Probability
Grade 6
Andrew Sundberg
andrew@laporte.k12.mn.us
Travis Whittington
twhittington@clbs.k12.mn.us
Justin Lundin
jlundin@kelliher.k12.mn.us
Executive Summary
This probability unit will take about ten class days. It will cover topics
such as chance, experimental and theoretical probability,
likely/unlikely and equally likely outcomes, and expected value.
Students will be taught how to use area diagrams and tree diagrams
to construct sample spaces. The goal of this unit is to give students
tools to better understand concepts of probability.
NCTM Standards Covered in Grades 6-8
- understand and use appropriate terminology to describe
complementary and mutually exclusive events
- use proportionality and a basic understanding of probability to
make and test conjectures about the results of experiments and
simulations
- compute probabilities for simple compound events, using such
methods as organized lists, tree diagrams, and area models
- use conjectures to formulate new questions and plan new
studies to answer them
Table of Contents
Day 1
Pretest, Introduction to Chance and Probability
Terminology
Day 2
Exploring Likely/Unlikely Outcomes and “Fair
Games”
Day 3
Using Maps/Mazes to explore area diagrams, tree
diagrams
Day 4
Experimental Probability vs. Theoretical
Probability
Day 5
Developing Strategies to find match Experimental
Probability to Theoretical Probability
Day 6-7
Expected Value
Days 8-9
Carnival – Make a Game and Find Expected Value
Day 10
Post-Test
Day 1 “Intro to Chance”
Objective: Students will introduced to chance and the terms complementary
and mutually exclusive events after they take the pretest.
Standards Covered:
• understand and use appropriate terminology to describe complementary
and mutually exclusive events
• use proportionality and a basic understanding of probability to make and
test conjectures about the results of experiments and simulations
Materials: “Probably Graphing” Worksheet, Pennies, Plastic Globe, Calculators
Procedures:
1. Start with Globe Activity to introduce probability to students. Guide them to a
method of determining the percentage/fraction of Earth that is covered by water,
without using prior knowledge. (Example- recording where our left thumb is when
we catch the globe) Talk about how the more we do our experiment, the more
valid our results become.
2. Move to Probably Graphing activity. Talk about the possible outcomes when we
flip a coin and introduce the terms “equally likely”, “complementary events”, and
“mutually exclusive events”.
3. Have students flip the pennies and record their data on Probably Graphing
worksheet. When finished, transfer data onto bottom of worksheet and record
running totals of heads and tails.
4. Put the whole class’s data together. It should reflect that our totals of heads and
tails should be relatively close even some individuals may have had much more of
one outcome than the other.
5. Introduce format –
Number of Favorable Outcomes
P(Event) =
Total Number of Outcomes
Summarize: Tie together the students expectation that we should get about half heads, half
tails with our experimental data. Does it reflect what we thought should happen? Did our
data of the globe reflect our prior knowledge that Earth is covered by about 70% water?
Discuss that heads and tails are complementary and mutually exclusive events.
Extension: Worksheet on pg. ____
Activity found @
http://illuminations.nctm.org/LessonDetail.aspx?ID=L790
Link to “Probably Graphing” worksheet
http://illuminations.nctm.org/Lessons/ProbExplorations/ProbablyGraphing-AS.pdf
Day 2 “Likely, Unlikely Outcomes”, “Fair Games?”
(Activity Modified from Texas Instruments, 1998)
Objective: Students will be asked to define likely, unlikely, and equally likely
outcomes and determine whether or not a game is “fair”.
Standards Covered:
• use proportionality and a basic understanding of probability to make and test
conjectures about the results of experiments and simulations
• compute probabilities for simple compound events, using such methods as organized
lists, tree diagrams, and area models.
Materials: Dice, worksheet
Procedures:
1. Write a number line on the board and label the ends 0 and 1. Talk about the fact that
if an event has the probability of 0 it is impossible and if it has a probability of 1 it is
certain to happen. Come up with some events and ask the students where they think
they fit on the number line.
2. Introduce Dice game. Students roll three dice and record the sums and products of
the three dice. “Hey, that’s not fair worksheet”
3. Have students record their results on worksheet for 20 trials.
4. Ask the students if they think the games were fair. Was one game fair and the other
unfair?
5. Introduce tree diagrams on board to show the theoretical probability of the possible
outcomes in the two games.
Summarize: Review the 0 to 1 number line and how that relates to the probability of events
occurring. Talk about how tree diagrams are a useful tool for finding sample spaces and
possible outcomes.
Extension/Homework: Roller Derby
(Activity found in Connected Math Program: How Likely Is It?)
Day 3 “Area Diagrams through Maps and Mazes”
(Activity Modified from Addison Wesley Publishing Company, 1986)
Objective: Students will learn about area diagrams through an activity.
Standards Covered:
• compute probabilities for simple compound events, using such methods as organized
lists, tree diagrams, and area models.
Materials: Maze Worksheet
Procedures:
1. Draw a map on the board with two possible destinations and a few possible ways of
getting to each destination. Give the hidden treasure example of “Where would you
hide the treasure?”
2. Explain the results of the map drawn on the board without showing tree diagrams and
area diagrams.
3. Hand out worksheet “The Maze”. Let students work on problem in groups of 2-3 for
10-15 minutes. (Some may use a tree diagram from yesterday’s learning)
4. Ask students to show their group’s solutions to the problem in front of class.
5. After everyone has shown solution, introduce area diagrams to class as a method to
solve this problem.
6. Reform student groups and have them solve Maze #2 using area diagrams and one
other method of their choice.
Summarize: Talk about the tools we now have to analyze probability problems (tree
diagrams and area diagrams) and move to the Counting Principle. (Two dice 6*6=36 possible
outcomes)
Extension: Play SKUNK if time allows.
(Activity from http://digitallesson.com/MathGames/GameSkunk.pdf)
Day 4 “Predicting Samples based on Experiments”
(Activity Modified from Investigations)
Objective: Students will be able to come up with a reasonable estimate for a
sample based upon experimental evidence.
Standards Covered:
• understand and use appropriate terminology to describe complementary and mutually
exclusive events;
• use proportionality and a basic understanding of probability to make and test
conjectures about the results of experiments and simulations
Materials: Hat/Box, Colored cubes
Procedure:
1. Prior to class, fill a box/hat with colored cubes. Keep track of the total number and
how many of each color cube there are.
2. Make tables on the board for the results of drawing the cubes.
3. Have students come up and draw one cube and replace it after each draw. Let them
know which colors are in the box, but not how many of each are in the box.
4. Repeat the experiment at least 3 times. When finished, have the students tally the
results individually.
5. Let the students come up with estimates of what they think is in the box. After some
time, tell them the total number of cubes to allow for a more reasonable estimate.
6. Let students come up in front of the class and explain their estimates. They must have
a reason why they came up with their estimates.
7. Tell the students what was in the box.
Summarize: Have a discussion on whether the experimental evidence we collected actually
reflected on what was in the box. Talk about how our predictions may be different after just
one trial compared to if we had done 100 trials. (The more an experiment is done, the closer
it should reflect to what is in the sample)
Extension: Worksheet on pg. -----
Day 5 “Matching Experimental to Theoretical”
Objective: Students will conduct experiments to compare theoretical and
experimental probability and use them to predict trends.
Standards Covered:
• understand and use appropriate terminology to describe complementary and mutually
exclusive events
Materials: Dice, Coins, Decks of Cards, Calculators, “What are my Chances?”
Worksheet
Procedures:
1. Start class with the following formula written on the board.
n
Number of Favorable Outcomes
P(Event) =
Total Number of Outcomes
2. Set up five experiments throughout the room that students will rotate through. They
are….
i. Flip a coin (P-heads)
ii. Roll a die (P-3)
iii. Draw a card (P-red card)
iv. Draw a card (P-diamonds)
v. Draw a card (P-7 of clubs)
3. Now, have the students write down what they think the theoretical probability is for
the above five experiments.
4. Distribute the “What are my Chances?” worksheet and let the students go around the
room and conduct the five experiments to find the experimental probability.
5. When finished, compile all the class’s data. Compare the experimental probability to
what we figured the theoretical probability would be.
Summarize: Again, point out that the experimental probability gets closer to matching the
theoretical probability (usually) with more and more data. Ask questions about trends that
could occur, such as “How many red cards would you expect to pull if you did this experiment
100 times?”, or “How many times would you expect to roll a 6 if you rolled the die 30 times?”
Extension: Homework Problem found on pg. ___
Activity found @
http://illuminations.nctm.org/ActivityDetail.aspx?ID=79
Link for “What are my Chances?” worksheet
http://illuminations.nctm.org/Lessons/ProbExplorations/WhatAreMyChances-AS.pdf
Days 6-7 “Expected Value”
(Activity Modified from Addison Wesley Publishing Company, 1986)
Objective: Students will be introduced to the concept of expected value by
combining probability with an assigned value.
Standards Covered:
• use conjectures to formulate new questions and plan new studies to answer them.
Materials: Newspaper Activity sheet, Ping Pong balls, Decks of Cards, Play
Money, Sacks or Boxes (cannot see inside of them)
Procedures:
1. Prior to class, separate the ping pong balls into groups of 6. Label 5 of the balls with a
1 and label the other with a 10. Split the cards in groups of 6. Make them in groups of
5 of one color and 1 of the other color. With the play money, put in groups of 5 $1
bills and 1 $10 bill.
2. Explain the story of the paper boy and his options of getting paid. He can take option
#1, which is get paid $5 per week every week. Option #2 is drawing two bills out of a
hat. In the hat is one $10 bill and 5 $1 bills. Explain that we are going to simulate
option #2 with the ping pong balls, cards, and play money.
3. Have a discussion about which deal the class thinks would be a better deal for the
paper boy. Ask if the length of time he’s a paper boy would change their decision.
4. After discussion, allow the class to conduct their experiments and collect data.
5. Compile the data (he can get paid either $11 or $2) of the whole class.
6. Lead into expected value, what are the chances he would come out ahead if he took
option 2?
Summarize: Show with a tree diagram that there are 30 possible outcomes in this
experiment, and 10 of them involve getting a $10 bill. So, the paper boy would have a 1/3
chance of getting $11 and a 2/3 chance of getting $2. Compare this to our experiment’s
results to see if this is reflected. Now, make the transition to expected value by adding the
value to the probability.
P($11)= 1/3 * $11 = 11/3
P($2)=2/3 * $2= 4/3
11/3 + 4/3= 15/3 (Expected Value)
So… the expected value over time is $5.
Discuss that over time, the paper boy’s choice doesn’t matter since he’s going to make $5
both ways if done over a long period of time.
Extension: If there’s time, give the students the same problem but with different numbers
(don’t have the same expected value this time). Ask them to find which option would be
better for the paper boy over time.
Days 8-9 “Carnival”
(Activity Modified from BSU Summer Math Institute)
Objectives: Students will create carnival games, calculate expected values of
the games, compare experimental evidence to theoretical
probability, and determine whether or not games were “fair”.
Standards Covered:
• understand and use appropriate terminology to describe complementary and mutually
exclusive events
• use proportionality and a basic understanding of probability to make and test
conjectures about the results of experiments and simulations
• compute probabilities for simple compound events, using such methods as organized
lists, tree diagrams, and area models.
Materials: Cards, Dice, Coins, Colored cubes, anything the students want to
make their carnival game, paper to tabulate results of carnival
games, Games of Chance handout
Procedures:
1. Start with a quick intro of “Games of Chance”. Ask if the games are fair games (ex.draw a club—pay 2-1). Have students calculate expected value for the games to figure
out their fairness.
2. Have students pair up and come up with their own game. The game has to have at
least 2 possible outcomes and the pair must come up with a payout for each outcome.
Each game costs $1. DO NOT figure out the expected value yet, we’ll do that after the
experiment.
3. Next, the students go around the carnival and play other group’s games. One group
member must stay with their game and the other can go around and play the game.
Switch off so both members have the opportunity to play the other games. The
student who stays with the game must keep track of the results of the game
(worksheet provided on pg. ___ of document) Students should play each game 10
times so we have sufficient data to judge our games.
4. After we’ve collected all the data, have students determine the expected value of
their individual games.
5. Then, have students figure out the actual payout of their games, i.e. their
experimental data.
Summarize: Draw connections between the expected value and the
experiments. Ask students if they charged about the right amount for the game
they created. Have a discussion about games at casinos and how the expected
value is ALWAYS in favor of the house.
Day 10 “Post Test”
Objective: Students will be able to demonstrate what they’ve learned and
apply it on problems in the test.
Standards Covered:
• understand and use appropriate terminology to describe complementary and mutually
exclusive events
• use proportionality and a basic understanding of probability to make and test
conjectures about the results of experiments and simulations
• compute probabilities for simple compound events, using such methods as organized
lists, tree diagrams, and area models.
Materials: Post Test
Procedures:
1. Discuss and review concepts covered in this unit.
2. Distribute assessment to students.
Assignment for Day 1
1.
Sally flipped a coin five times in a row and got tails every time. She told her teacher there must
be something wrong with the coin she was flipping. Do you think there was something wrong
with the coin? Why or why not?
2.
Leo flipped a coin and got heads three times in a row. What are the chances he will get a head
on the next toss? Explain your reasoning.
3.
Is it possible to flip a coin 20 times and get tails 20 times? Is it likely to happen? Explain your
reasoning.
Assignment for Day 4
1.
There is a piggy bank with coins in it. You cannot see inside of the piggy bank. You decide to
conduct an experiment to see if you can determine what’s in the bank without dumping out the
coins. You draw and replace your coin 30 times. Here’s the results:
- 12 pennies
- 4 nickels
- 6 dimes
- 8 quarters
a.
Based upon your experiment, write down two observations that you think are true about
the sample.
b.
Later, you find out there were 100 coins in the piggy bank. Based on your experiment,
write down reasonable estimates for the number of each coin. Explain your reasoning.
Pennies –
Nickels –
Dimes –
Quarters –
Homework for Day 5
1.
When you play Monopoly, you sometimes end up in jail. One way to get out of jail is to roll
doubles with your dice. What is the probability of rolling doubles and getting out of jail? Prove
your answer by either using an area diagram or tree diagram.
My Game’s Results
Possible Outcomes
Outcome ______
Outcome ______
Name_________________
Probability Pretest
0
.5
1
1. Underneath each of the numbers on the number line, label an
event that would occur at that probability.
2. The probability of a certain event is 3/8. What is the probability
of that event NOT happening?
3. A bag contains one green marble, two yellow marbles, four blue
marbles, and five red marbles.
a. What is the probability of randomly drawing a red marble
from the bag?
b. What is the probability of NOT drawing a blue marble?
c. If you double the number of each kind of marble, what is
the probability of drawing a blue marble?
d. How many red marbles would you need to add to the bag
to make the probability of drawing a red marble ½?
4. Sam tossed a quarter 10 times. He says it landed on tails every
time.
a. Is this possible? Explain.
b. Is this probable? Explain.
5. You have a deck of 52 cards. What is the probability you will
draw….
a. P(black card)=
b. P(Ace)=
c. P(spade)=
d. P(face card)=
e. P(2 of diamonds)=
6. Using BOTH an area diagram and a tree diagram, show the
sample size for the following experiment. What are the
possibilities when you flip a coin, then roll a die?
Area Diagram
Tree Diagram
a. What is the probability of flipping a heads and rolling a 3?
b. What is the probability of flipping tails and rolling an odd
number?
c. What is the probability of flipping heads and rolling
anything but a 2?
7. Your friend makes up a game with a number cube. He pays
whatever amount in dollars that shows up on the number cube.
a. What is the expected value of your friend’s game?
b. How much should he charge to make it a “fair game”?
8. Fifty students at Forbes Middle School were surveyed about
their favorite sandwich. Here are the results of the survey:
Peanut Butter
Ham & Cheese
Tuna
Roast Beef
22
15
7
6
a. If a student is picked randomly, what is the probability the
student’s favorite sandwich is peanut butter?
b. If a student is picked at random, what is the probability their
favorite sandwich is roast beef?
c. If there are 400 students in the school, how many would you
expect to say ham & cheese is their favorite sandwich?
Name_________________
Probability Post-Test
0
.5
1. Underneath each of the numbers on the number line, label
an event that would occur at that probability. Also, put an
event between 0 and .5 and between .5 and 1.
2. The probability of a certain event is 3/8. What is the
probability of that event NOT happening?
1
3. A bag contains one green marble, three yellow marbles,
three blue marbles, and three red marbles.
a. What is the probability of randomly drawing a red
marble from the bag?
b. What is the probability of NOT drawing a blue marble?
c. If you double the number of each kind of marble, what
is the probability of drawing a blue marble?
d. How many red marbles would you need to add to the
bag to make the probability of drawing a red marble ½?
4. Sam tossed a quarter 10 times. He says it landed on tails
every time.
a. Is this possible? Explain.
b. Is this probable? Explain.
5. You have a deck of 52 cards. What is the probability you will
draw….
a. P(black card)=
b. P(8)=
c. P(heart)=
d. P(black 8)=
e. P(Q of diamonds)=
6. Using BOTH an area diagram and a tree diagram, show the
sample size for the following experiment. What are the
possibilities when you flip a coin, then roll a die?
Area Diagram
Tree Diagram
a. What is the probability of flipping a heads and rolling a
3?
b. What is the probability of flipping tails and rolling an
odd number?
c. What is the probability of flipping heads and rolling
anything but a 2?
7. Your friend makes up a game with a number cube. He pays
whatever amount in dollars that shows up on the number
cube.
a. What is the expected value of your friend’s game?
b. How much should he charge to make it a “fair game”?
8. Fifty students at Forbes Middle School were surveyed about
their favorite sandwich. Here are the results of the survey:
Peanut Butter
Ham & Cheese
Tuna
Roast Beef
13
20
11
6
d. If a student is picked randomly, what is the probability the
student’s favorite sandwich is peanut butter?
e. If a student is picked at random, what is the probability their
favorite sandwich is roast beef?
f. If there are 400 students in the school, how many would you
expect to say ham & cheese is their favorite sandwich?