that it is not red?

Probability:

 P(not red) = 1 - P(red)
 = 1 - 1
 = 0

Note: This is an impossible event.

Summary: The probability of an event is the measure of the chance that the event will occur as a result of the experiment. The probability of an event A, symbolized by P(A), is a number between 0 and 1, inclusive, that measures the likelihood of an event in the following way:

• If P(A) > P(B) then event A is more likely to occur than event B.
• If P(A) = P(B) then events A and B are equally likely to occur.
• If event A is impossible, then P(A) = 0.
• If event A is certain, then P(A) = 1.
• The complement of event A is .    P( ) = 1 - P(A)

### Exercises

Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, choose a different button.

 1. A glass jar contains 5 red, 3 blue and 2 green jelly beans. If a jelly bean is chosen at random from the jar, what is the probability that it is not blue?   None of the above. RESULTS BOX:
 2. A student is chosen at random from a class of 16 girls and 14 boys. What is the probability that the student chosen is not a girl?  1 None of the above. RESULTS BOX:
 3. A number from 1 to 5 is chosen at random. What is the probability that the number chosen is not odd?  0 None of the above. RESULTS BOX:
 4. If a number is chosen at random from the following list, what is the probability that it is not prime? 2, 3, 5, 7, 11, 13, 17, 19 1 0 None of the above. RESULTS BOX:
 5. If a single 6-sided die is rolled, what is the probability of rolling a number that is not 8? 1 0 None of the above. RESULTS BOX: