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3.3 The Complement Rule

LEARNING OBJECTIVES

  • Calculate probabilities using the complement rule.

The complement of an event [latex]A[/latex] is the set of all outcomes in the sample space that are not in [latex]A[/latex]. The complement of [latex]A[/latex] is denoted by [latex]A^C[/latex] and is read “not [latex]A[/latex].”

A diagram illustrating the complement of an event A. A rectangle represents the sample space. Inside the rectangle, a circle represents event A. The complement of A is everything inside the rectangle and outside the circle.

EXAMPLE

Suppose a coin is flipped two times. Previously, we found the sample space for this experiment:  [latex]S=\{HH, HT, TH, TT\}[/latex] where [latex]H[/latex] is heads and [latex]T[/latex] is tails.

  1. What is the complement of the event “exactly one head”?
  2. What is the complement of the event “at least one tail.”

Solution

  1. The event “exactly one head” consists of the outcomes [latex]HT[/latex] and [latex]TH[/latex]. The complement of “exactly one head” consists of the outcomes [latex]HH[/latex] and [latex]TT[/latex]. These are the outcomes in the sample space [latex]S[/latex] that are NOT in the original event “exactly one head.”
  2. The event “at least one tail” consists of the outcomes [latex]HT[/latex], [latex]TH[/latex], and [latex]TT[/latex]. The complement of “at least one tail” consists of the outcomes [latex]HH[/latex]. These are the outcomes in the sample space [latex]S[/latex] that are NOT in the original event “at least one tail.”

TRY IT

Suppose we roll a fair six-sided die with the numbers [latex]1, 2, 3, 4, 5, 6[/latex] on the faces. Previously, we found the sample space for this experiment:  [latex]S=\{1,2,3,4,5,6\}[/latex]

  1. What is the complement of the event “rolling a 4”?
  2. What is the complement of the event “rolling a number greater than or equal to 5”?
  3. What is the complement of the event “rolling a even number”?
  4. What is the complement of the event “rolling a number less than 4”?

 

Click to see Solution
  1. The complement is [latex]\{1,2,3,5,6\}[/latex].
  2. The complement is [latex]\{1,2,3,4\}[/latex].
  3. The complement is [latex]\{1,3,5\}[/latex].
  4. The complement is [latex]\{4, 5, 6\}[/latex].

The Probability of the Complement

In any experiment, an event [latex]A[/latex] or its complement [latex]A^C[/latex] must occur. This means that [latex]\displaystyle{P(A)+P(A^C)=1}[/latex].  Rearranging this equation gives us a formula for finding the probability of the complement from the original event:

[latex]\begin{eqnarray*}P(A^C)&=&1-P(A)\\\\\end{eqnarray*}[/latex]

EXAMPLE

An online retailer knows that [latex]30\%[/latex] of customers spend more than [latex]\$100[/latex] per transaction. What is the probability that a customer spends at most [latex]\$100[/latex] per transaction?

Solution

Spending at most [latex]\$100[/latex] ([latex]\$100[/latex] or less) per transaction is the complement of spending more than [latex]\$100[/latex] per transaction.

[latex]\begin{eqnarray*}P(\text{at most \$100})&=&1-P(\text{more than \$100})\\&=&1-0.3\\&=&0.7\end{eqnarray*}[/latex]

TRY IT

At a local college, a statistics professor has a class of [latex]80[/latex] students. After polling the students in the class, the professor finds out that [latex]15[/latex] of the students play on one of the school’s sports teams and [latex]60[/latex] of the students have part-time jobs.

  1. What is the probability that a student in the class does not play on one of the school’s sports teams?
  2. What is the probability that a student in the class does not have a part-time job?
Click to see Solution
  1. [latex]\displaystyle{P(\text{no sports team})=1-P(\text{sports team})=1-\frac{15}{80}=0.8125}[/latex]
  2. [latex]\displaystyle{P(\text{no part-time job})=1-P(\text{part-time job})=1-\frac{60}{80}=0.25}[/latex]

Exercises

  1. Suppose a coin is flipped three times.
    1. Find the sample space for this experiment.
    2. What is the complement of the event “exactly 1 tail”?
    3. What is the complement of the event “at most 2 heads”?
    4. What is the complement of the event “2 or more tails”?
    Click to see Answer
    1. [latex]\{HHH, HHT, HTH, THH, TTH, THT, HTT, TTT\}[/latex]
    2. [latex]\{HHH, TTH, THT, HTT, TTT\}[/latex]
    3. [latex]\{HHH\}[/latex]
    4. [latex]\{HHH, HHT, HTH, THH\}[/latex]

     

  2. A 12-sided die is in the shape of a regular dodecahedron. The faces of the 12-sided die are labelled with the numbers 1 to 12. Suppose the 12-sided die is rolled one time.
    1. Find the sample space of this experiment.
    2. What is the complement of the event “rolling a 7”?
    3. What is the complement of the event “rolling a number less than or equal to 9”?
    4. What is the complement of the event “rolling a number that is a multiple of 3”?
    5. What is the complement of the event “rolling a 5 or 9 or 12”?
    6. What is the complement of the event “rolling a number greater than 8?”?
    7. What is the complement of the event “rolling an odd number”?
    Click to see Answer
    1. [latex]\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}[/latex]
    2. [latex]\{1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12\}[/latex]
    3. [latex]\{10, 11, 12\}[/latex]
    4. [latex]\{1, 2, 4, 5, 7, 8, 10, 11\}[/latex]
    5. [latex]\{1, 2, 3, 4, 6, 7, 8, 10, 11\}[/latex]
    6. [latex]\{1, 2, 3, 4, 5, 6, 7, 8\}[/latex]
    7. [latex]\{2, 4, 6, 8, 10, 12\}[/latex]

     

  3. A recent survey asked people about home ownership and annual income. A total of [latex]750[/latex] people were surveyed. Of the [latex]750[/latex] people surveyed, [latex]425[/latex] owned a home. Of the [latex]750[/latex] people surveyed, [latex]338[/latex] people had an annual income of [latex]\$60,000[/latex] or more.
    1. What is the probability that one of the people in the survey does not own a home?
    2. What is the probability that one of the people in the survey has an annual income of less than [latex]\$60,000[/latex]?
    Click to see Answer
    1. [latex]0.4333[/latex]
    2. [latex]0.5493[/latex]

     

  4. A local college surveyed its recent graduates about their overall satisfaction with their college experience and employment status post-graduation. In the survey, [latex]75\%[/latex] of respondents said they were satisfied with their college experience, and [latex]64\%[/latex] of respondents said they found full-time jobs after graduation.
    1. What is the probability that a respondent was not satisfied with their college experience?
    2. What is the probability that a respondent did not find full-time jobs after graduation?
    Click to see Answer
    1. [latex]0.25[/latex]
    2. [latex]0.36[/latex]

     

“3.4 The Complement Rule” and “3.8 Exercises” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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