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3.3: The Addition and Complement Rules

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The Addition Rule

Two events are considered mutually exclusive if they cannot both occur. If events A and B are mutually exclusive, the probability that event A or event B will occur is the sum of the individual probabilities. Formulaically speaking, If events A and B are mutually exclusive, P(A or B)=P(AB)=P(A)+P(B).

If events are not mutually exclusive, they can both occur. Here we add the probability of each event, and subtract the overlap (A and B) because it was counted twice. Formulaically speaking, If events A and B are not mutually exclusive, P(A or B)=P(AB)=P(A)+P(B)P(AB).

These make up the addition rule.

  1. Let’s return to the census data on sex and marital status.

    Marital Status

    Married

    Widowed

    Divorced

    Separated

    Never Married

    Total

    Male

    64.5

    3.4

    12.2

    2.1

    47.5

    129.7

    Female

    63.4

    11.8

    16.5

    2.9

    41.5

    136.1

    Total

    127.9

    15.2

    28.7

    5

    89

    265.8

    1. Are the events “divorced” and “separated” mutually exclusive? Explain.






       
    2. What is the probability that a randomly selected adult in the US is divorced or separated? Round your answer to three decimal places. Show your thinking using the table.
       

      Marital Status

      Married

      Widowed

      Divorced

      Separated

      Never Married

      Total

      Male

      64.5

      3.4

      12.2

      2.1

      47.5

      129.7

      Female

      63.4

      11.8

      16.5

      2.9

      41.5

      136.1

      Total

      127.9

      15.2

      28.7

      5

      89

      265.8

      P( divorced  separated )=        _+        _






       

    3. Are the events “separated” and “female” mutually exclusive? Explain.




       
    4. What is the probability that a randomly selected adult in the US is separated or female? Use probability notation in your answer and round your answer to three decimal places. Show your thinking using the table.

      Marital Status

      Married

      Widowed

      Divorced

      Separated

      Never Married

      Total

      Male

      64.5

      3.4

      12.2

      2.1

      47.5

      129.7

      Female

      63.4

      11.8

      16.5

      2.9

      41.5

      136.1

      Total

      127.9

      15.2

      28.7

      5

      89

      265.8









       
  2. Here's a new game with two spinners. For this game, we say the spinners "match" if they land on the same color (e.g., both red, or both blue).

    Two simple circular spinners. Spinner #1 has one-quarter red area and three-quarter blue, and spinner #2 has one-quarter blue area and three-quarter red.
    Images are created with the graphing calculator, used with permission from Desmos Studio PBC.

    1. What is the probability that both spinners land on red?




       
    2. What is the probability that both spinners land on blue?




       
    3. What is the probability that the spinners match (both red or both blue)? Use probability notation in your answer.





       

The Complement Rule

The complement of an event A is “not A” and is denoted Ac. Because one of these events must occur, their probabilities must add to one. Therefore, we can compute the probability of event A using the rule of complements.

  1. Marjorie forgets to prepare for a multiple choice quiz so she randomly guesses on all 3 questions. Each question has 5 answer options.
     
    1. What is the probability that Marjorie gets a question correct on the quiz?








       
    2. What is the probability that Marjorie gets a question incorrect on the quiz?







       
    3. What is the probability that Marjorie gets all 3 questions correct?








       
    4. What is the probability that Marjorie gets at least 1 question incorrect?



       

       

       

       

       

       

       

       


       

This page titled 3.3: The Addition and Complement Rules is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Hannah Seidler-Wright.

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