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The Complement
of an Event |
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Unit 6 |
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Experiment 1:
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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?
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Sample Space:
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{yellow, blue, green, red}
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Probability:
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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.
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| P(not red) |
=
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1
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+
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1
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+
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1
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=
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3
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4
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4
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4
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4
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In Experiment 1, landing on a sector that is not red is the complement of landing on a
sector that is red.
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Definition:
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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).
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Rule:
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Given the probability of an event, the probability of its complement can be found by subtracting the
given probability from 1.
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P() = 1 - P(A)
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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.
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Experiment 1:
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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?
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Sample Space:
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{yellow, blue, green, red}
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Probability:
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| P(not red) |
=
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1
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-
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P(red)
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=
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1
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-
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1
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4
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=
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3
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4
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Experiment 2:
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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?
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![[IMAGE]](images/poker_hand.gif)
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Probability:
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| P(not king) |
=
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1
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-
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P(king)
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=
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1
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-
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4
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52
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=
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48
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52
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=
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12
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13
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Experiment 3:
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A single 6-sided die is rolled. What is the probability of rolling a
number that is not 4?
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Probability:
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| P(not 4) |
=
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1
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-
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P(4)
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=
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1
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-
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1
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6
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=
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5
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6
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Experiment 4:
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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?
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Probability:
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| P(not club) |
=
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1
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-
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P(club)
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=
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1
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-
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13
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52
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=
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39
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52
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=
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3
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4
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Experiment 5:
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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?
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![[IMAGE]](images/red_marbles.gif)
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Probability:
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| P(not red) |
=
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1
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-
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P(red)
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=
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1
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-
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1
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=
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0
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| Note: This is an impossible event. |
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| 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)
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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? |
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| 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? |
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| 3. |
A number from 1 to 5 is chosen at random. What is the probability that the number chosen is not odd? |
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| 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 |
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| 5. |
If a single 6-sided die is rolled, what is the probability of rolling a number that is not 8? |
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