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? |
| 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? |
| 3. | A number from 1 to 5 is chosen at random. What is the probability that the number chosen is not odd? |
| 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 |
| 5. | If a single 6-sided die is rolled, what is the probability of rolling a number that is not 8? |