Lesson 1 Visualizing Probability with Area Models Develop Understanding

Jump Start

1.

For each of the following spinners, what is the probability that the arrow of the spinner will point to a region labeled “A” on the first spin?

a.

Spinner 1:

b.

Spinner 2:

$R S T U V$ is a regular pentagon.

c.

Spinner 3:

$\overset{―}{M N} ⊥ \overset{―}{P Q}$

$O$ is the midpoint of segment $P K$ .

$K$ is the midpoint of segment $P Q$ .

2.

If a dart is equally likely to hit any point on the following dartboards (that is, inside the outer circle or square), what is the probability that the dart will hit the shaded region?

a.

Dartboard 1:

b.

Dartboard 2:

$\overset{―}{M N} ⊥ \overset{―}{P Q}$

$O$ is the midpoint of segment $P K$ .

$K$ is the midpoint of segment $P Q$ .

Learning Focus

Create and use area models to determine theoretical probability of an outcome.

How can the probability of an event be represented geometrically?

How can an area model be used to calculate the probability of two events that happen at the same time or one after the other?

Open Up the Math: Launch, Explore, Discuss

A compound event involves the probability of two or more independent events occurring together.

For example, we may want to find the probability of both spinner 1 and spinner 2 pointing to a region marked “A” if the two spinners are spun simultaneously. Although we know the probability of each independent event, how might we find the probability of the two outcomes occurring together?

Spinner 1

Spinner 2

$R S T U V$ is a regular polygon.

1.

Use an organized list, a table, a tree diagram, or some other representation to determine the theoretical probability that both spinners point to a region marked “A” when Spinner 1 and Spinner 2 are spun at the same time. Your model should find the theoretical probability of the desired outcome for this compound event, and convince others that your conclusion is correct.

2.

Dez started creating the following Area Model to help solve problem 1.

a.

How does the probability that both spinners point to a region marked “A” show up in this model?

b.

What is the probability that both spinners point to a region marked “A”?

c.

What is the probability that both spinners point to a region marked with the same letter?

d.

Is there a numerical way to predict the probabilities you found in parts b and c? If so, tell what it is and explain why it works using your model.

3.

Consider the following two spinners:

Spinner 4:

$R S T U$ is a square.

$P$ is the midpoint of segment $R U$ .

$Q$ is the midpoint of segment $S T$ .

$K$ is the midpoint of segment $P Q$ .

$\overset{―}{K J} ⊥ \overset{―}{P Q}$

Spinner 5:

$R S T U$ is a square.

$M$ is located at the intersection of the two diagonals.

Which of the following representations best illustrates the probability that both Spinner 4 and Spinner 5 point to a region marked “A”?

A.

Theoretical Probability Model I

B.

Theoretical Probability Model II

4.

Based on the model you chose in problem 3, what is the theoretical probability that both spinners 4 and 5 point to a region marked “A”? How can you use the model to determine the probability?

5.

What are some advantages of using an area model to solve problem 3, rather than a table, organized lists, or a tree diagram?

6.

Sometimes rectangular models can be used to organize information and make sense of contexts that may involve choices that depend on other choices.

Commuting to School with Friends: Each morning Xavier goes to school with one of his friends: Adam, Ben or Charles. If he meets with Adam, they will either ride the bus or take the subway to school. If he meets with Ben, they will either walk or take the bus. If he meets with Charles, they can choose to take the subway, the bus, or go by car with Charles’ mother. If each choice is totally random, what is the probability that Xavier arrives at school by riding the subway?

a.

Create a rectangular model of this context.

b.

What is the probability that Xavier arrives at school by riding the subway?

c.

Is this an area model?

Ready for More?

Area models can be used to solve problems that involve successive events that depend on the outcome of the previous event, such as shooting successive foul shots in a basketball game.

1.

A one-and-one situation occurs in basketball when a player gets to shoot a second foul shot if the player makes the first shot. Consequently, the player can score 0, 1, or 2 points in a one-and-one foul situation. What is the probability that an $80 %$ foul shooter scores 0 points, 1 point, or 2 points?

a.

Draw an area model to represent the first shot: (Since the probability is given in percent, a 10 x 10 grid will be helpful.)

b.

Since the player takes a second shot only if the first shot was successful, use the area representing a successful first shot as the region for representing the two possible outcomes of the second shot. Modify your area model to represent both the first successful shot and the possible outcomes of the second shot.

c.

Interpret your model to determine the theoretical probability for each of the following outcomes:

0 points:

1 point:

2 points:

2.

Online card trading games advertise the odds of getting a special card in one of the packs. Many players post reviews online complaining that the games aren’t fair. Their arguments often sound like this: “They said the odds of getting a special card was 1 in 3 packs, but I opened 3 packs and didn’t get a special card.”

a.

What do you think of this argument?

b.

Draw an area model to represent the possible outcomes after opening 3 packs. Assume that each of the three draws is unsuccessful, since you wouldn’t open another pack if you already got the special card. Therefore, the unsuccessful regions of your model will represent the area for the outcomes of the next opening of a pack.

c.

Interpret your area model. What is the probability of getting a special card after opening $3$ packs if the odds are $1$ in $3$ ?

Takeaways

A probability area model is useful for:

To show that outcomes of an event are not equally likely in a probability model:

Vocabulary

area model
compound events
event
theoretical probability
Bold terms are new in this lesson.

Lesson Summary

In today’s lesson, students used area models to determine the probabilities of compound events. Area models visually represent the probability of an outcome as a fraction of the total area of a geometric figure representing the sample space of all possible outcomes of an event. The probability of an outcome can be calculated using area formulas.

Retrieval

1.

Evaluate each expression, rewrite as one number.

a.

$\frac{1}{7} \cdot \frac{3}{4} =$

b.

${(\frac{5}{9})}^{2} =$

2.

Determine the type of function as linear, exponential or quadratic. Create the representation desired.

$f (x) = x^{2} - 5$

Type of function:

Create a table for the function: