3.1.1 Using an Area Model

3.1.1 Using an Area Model

by Emily Joosse

It's in the Genes​

​Can you bend your thumb backwards at the middle joint?

​

​Can you roll your tongue into a "U" shape?

​

​There are special genes that allow you to be able to do either one of these things.

​Copy the table on your own paper.

​Assume that half the U.S. population can bend their thumbs backwards, and half can roll their tongues.

​Also assume that these genes are independent (having one gene does not affect whether or not you have the other) and randomly distributed (spread out) throughout the population.

Open Ended

List all outcomes in the sample space for these events. That is, list all the different combinations you can have of these two traits. Write your list using "curly brackets". For example, it should start as {YesYes, ..., ..., ...}.

Fill in the Blanks

Open Ended

According to this table, what is the probability that the person has only one of these special traits? Justify your conclusion.

​What if...?

​This table is useful because every cell in the table is equally likely. Therefore, each possible outcome has a 1/4 probability.

​

​In reality, however, only 1/4 of the population can bend their thumb, and 7/10 can roll their tongue. Draw a new table to represent these new probabilities.

Open Ended

The areas of the regions represent the relative probabilities for different outcomes. For example, the portion of the probability area model representing people with both special traits is a rectangle with a width of 1/4 and a height of 7/10.

What is the area of this rectangle? What does the area mean in terms of the two traits? What should the area of the entire rectangular model be? Why?

Open Ended

Based on your new probability area model, what is the probability that a randomly selected person can roll his or her tongue but not bend his or her thumb back? Explain how you got this probability.

​Probabilities in Vein

​You and your best friend may not only look different, you may also have different types of blood! For instance, members of the American Navajo population can be classified into two groups (73% has type O blood, while 27% has type A blood.

​(Blood types describe certain chemicals, called antigens, that are found in a person's blood).

​COPY THIS TABLE AND COMPLETE IT.

Multiple Choice

What is the probability that two Navajo individuals selected at random have type A blood?

• 19.71%

• 27%

• 7.29%

• 53.29%

Multiple Choice

What is the probability that person #1 has type A and person #2 has type O?

• 19.71%

• 27%

• 7.29%

• 53.29%

Multiple Choice

What is the probability that two Navajo individuals have different blood types?

• 19.71%

• 60.58%

• 7.29%

• 39.42%

Multiple Choice

What is the probability that two Navajo individuals selected at random have the same blood type?

• 19.71%

• 60.58%

• 7.29%

• 53.29%

​SHIPWRECKED!

​Zack and Nick (both from Vermont) are shipwrecked on a desert island! Zack has been injured and is losing blood rapidly, and Nick is the only person around to give him a transfusion.

​Most populations can be classified into four blood types: O (45%), A (40%), B (11%), AB (4%). These percentages are based on Vermont populations.

​We can make an area model to represent the sample space

​Copy and complete this area model on your own paper. If the outside probabilities are percentages, then the inside needs to be percentages as well.

Fill in the Blanks

Fill in the Blanks

Open Ended

To create that area model, we assumed that Nick and Zack weren't related. Blood type, however, is affected by genetic inheritance.

1. How would our possible outcomes be affected if Zack and Nick were related to each other (for example, father/son, or brothers)?

2. What has to be true in order to assume a probability area model will give an accurate theoretical probaility?