# Probability and Expected Value

**Goal:** The goal of this lesson is to introduce
students to the methods and ideas behind

probability theory.

**Theoretical Probability**

Students will learn how to find all the possible outcomes of a trial, including

repeated outcomes, which are often overlooked. In order to do this, they will
make

a tree diagram and learn how to make an area model. Once they have determined

the possible outcomes, they will compute theoretical probabilities.

**Expected Value**

Students will make predictions about an experiment based on the theoretical

probabilities they found.

**Experimental Probability**

Finally, students will perform an experiment and calculate the experimental

probabilities. They will compare these experimental probabilities with the

theoretical probabilities they found earlier, and they will compare the results
of the

experiment with the expected results

**Grade and Course:** 7^{th} grade math

**KY Standards:
MA-07-4.4.1
Students will apply counting techniques to determine the size of a sample space
for a realworld
or mathematical situation.**

**MA-07-4.4.2**

Students will:

• determine theoretical probabilities of simple events

• determine probabilities based on the results of an experiment and

• make inferences from probability data.

Students will:

• determine theoretical probabilities of simple events

• determine probabilities based on the results of an experiment and

• make inferences from probability data.

MA-07-4.4.3

Students will tabulate experimental results from simulations and explain how theoretical

and experimental probabilities are related.

**Objectives:**

After this lesson students will be able to:

o Differentiate between a theoretical and an experimental probability.

o Draw a tree diagram to find the possible outcomes of a trial.

o Draw an area model to find the possible outcomes of a trial.

o Compute theoretical probabilities.

o Make inferences based on theoretical probabilities.

o Compute experimental probabilities.

**Resources/materials needed:**

o One quarter

o 1 whiteboard, blackboard, or overhead projector

o 2 buckets or bags per group, labeled “Bucket 1” and “Bucket 2”

o 1 copy of worksheet per group (attached at end of document)

o 2 red, 1 yellow, 1 blue, and 3 green blocks per group, arranged as follows:

**Description of Plan:
Prior Knowledge**

Before this lesson, students know that the probability of an event is the number of ways

that event can happen, divided by the total number of possible outcomes. They are

familiar with reducing fractions.

**Introduction to Class**

When students enter the classroom, the buckets with the cubes in them are sitting in the

front of the room to capture the students’ curiosity. The lesson begins with a series of

class discussion questions intended to get the students thinking. For example:

- What are some carnival games that involve probability?

- What games could we play by flipping this quarter?

- What games could we play with the color cubes in these buckets? (Hold

up Bucket 1 and Bucket 2.)

Tell the class that today they are learning about probabilities that can be used in games

just like the ones they discussed and elsewhere, and they are going to get the opportunity

to perform two experiments, one with a quarter and one with the color cubes.

**Introduction to Activity 1 (Coin Flip)**

Define THEORETICAL PROBABILITY and EXPERIMENTAL PROBABILITY in

writing on the board. If the students usually keep their definitions in the same
place, have

them write it there. Talk about what the word theoretical sounds like (theory).
Talk

about what word experimental sounds like (experiment). Give a couple examples of
each

type of probability and have the students decide whether they are experimental
or

theoretical.

Now propose the following situation to the students: We flip a coin twice. How
many

possible outcomes are there? What are they? Write them on the board:

Possible Outcomes: 2 Heads, 2 Tails, A Head and a Tail

Ask the students, What is the probability of each outcome? (Students will
probably say

1/3.) Explain to the students that 1/3 is not the correct probability in any of
the three

cases. It seems to make sense, but there is a problem. Sometimes there are
hidden

outcomes that are repeated, and when dealing with probability, one must be
careful to

count ALL possible outcomes using a counting technique.

**Activity 1 (Coin Flip) Procedure**

Show the students how to make a factor tree to list all the possible outcomes of
flipping a

coin twice on the board:

List the possible outcomes: HH, HT, TH, TT

Now introduce the notation “P(event)” which means “the probability of an event.”
Find

P(HH), P(TT), P(HT), and P(HT) on the board as a class. Discuss how these

probabilities were found. Also find P(coins match) and P(coins do not match).

Ask the class, Are these probabilities theoretical or experimental? Why?

Announce to the students, "We are going to do our own experiment. I have a
quarter, and

each person in the class is going to flip it twice. We are going to be divided
into two

teams like so (girls vs. boys, left half vs. right half, etc.). Each time
someone on your

team gets a match (two heads or two tails), your team gets a point. If the two
flips do not

match, you get no point. The team with the most points wins. Since there are ___

students in the classroom, how many people do you expect to get a match? Not a
match?

How many people do you expect to get HH? TT? HT? TH?" (Write all of the expected

values on the board and how the students calculated them. Dwell on this subject
for a

while and make sure students understand where the expected values come from.)

At this point, go around the room and have each student
flip the quarter twice. Have

someone at the board record each student's outcome under that student's team
name. If

pressed for time, all students can flip a penny twice at the same time, but
going around

the room slowly builds anticipation and gives the students a break in this long
lesson.

When the game is done, congratulate the winners, console the losers, and have
the

students calculate the experimental probabilities of HH, HT, TH, TT, Match, and
No

Match on their own paper using all of the data, not just their team's data. Make
sure they

use the notation "P(HH) = ___" rather than just writing down the answer. Have
them

write a sentence comparing the theoretical probabilities to the experimental
probabilities.

Were they the same? Were they different? In what way?

Before transitioning to the next activity, discuss why just using an individual
team's data

would be less reliable than using the entire classroom's data if you were
interested in

estimating the theoretical probability.

**Introduction to Activity 2 (Drawing Blocks)**

Tell the class that the next activity will involve the buckets of color cubes
that the class

talked about earlier. Pose the following questions:

- If you close your eyes and draw a cube from Bucket 1, what is the

probability of drawing the red cube? A green cube?

- Now imagine that you close your eyes and draw one cube from Bucket 1

and one cube from Bucket 2. What is the probability of drawing a green

and a yellow? Why do you think that?

The concept of an area model is introduced. A square is drawn on the board or
overhead,

labeled “Bucket 1,” and cut into three equal pieces. One third is labeled “R,”
and the

other two are labeled “G.” This area model represents Bucket 1. Since there is a
1/3

chance of drawing a red from Bucket 1, “R” gets 1/3 of the area model. Since
there is a

2/3 chance of drawing a green from Bucket 1, “G” gets 2/3 of the area model. It
should

look like this:

Bucket 1