The Addition Principle of Counting Question: What is the addition principle of counting? To what type of situation is it applied?
Launch: Consider the set of all integers from 1 and 20. Answer the following questions about this set: a) Fill in the Venn diagram below with all 20 integers in the set. Place multiples of 2 in circle A, multiples of 3 in circle B. Place values which are neither multiples of 2 nor multiples of 3 inside the rectangle and outside of both circles.
a) How many members of the set are multiples of 2? b) How many members of the set are multiples of 3? c) How many members of the set are multiples of both 2 and 3? d) Suppose you are asked to choose an element from the rectangle that is either a multiple of two or a multiple of 3 (or both). How many numbers are there to choose from? e) Is your answer to part d) the sum of the answers to parts a) and b)? If not, how does it relate to the answers to for a), b) and c)?
Example 1: Below is a membership roster for a school club. The roster contains the first name, gender and class of each club member. First Name Gender Class Beth Female Freshman Caleb Male Freshman John Male Freshman Jorge Male Sophomore Mary Female Sophomore Lucy Female Sophomore Abby Female Junior Rob Male Junior Sam Male Jnnior Will Male Senior Jada Female Senior Joe Male Senior Kelly Female Senior A person in the club is to be chosen to travel to a national convention. 1.
How many females are there to choose from?
How many seniors are there to choose from?
How many senior females are there to choose from?
After the choice is made, you are told that the chosen person was a senior or a female (or both). How many club members are there who fit this description?
Special Note: In mathematics, the word “or” is most always used to mean “one or the other or both”. This is known as an inclusive “or”. Throughout the rest of this activity, always use this interpretation when asked to answer questions about the occurrence of one event or another. 5.
Explain the relationship between the answer to question 4 and the answers to questions 1 through 3.
The relationship discovered above is the Addition Principle of Counting. Use it to answer the following question: Question: Suppose event A can occur in 7 ways and event B can occur in 4 ways. Further, suppose that both events A and B can occur together in 2 ways. How many ways can either event A or event B occur? Fill in the blank in the following explanation of this principle. Form 1 of Addition Principle of Counting: Suppose an event A can occur in m ways and an event B can occur in n ways. Further, both A and B can occur together in k ways. Then the number of ways that either A or B can occur is ______________________________. Definition: Two events A and B are said to be disjoint or mutually exclusive if both A and B cannot occur together in a single event. Example: Look again at the club membership roster. Suppose SF is the event that a senior female is chosen for the trip. Suppose JM is the event that a junior male is chosen for the trip. 1.
Explain why events SF and JM would be disjoint.
How many ways can SF occur?
How many ways can JM occur?
How many ways can both SF and JM occur together?
How many ways can either SF or JM occur?
How did your answer to 5 relate to your answers to 2 and 3?
You have discovered a special form of the addition principle of counting that applies to disjoint events. Use what you have discovered to answer the following question:
Question: Events A and B are known to be disjoint. Event A can occur in 14 ways. Event B can occur in 7 ways. How many ways can either A or B occur? Fill in the blank in the following form of the addition principle of counting. Form 2 of Addition Principle of Counting: Suppose an event A can occur in m ways and an event B can occur in n ways. Further, events A and B are disjoint. Then the number of ways that either A or B can occur is ______________________________. Form 1 is the most general version of the addition principle of counting because it can be applied to problems where the events A and B are disjoint as well as to problems where they are not disjoint. Explain why Form 1 works even when the two events are disjoint.
Venn diagrams are sometimes useful in problems like those encountered above. In the Venn diagram below, let circle S represent the seniors. Let circle F represent the females. Place each seniors name in circle S. Place each females name in circle F. Make sure to place senior females in both circles, BUT only write each name once. Place all other club members’ names outside the circle yet inside the box.
Count the number of members whose names appear in either S or F. Does this correspond with your answer to the question about the number who are either seniors or females?
Complete the Venn diagram below by letting circle SF represent the senior females and circle JM represent the junior males. Write the names of the members in the appropriate circles. You can write the other club members’ names outside the circles if you choose to do so, but it will get a little crowded.
In this problem SF and JM were disjoint or mutually exclusive sets/events. What property of the filledin diagram, relates to the fact the sets are disjoint? In fact, since the SF and JM are disjoint, we could have filled out the same information on the following Venn diagram.
Why would the Venn diagram such as the one above not be adequate for displaying the event of choosing a senior or a female?
In what type of situation is the addition principle of counting applied?
What is the most general form of the addition principle of counting?
What are disjoint events?
What is the addition principle of counting for disjoint events?
In Class Problems: 1. A popular nighttime game show has contestants choosing cases that are marked with the numbers 1 through 26. Below is a table where the 26 cases are represented. The case numbers are displayed in the table below. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 1) How many cases display numbers less than or equal to 8? 2) How many cases display numbers greater than 23? 3) Are the events described in questions 1) and 2) disjoint? Explain.
4) If a contestant decides to choose one case that either display a number less than or equal to 8 or greater than 23, then how many cases does she have to choose from? 5) Explain how the answer to question 4 is related to the answers for questions 1 and 2. 6) How many cases are there with even numbers on them? 7) How many cases are there with numbers less than 7 on them? 8) Suppose a contestant decides to choose a case with an even number or a case with a number less than 7. Go through the cases and mark the ones with numbers that are either even or less than 7. Mark a case twice if it satisfies both conditions. 9) How many cases display even numbers that are less than 7? 10) How many cases were marked at least once in question 8? 11) How does your answer to question 10) relate to your answers to 6), 7) and 9)?
Sometimes the number of members in sets can be quite large and listing all the individual outcomes can make the situation more confusing. However, the addition principle of counting can still be applied if one knows how many elements are in each set. Below is such a situation. Suppose some data was collected on some students from a high school. The students were asked to choose one favorite fast food from among 3 choices and to specify one favorite subject from among 4 choices. Students were not allowed to choose more than one favorite in each category. The table below is called a contingency table. It shows the counts for each set of choices. For example, 45 students chose pizza as their favorite fast food and math as their favorite subject, so 45 appears in the cell of the table in the Math row and the Pizza column. Pizza Burgers Mexican Total Math 45 40 30 English 32 42 34 History 36 30 30 Science 28 20 16 Total a) Look at the table to determine how many students chose burgers as their favorite fast food and science as their favorite subject? b) Compute the row and column totals to fill in the blank cells in the table. In the last cell of the bottom row, place the sum of all the interior cells. This value will be the total number of students who participated in the survey. It can also be found by summing all the row totals or summing all the column totals. c) Use the completed table to determine ow many students chose pizza as their favorite fast food? d) How many students chose Mexican food as their favorite fast food? e) Are the events described in c) and d) disjoint? Explain.
f) Use the addition principle of counting to state how many students chose either pizza or Mexican food as their favorite fast food. Explain your result.
g) How many students chose math as their favorite subject?
h) Is the event of choosing math as the favorite subject disjoint from the event of choosing pizza as the favorite fast food? Why or why not?
i) Use the addition principle of counting to determine how many students chose math as their favorite subject or pizza as their favorite fast food. Explain your result.
Closure: Complete the following: If two events A and B are disjoint then _________________________________. What is another term for disjoint? _____________________________________. If events A and B are disjoint, and event A can occur in m ways and event B can occur in n ways then the number of ways either A or B can occur is _________________. If event A can occur in m ways and event B can occur in n ways and both events can occur together in k ways then the number of ways either A or B can occur is ________________.
A look into Veronica’s closet shows that she has 10 red tshirts. She has 17 tshirts with sports logos on them. 4 of the red tshirts have sports logos on them. Use the addition principle of counting to count the number of choices in Veronica’s closet which are either a red tshirt or a tshirt with a sports logo.
Max is a waiter in a restaurant. In one week, 35 people ordered shrimp and 45 people ordered steak. Max noted that 15 of these people ordered a platter that had both shrimp and steak on it. Use the addition principle of counting to count the number of people served in that week who ordered either shrimp or steak.
A market research firm asked some customers at a local mall to choose between three cola flavors. The gender of each customer was recorded. It turned out that 5 males liked cherry, 3 females liked cherry, 7 males liked vanilla, 5 females liked vanilla. 6 females liked lime. 4 males liked lime. Fill in the following contingency table to display this information: Cherry
Male Female Totals a) How many males were surveyed? Females? b) How many people surveyed preferred cherry cola? Vanilla? Lime? c) How many people surveyed preferred either cherry or vanilla cola? d) How many people surveyed were either female or preferred vanilla cola/ 4. A class is surveyed on where they want to take their senior trip. It turns out that 37 males want to go to Jamaica. 45 females want to go to Jamaica. 28 males want to go to Florida. 31 females want to go to Florida. 17 males want to go to Mexico. 19 females want to go to Mexico. a) Use this information to make a contingency table.
b) How many students surveyed were either male or wanted to go to Jamaica? c) How many students surveyed either wanted to go to Jamaica or to Mexico? d) Which problem above (b) or c)) involves mutually exclusive events? 5.
(Challenge problem) A club has designed three tshirts for its members and is taking orders from students. The three designs include a longsleeved tshirt, a shortsleeved t shirt and a tankstyle tshirt. Some members buy no tshirts, some buy one style, some buy two styles and some buy all three styles. Here are the details of the purchases: Note: The information given for each category includes all purchases for that category. For example, when you are told that 13 students purchased a longsleeved tshirt and a tank, this count includes the 6 students who purchased all three styles.
6 students purchased all three styles 13 students purchased a longsleeved tshirt and a tank 9 students purchased a longsleeved and a shortsleeved tshirt. 16 students purchased a tank and a shortsleeved tshirt. 28 students purchased a longsleeved tshirt 34 students purchased a shortsleeved tshirt 25 students purchased a tank 5 students purchased no tshirt
Fill in the Venn diagram below with the correct number of students. The first piece of information has been added to the diagram already.
How many students purchased either a tank or a shortsleeved tshirt? How many students are in the club?