UP - Using Diagrams and Models in Probability Lesson

Using Diagrams and Models in Probability Lesson

Adapted from Course materials (II.A Student Activity Sheet 1, 2, & 3) for AMDM developed under the leadership of the Charles A. Dana Center, in collaboration with the Texas Association of Supervisors of Mathematics and with funding from Greater Texas Foundation.

Ms. Sweet, a third-grade teacher, loves the fall season. She loves pumpkins, costumes, and candy. Ms. Sweet is the fall festival coordinator for a small town in middle Georgia. A team of organizers volunteer each year to plan a week-long festival. The community enjoys many activities to celebrate the fall season.

Ms. Sweet conducted a survey of her classroom to help determine candy preferences. She asked her 29 students to pick which type of candy they like given the choice of chocolate candy, chewy candy, or both (notice that there were 4 children who did not like candy, they were placed outside the circle). She put the results in a Venn Diagram:

Image of venn diagram showing 10 chocolate, 8 chewy, and 7 both chocolate and chewy.

 

Answer the following questions concerning her survey results.

  1. List 5 facts about Ms. Sweet's students: There are many facts you can choose from. Here is a list.  
  • There are 29 students in Ms. Sweet's classroom.
  • There are 25 students who like candy.
  • There are 4 students who do not like candy.
  • There are 10 students who only like chocolate candy. 
  • There are 7 students who like chocolate and chewy candy.
  • There are 17 students who like chocolate candy. 
  • There are 15 students who like chewy candy.
  • There are 8 students who only like chewy candy.
  1. If a student is selected at random from Ms. Sweet's classroom, what is the probability that the student likes both chocolate and chewy candy? This probability question can be written to look like P(Chocolate and Chewy)

Look at the Venn Diagram, students who like both chocolate and chewy are found in the center. You should see 7 students, so 7 is the chance of getting the desired outcome. Place that number over the total number of possible outcomes. In this case, it's 29 total students - then divide. Then multiply by 100 to change the decimal into a percentage. The answer is 24.1%.

 

  1. If a student is selected at random from Ms. Sweet's classroom, what is the probability that the student does not like candy at all? This probability question can be written to look like P(not Chocolate and not Chewy)

This time the Venn Diagram doesn't really help us, but we do know that there are 29 students in the class and only 25 of them are on the Venn Diagram. The other 4 did not like candy. Set up your math problem as in number 2 above.

 

  1. Find the probability P(Chocolate or Chewy)

Here you are looking for students that like chocolate candy or that like chewy candy. Referring back to the Venn Diagram, 17 students like chocolate and 15 students like chewy, but we have to be careful and not double count the ones who like both chocolate and chewy (the 7 in the middle).


A mathematical probability calculation determining the probability of either chocolate or chewy, using the formula P(chocolate or chewy) = P(chocolate) + P(chewy) - P(chocolate and chewy). The steps include calculating the probabilities as fractions, with P(Chocolate or Chewy) = 17/29 + 15/29 - 7/29, which simplifies to 25/29. The result is approximately 0.862, and when multiplied by 100, the probability is 86.2%.

We can also solve this problem using the complement of the probability of the students that do not prefer chewy or chocolate. To find the probability of the complement of an event, you must subtract 1 from the probability of the event.

A probability calculation that subtracts the probability of an event not being chocolate and not being chewy from 1. The equation is written as 1 - P(not chocolate and not chewy), followed by the calculation 1 - 0.138 = 0.862. The final result is multiplied by 100, leading to 86.2%.

  1. Find the probability of a student that only likes Chewy candy, given that the student does not like Chocolate Candy, or P(Chewy| not Chocolate). The | that you see in the probability statement is read as "given." So P(A|B) would be read as "the probability of A, given B." What this does is reduce the total pool of students. There are 29 students in the class, but only 12 that don't like chocolate candy, the 8 that like chewy and the 4 that don't like any candy.

A probability calculation determining the probability of an event being chewy given that it is not chocolate. The equation is written as P(Chewy | not Chocolate) = 8/12 = 0.678. The result is then multiplied by 100, leading to 67.8%.

Tree Diagrams

Venn diagrams and tree diagrams can be used to find probability. Venn diagrams can help us make comparisons by organizing the data using intersecting circles. The sample space includes all events inside the circles and outside of the circles. An intersection is the overlap between one or more events. Venn Diagram can easily show us the probability of event A and event B occurring at the same time.

We can create a tree diagram to illustrate the sequence of events, compound events, and all the possible outcomes. To calculate the probability that both events will occur we must multiply the probability of event A by the probability of event B.

The festival organizers decided to construct a corn maze in a field and charge customers to walk through the maze. Customers can only walk forward. If the customers end up at an exit with a prize, they win a pumpkin.

Corn Maze showing three paths, with one entry. All three paths are connected by other, smaller pathways.

What are the probabilities that a customer, entering the maze on the upper,

middle or lower path can proceed to an exit with or without a pumpkin?

Four orange circles represent a place on the corn maze where users have to choose a path

A graphic representation of a corn maze with a map and a probability question. The map shows two paths labeled ‘Upper Path’ and ‘Middle Path,’ starting from an entrance at the bottom left corner and leading to an exit at the top right corner. Three pumpkin icons are placed along the Upper Path, while one is on the Middle Path. There are two forks in the path where decisions must be made, indicated by arrows pointing in different directions. Accompanying text explains a probability problem related to choosing paths within the maze.

This information can also be expressed using a tree diagram. The diagram below is a tree diagram based on this corn maze. Notice that if you took the lower path, you would have a second choice of three paths. If you took the upper one of those, that choice would then be followed by another fork. If you then took the upper fork, you would get a pumpkin.

The tree diagram below shows this, except the second fork is repeated to reflect the possibility that you could get there from the lower path.

A tree diagram labeled ‘Tree Diagram’ at the top. It starts with a single point labeled ‘Start’ with an arrow pointing to three branches. Each branch is marked with a fraction: the first and third branches are marked with 1/3, and the middle branch is marked with 1/2. From each of these branches, two more branches emerge, each marked with 1/2. At the end of each branch, there are images of either a pumpkin or text saying ‘No Pumpkin.’ The pumpkins appear at the ends of the top two branches and one bottom branch from the middle section. All other endpoints have ‘No Pumpkin’ text.

Area Model

We have discussed different models to help us find probability. Another model to consider is called an area model. An area model enables you to use fractions of the area of a diagram to represent corresponding probabilities. Area models are helpful when the outcomes of the events are not equally likely. The model can show that some events occupy more area than others, which indicates that these events are not equally likely. Tree diagrams can also be used to model these situations; the branches of the tree, however, are weighted to show the likelihood of the occurrences.

Watch the following videos to learn how to make an area model using a more complicated maze.

Part 1

Part 2

Part 3

Self-Assessment: Create your own Area Model

Using our original corn maze, create your own area diagram to determine what the probability would be of a customer winning a pumpkin. When you have finished, check your work using the answer key linked below. 

Download iconDownload the answer key to check your answers.

IMAGES CREATED BY GAVS