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3.4 The Addition Rule

LEARNING OBJECTIVES

  • Calculate “or” probabilities using the addition rule.
  • Determine if two events are mutually exclusive.

For two events [latex]A[/latex] and [latex]B[/latex] we might want to know the probability that at least one of the two events occurs. For example, we might want to find the probability of rolling a 2 or a 5 in a single roll of a die, or we might want to find the probability that someone has a smartphone or a tablet. In probability terms, we want to find [latex]P(A\text{ or }B)[/latex], the probability that either [latex]A[/latex] or [latex]B[/latex] occurs. In probability, “or” is always an inclusive “or,” which means that either [latex]A[/latex] occurs, or [latex]B[/latex] occurs, or both occur.

The Addition Rule for Probabilities

To find [latex]P(A\text{ or }B)[/latex], we start by adding the individual probabilities, [latex]P(A)[/latex] and [latex]P(B)[/latex]. But this means that the overlap between the two events [latex]A[/latex] and [latex]B[/latex] is counted twice:  once by [latex]P(A)[/latex] and once by [latex]P(B)[/latex]. To correct for this double counting, we need to subtract [latex]P(A\text{ and }B)[/latex], the probability of both events occurring. This gives us the addition rule to find [latex]P(A\text{ or }B)[/latex]:

[latex]\begin{eqnarray*}P(A\text{ or }B)&=&P(A)+P(B)-P(A\text{ and }B)\\\\\end{eqnarray*}[/latex]

EXAMPLE

At a local language school, [latex]40\%[/latex] of the students are learning Spanish, [latex]20\%[/latex] of the students are learning German, and [latex]8\%[/latex] of the students are learning both Spanish and German. What is the probability that a randomly selected student is learning Spanish or German?

Solution

[latex]\begin{eqnarray*}P(\text{Spanish or German})&=&P(\text{Spanish})+P(\text{German})-P(\text{Spanish and German})\\&=&0.4+0.2-0.08\\&=&0.52\end{eqnarray*}[/latex]

EXAMPLE

There are [latex]50[/latex] students enrolled in the second year of a business degree program. During this semester, the students have to take some elective courses. [latex]18[/latex] students decide to take an elective in psychology, [latex]27[/latex] students decide to take an elective in philosophy, and [latex]10[/latex] students decide to take an elective in both psychology and philosophy. What is the probability that a student takes an elective in psychology or philosophy?

Solution

[latex]\begin{eqnarray*}P(\text{psychology or philosophy})&=&P(\text{psychology})+P(\text{philosophy})\\&&-P(\text{psychology and philosophy})\\&=&\frac{18}{50}+\frac{27}{50}-\frac{10}{50}\\&=&0.7\end{eqnarray*}[/latex]

TRY IT

At a local basketball game, [latex]70\%[/latex] of the fans are cheering for the home team, [latex]25\%[/latex] of the fans are wearing blue, and [latex]12\%[/latex] of the fans are cheering for the home team and wearing blue. What is the probability that a randomly selected fan is cheering for the home team or wearing blue?

 

Click to see Solution

 

[latex]\begin{eqnarray*}P(\text{home team or blue})&=&P(\text{home team})+P(\text{blue})-P(\text{home team and blue})\\&=&0.7+0.25-0.12\\&=&0.83\end{eqnarray*}[/latex]

EXAMPLE

Suppose a study of speeding violations and drivers who use cell phones produced the following fictional data:

Cell phone use Speeding violation in the last year No speeding violation in the last year Total
Cell phone user 25 280 305
Not a cell phone user 45 405 450
Total 70 685 755
  1. What is the probability that a randomly selected person is a cell phone user or has no speeding violations in the last year?
  2. What is the probability that a randomly selected person had a speeding violation in the last year or does not use a cell phone?

Solution

  1. [latex]\begin{eqnarray*}P(\text{cell phone or no violations})&=&P(\text{cell phone})+P(\text{no violations})\\&&-P(\text{cell phone and no violations})\\\\&=&\frac{305}{755}+\frac{685}{755}-\frac{280}{755}\\\\&=&\frac{710}{755}\end{eqnarray*}[/latex]
  2. [latex]\begin{eqnarray*}P(\text{violations or no cell phone})&=&P(\text{violations})+P(\text{no cell phone})\\&&-P(\text{violations and no cell phone})\\\\&=&\frac{70}{755}+\frac{450}{755}-\frac{45}{755}\\\\&=&\frac{475}{755}\end{eqnarray*}[/latex]

TRY IT

This table shows the number of athletes who stretch before exercising and how many had injuries within the past year.

Stretching Practice Injury in last year No injury in last year Total
Stretches 55 295 350
Does not stretch 231 219 450
Total 286 514 800
  1. What is the probability that a randomly selected athlete stretches before exercising or had an injury last year?
  2. What is the probability that a randomly selected athlete does not stretch before exercising or had no injuries in the last year?
Click to see Solution
  1. [latex]\displaystyle{\text{Probability}=\frac{350}{800}+\frac{286}{800}-\frac{55}{800}=0.72625}[/latex]
  2. [latex]\displaystyle{\text{Probability}=\frac{450}{800}+\frac{514}{800}-\frac{219}{800}=0.93125}[/latex]

Mutually Exclusive Events

Two events, [latex]A[/latex] and [latex]B[/latex], are mutually exclusive if the two events cannot happen at the same time. That is, the events [latex]A[/latex] and [latex]B[/latex] do not share any outcomes, and so [latex]P(A\text{ and }B)=0[/latex]. For example, in the experiment of flipping a coin, the events heads and tails are mutually exclusive because it is not possible to have both heads and tails on the top face. In the case of mutually exclusive events, the addition rule is [latex]\displaystyle{P(A\text{ or }B)=P(A)+P(B)}[/latex].

EXAMPLE

Suppose a bag contains [latex]20[/latex] balls.  [latex]10[/latex] of the balls are white, [latex]7[/latex] of the balls are red, and [latex]3[/latex] of the balls are blue.  Suppose one ball is selected at random from the bag.

  1. Are the events “selecting a white ball” and “selecting a red ball” mutually exclusive? Why?
  2. What is the probability of selecting a white or red ball?

Solution

  1. The events “selecting a white ball” and “selecting a red ball” are mutually exclusive because the events cannot happen at the same time. It is not possible for the selected ball to be both white and red.
  2. [latex]\displaystyle{P(\text{white or red})=P(\text{white})+P(\text{red})=\frac{10}{20}+\frac{7}{20}=0.85}[/latex]

NOTE

In the calculation of the probability in part 2, there is nothing to subtract. Because the events are mutually exclusive, [latex]\displaystyle{P(\text{white and red})=0}[/latex].

TRY IT

At a local college, [latex]60\%[/latex] of the students are taking a math class, [latex]50\%[/latex] of the students are taking a science class, and [latex]30\%[/latex] of the students are taking both a math and a science class.

  1. Are the events “taking a math class” and “taking a science class” mutually exclusive? Explain.
  2. What is the probability that a randomly selected student is taking a math class or a science class?
Click to see Solution
  1. The events “taking a math class” and “taking a science class” are not mutually exclusive because the events can happen at the same time (i.e. a student can be taking both a math class and a science class). As stated in the question, [latex]\displaystyle{P(\text{math and science})=0.3\neq 0}[/latex].
  2. [latex]\displaystyle{P(\text{math or science})=P(\text{math})+P(\text{science})-P(\text{math and science})=0.6+0.5-0.3=0.8}[/latex]

TRY IT

A fair, six-sided die is rolled one time.

  1. Are the events “rolling a 4” and “rolling an even number” mutually exclusive?
  2. Are the events “rolling a 4” and “rolling an odd number” mutually exclusive?
  3. What is the probability of rolling a 4 or rolling an odd number?
Click to see Solution
  1. The events “rolling a 4” and “rolling an even number” are not mutually exclusive because the events can happen at the same time (i.e. 4 is an even number).
  2. The events “rolling a 4” and “rolling an odd number” are mutually exclusive because the events cannot happen at the same time. It is not possible to roll a die and get a 4 (an even number) and an odd number on the top face at the same time
  3. [latex]\displaystyle{P(\text{4 or odd})=P(\text{4})+P(\text{odd})=\frac{1}{6}+\frac{3}{6}=\frac{4}{6}}[/latex]

Video: “Addition rule for probability | Probability and Statistics | Khan Academy” by Khan Academy [10:43] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.

Exercises

  1. 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. Are the events “rolling an even number” and “rolling a multiple of 3” mutually exclusive? Explain
    2. What is the probability of “rolling an even number” or “rolling a multiple of 3”?
    3. Are the events “rolling a number greater than 9” and “rolling a number less than 5” mutually exclusive? Explain.
    4. What is the probability of “rolling a number greater than 9” or “rolling a number less than 5”?
    Click to see Answer
    1. Not mutually exclusive because the events can happen at the same time (i.e. 6 is an even number and is a multiple of 3).
    2. [latex]0.6667[/latex]
    3. Mutually exclusive because it is not possible to get a number greater than 9 (i.e. 10, 11, 12) and a number less than 5 (i.e. 1, 2, 3, 4) at the same time.
    4. [latex]0.5833[/latex]

     

  2. 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, [latex]338[/latex] people had an annual income of [latex]\$60,000[/latex] or more, and [latex]293[/latex] people owned a home and had an annual income of [latex]\$60,000[/latex] or more.
    1. Are the events “owned a home” and “annual income of [latex]\$60,000[/latex] or more” mutually exclusive? Explain.
    2. What is the probability that one of the people in the survey owned a home or had an annual income of [latex]\$60,000[/latex] or more?
    Click to see Answer
    1. Not mutually exclusive because [latex]293[/latex] people fall into both categories.
    2. [latex]0.6267[/latex]

     

  3. 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, [latex]64\%[/latex] of respondents said they found full-time jobs after graduation, and [latex]52\%[/latex] of respondents said they were satisfied with their college experience and found full-time jobs after graduation.
    1. Are the events “satisfied with college experience” and “found full-time job” mutually exclusive? Explain.
    2. What is the probability that a respondent was satisfied with their college experience and found full-time jobs after graduation?
    Click to see Answer
    1. Not mutually exclusive because [latex]52\%[/latex] fall into both categories.
    2. [latex]0.87[/latex]

     

  4. [latex]U[/latex] and [latex]V[/latex]are mutually exclusive events. [latex]P(U)=0.26[/latex]; [latex]P(V)=0.37[/latex]. Find:
    1. [latex]P(U\text{ and }V)[/latex]
    2. [latex]P(U\text{ or }V)[/latex]
    Click to see Answer
    1. [latex]0[/latex]
    2. [latex]0.63[/latex]

     

  5. At a local college, [latex]20\%[/latex] of the students are studying business, [latex]40\%[/latex] of the students are studying mathematics, and [latex]8\%[/latex] of the students are studying both business and mathematics.
    1. What is the probability that a randomly selected student studies business or mathematics?
    2. Are the events “business” and “mathematics” mutually exclusive? Explain.
    Click to see Answer
    1. [latex]0.52[/latex]
    2. Not mutually exclusive because [latex]8\%[/latex] of the students are studying both.

     

  6. In a collection of eight cards, five cards are green, and three cards are yellow. The five green cards are numbered 1, 2, 3, 4, and 5. The three yellow cards are numbered 1, 2, and 3. The cards are well shuffled. One card is selected at random.
    1. What is the probability the card is green?
    2. What is the probability the card is green or has an even number on it?
    3. What is the probability the card is green and has an even number on it?
    4. Are the events “green” and “even” mutually exclusive? Explain.
    5. What is the probability the card is yellow or has a number greater than 3 on it?
    6. Are the events “yellow” and “number greater than 3” mutually exclusive? Explain.
    Click to see Answer
    1. [latex]0.625[/latex]
    2. [latex]0.75[/latex]
    3. [latex]0.25[/latex]
    4. Not mutually exclusive because there are green cards with even numbers.
    5. [latex]0.625[/latex]
    6. Mutually exclusive because there are no yellow cards with numbers greater than 3.

     

  7. Canadian Blood Services collects blood donations. A person with type O blood and a negative Rh factor (Rh-) can donate blood to any person with any blood type. Data shows that [latex]43\%[/latex] of people have type O blood, [latex]15\%[/latex] of people have Rh- factor and [latex]52\%[/latex] of people have type O or Rh- factor.
    1. Find the probability that a person has both type O blood and the Rh- factor.
    2. Find the probability that a person does NOT have both type O blood and the Rh- factor.
    Click to see Answer
    1. [latex]0.06[/latex]
    2. [latex]0.48[/latex]

     

  8. At a college, [latex]72\%[/latex] of courses have final exams, [latex]46\%[/latex] of courses require research papers, and [latex]32\%[/latex] of courses have both a research paper and a final exam.
    1. Find the probability that a course has a final exam or a research project.
    2. Find the probability that a course has NEITHER of these two requirements.
    Click to see Answer
    1. [latex]0.86[/latex]
    2. [latex]0.14[/latex]

     

  9. In a box of assorted cookies, [latex]36\%[/latex] contain chocolate, [latex]12\%[/latex] contain nuts, and [latex]8\%[/latex] contain both chocolate and nuts. Sean is allergic to both chocolate and nuts.
    1. Find the probability that a cookie contains chocolate or nuts (Sean can’t eat it).
    2. Find the probability that a cookie does not contain chocolate or nuts (Sean can eat it).
    Click to see Answer
    1. [latex]0.4[/latex]
    2. [latex]0.6[/latex]

     

  10. A previous year, the weights, in pounds, of the members of the San Francisco 49ers and the Dallas Cowboys were published in the San Jose Mercury News. The factual data are compiled into the table.
    Shirt Number At most 210 211–250 251–290 More than 290 Total
    1–33 21 5 0 0 26
    34–66 6 18 7 4 35
    67–99 6 12 22 5 45
    Total 33 35 29 9 106

    For the following, suppose that one player from these two teams is selected at random.

    1. What is the probability that the player’s weighs at most 210 pounds or has a shirt number in the 67-99 category?
    2. What is the probability that the player weighs between 251 and 290 pounds or has a shirt number in the 34-66 category?
    3. Are the events “1-33” and “more than 290” mutually exclusive? Explain
    Click to see Answer
    1. [latex]0.6792[/latex]
    2. [latex]0.5377[/latex]
    3. Mutually exclusive because there are no players that fall into both of those categories.

     

  11. The following table of data obtained from www.baseball-almanac.com shows hit information for four players. Suppose that one hit from the table is randomly selected.
    Name Single Double Triple Home Run Total Hits
    Babe Ruth 1,517 506 136 714 2,873
    Jackie Robinson 1,054 273 54 137 1,518
    Ty Cobb 3,603 174 295 114 4,189
    Hank Aaron 2,294 624 98 755 3,771
    Total 8,471 1,577 583 1,720 12,351
    1. What is the probability that a hit was made by Jackie Robinson or was a double?
    2. What is the probability that the hit was a single or made by Hank Aaron?
    3. Are the events “Babe Ruth” and “home run” mutually exclusive? Explain.
    Click to see Answer
    1. [latex]0.2285[/latex]
    2. [latex]0.8054[/latex]
    3. Not mutually exclusive because Babe Ruth hit [latex]714[/latex] home runs.

     

  12. The table shows a random sample of musicians and how they learned to play their instruments.
    Gender Self-taught Studied in School Private Instruction Total
    Female 12 38 22 72
    Male 19 24 15 58
    Total 31 62 37 130
    1. Find the probability a musician is male or is self-taught.
    2. Find the probability that a musician studied in school or is female.
    Click to see Answer
    1. [latex]0.5385[/latex]
    2. [latex]0.7385[/latex]

     

  13. The table shows the political party affiliation of each of [latex]67[/latex] members of the US Senate in June 2012 and when they are up for reelection.
    Up for reelection: Democratic Party Republican Party Other Total
    November 2014 20 13 0 33
    November 2016 10 24 0 34
    Total 30 37 0 67
    1. What is the probability that a randomly selected senator is a Republican or is up for reelection in November 2016?
    2. What is the probability that a randomly selected senator is a Democrat or is up for reelection in November 2014?
    Click to see Answer
    1. [latex]0.7015[/latex]
    2. [latex]0.6418[/latex]

     

  14. The table below identifies a group of children by one of four hair colours and by type of hair.
    Hair Type Brown Blond Black Red Totals
    Wavy 20 5 15 3 43
    Straight 80 15 65 12 172
    Totals 100 20 80 15 215
    1. What is the probability that a randomly selected child has wavy hair or black hair?
    2. What is the probability that a randomly selected child has blond hair or straight hair?
    3. Are the events “wavy” or “brown” mutually exclusive? Explain.
    Click to see Answer
    1. [latex]0.5023[/latex]
    2. [latex]0.8233[/latex]
    3. Not mutually exclusive because [latex]20[/latex] children fall into both categories.

     

“3.5 The Addition 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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