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?

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  a_bar_black.gif(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(a_bar_black.gif) = 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.

SpinnerExperiment 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

Poker HandExperiment 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?

  dice

  

Probability:

P(not 4)  =  1  -  P(4)
   =  1  -  1
6
   =  5
6

Poker HandExperiment 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

Red MarblesExperiment 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 a_bar_black.gif.    P(a_bar_black.gif) = 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?
   
 
1_over_2_0.gif  

3_over_10_0.gif

7_over_10_0.gif

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?
   
 
8_over_15_0.gif
 

7_over_15_0.gif

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?
   
 
2_over_5_0.gif
 

3_over_5_0.gif

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

1_over_8_1.gif

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?
   
 

5_over_6_0.gif

1

0

None of the above.

RESULTS BOX: