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By: Lynnea Salscheider
Lynnea82@yahoo.com
By: Jess Stuewe
jstuewe@detlakes.k12.mn.us
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Table of Contents
Day 1 – Pretest------------------------------------------------------------------------ pages 20-22
Day 2 and 3 – Probability Maze------------------------------------------------------ pages 3-7
Day 4 and 5 – Geometric Probability----------------------------------------------- pages 8-9
Day 6 and 7 – The Long Flight Home-------------------------------------------------- page 10
Day 8 – Probability Misconceptions--------------------------------------------- pages 11-12
Day 9 and 10 – Roller Derby------------------------------------------------------- pages 13-14
Day 11 – Matrix Activity------------------------------------------------------------ pages 15-19
Day 12 – Posttest--------------------------------------------------------------------- pages 20-22
Sample MCA-II and Pass Questions----------------------------------------------------page 23
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Probability Mazes
(Ideas and worksheets taken from MATH 6602.)
Standard:
8-6.7 Analyze probability using area models.
8-6.7 Apply procedures to compute the odds of a given event.
7.4.3. Calculate probabilities and reason about probabilities using proportions to solve real world and
mathematical problems.
Practice with some simple area models:
You are given two baskets, two white marbles, and two red marbles. Which arrangement of the two
white and two red marbles in the baskets gives the best chance of drawing a white marble?
Make a copy of the basket activity for each student. Go over each probability model so that students are
comfortable with simple probability. This is a warm-up for the Pirate Island activity.
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Launch:
On Pirate Island, students will be hiding their treasure in cave A or in cave B. Try to make your treasure
as difficult as possible to find. After students have decided on which cave, have students calculate the
probability using an area model and tree diagram.
P(A)= 14/36
P(B)= 22/36
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Explore:
Have students work together in pairs to find the probability of next two mazes. Students need to use an
area model and tree diagram to support their conclusions.
Share:
Have several groups put up area models and tree diagrams on board for discussion. Check to be sure
students agree on probability for the first maze of P(A)= 14/36, and P(B)= 22/36, and the second maze of
P(A)= 36/72 and P(B)= 36/72. Have students share which model they like better and why.
Summarize:
Remind students that probability is the favorable number of outcomes over the total number of
outcomes. Remind students that standardized tests will probably want all answers in simplest form.
Students can also find the odds of arriving on section A and section B after the probability has been
determined.
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Geometric Probability
(Ideas taken from Glencoe’s Pre-Algebra Textbook, 2010 edition, pages 880-881.)
Standard:
8-6.7 Analyze probability using area models.
7.4.3. Calculate probabilities and reason about probabilities using proportions to solve real world and
mathematical problems.
Launch:
Students will work in small groups to find the probability of landing on various area models. Students
need access to the 2010 edition of Glencoe’s Pre-Algebra book, pages 880-881. Have students work in
small groups to find the probability of landing in the shaded region for Example a, and the probability of
landing in the circle for Example b.
Explore:
Have several groups write their findings for the examples listed above. Remind students that they need
to write the ratio of the area of the specific region to the area of the entire target. Students should find
P(shaded region)= 60%, and P(pink circle)= 38%. Next, students will complete page 881 in small groups.
Share:
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After probabilities have been found for page 881, each group will put up a problem so that all 12 are
modeled on the board. Discuss student probabilities and explanations. Have students share which
problems they found were the most difficult, and why they thought so.
Summarize:
Remind students that the concepts of simple probability and geometric probability are the same, except
that geometric probability involves the ratio of the area of a region to the total area.
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The Long Flight Home
(From Navigating through Probability in Grades 6-8, NCTM)
Standard:
8-6.3 Use theoretical and experimental probability to make inferences and convincing arguments about
an event or events.
7.4.3. Calculate probabilities and reason about probabilities using proportions to solve real world and
mathematical problems.
Launch:
Students need a copy of pages 123 and 124 from the Navigating through Probability workbook. Each
pair of students will also need a paper cup, and two red cubes and two blue cubes. Working in groups,
have the students read the problem, and set up the simulation by asking if the two events “same
gender” and “different genders” are equally likely.
Explore:
Students will work in pairs to complete the table on page 123.
Share:
After the students have completed the first activity sheet, ask, “Which outcome occurred more often?”
Record the results for the class on the board. It is likely that “different genders” will have occurred more
often in most of the simulations. Ask students to explain the reason for the results.
Extension:
Have students generate the complete sample space if one of the blue cubes has a “1” on it and the
other blue cube has a “2’ on it. Same goes for the red. Students should use subscripts when labeling.
Summarize:
The students should find that P(same gender)= 1/3, and P(different genders)= 2/3. Remind students that
the likelihood that one male and one female will sit together on the flight home is twice the likelihood
that both friends will be of the same gender. After determining the sample space in question 1, the
students can compare the experimental probabilities and the theoretical probabilities.
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Probability Misconceptions
(Ideas taken from CIMT, University of Exeter Probability Unit.)
Standard:
8-6.3 Use theoretical and experimental probability to make inferences and convincing arguments about
an event or events.
7.4.3. Calculate probabilities and reason about probabilities using proportions to solve real world and
mathematical problems.
Launch:
This activity is intended to provide an opportunity to discuss common probability misconceptions. The
statements given are all incorrect. They can be copied onto cardstock, cut out and distributed to small
groups of students.
Explore:
After students have been given enough time to think about and discuss common misconceptions,
facilitate a whole group discussion on each statement.
Share:
Students will share their finding with the class.
Summarize:
Remind students that probability can be a bit counterintuitive! Even though we may tend to think that
after flipping 10 heads in a row we are more likely to get a tail, but the next flip is independent of the
previous flips, and is therefore still a 50% chance.
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1. I flipped a fair coin three times
and got 3 tails in a row. It is more
likely that I flip a head if I flip
again.
2. There are 3 red blocks and 5 blue
blocks in a bag. I choose a block
at random. The probability that
the block is red is 60%.
3. Anytown Rovers play a soccer
game against Betown United.
Anytown can win, lose, or draw,
so the probability Anytown will
win is 1/3.
4. It is harder to roll a 6 than it is to
roll a 3 on a number cube.
5. John buys 2 raffle tickets. If he
chooses two tickets from
different places in the book he is
more likely to win than if he
chooses two consecutive tickets.
6. My Grandpa smoked 20
cigarettes a day for 60 years and
lived to be 90, so smoking can’t
be bad for you.
7. 13 is an unlucky number so you
are less likely to win a raffle with
a ticket number of 13 than with a
different number.
8. If 6 fair number cubes are rolled
at the same time, I am less likely
to get 1, 1, 1, 1, 1, 1, than 1, 2, 3,
4, 5, 6.
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Roller Derby
(Connected Math Investigation 5, class handout.)
Standard:
8-6.3 Use theoretical and experimental probability to make inferences and convincing arguments about
an event or events.
7.4.3. Calculate probabilities and reason about probabilities using proportions to solve real world and
mathematical problems.
Launch:
Students will be playing Roller Derby. Each pair of students will compete using a game board with
columns numbered 1-12, a pair of dice, and 12 markers. Each student places markers on the columns of
choice. To see who starts, each player rolls a dice, the player with the highest roll goes first. Players take
turns rolling and removing markers from the column that the sum matches. The first player to remove
all markers wins.
Explore:
Allow time for students to play a round of roller derby. Observe student strategies. Ask students if they
need to revise their strategy after playing one game. Most students will move more markers towards
the center, realize it’s impossible to get a sum of 1, etc.
Summarize:
After students have played a few rounds of roller derby, discuss winning strategies. Assist students with
representing the sample space several different ways, such as an addition table or line plot. After
completing sample space, ask the following questions. When we roll two number cubes, how many
different number pairs are possible? (36) Are these pairs equally likely? (yes) How many different sums
are possible? (11) Are these sums equally likely? (no) When you roll two number cubes, what is the
probability that the sum will be 12? (1/36) What is the probability that the sum will be 1? (0) What is the
probability that the sum will be 0? (0)
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Matrix Activity
(Hand-out from Dr. Webb in MATH 6602)
Standard:
8-6.2 Organize data in matrices or scatterplots as appropriate.
7.4.3. Calculate probabilities and reason about probabilities using proportions to solve real world and
mathematical problems.
Launch:
Have copies of the 3 contingency tables for each student. Begin with the anthropology example and
solve as a whole class activity on the smart board. Find the sum of all people that received the present.
Add the female and male columns, as well as the high and low rows. Answer all 4 questions on
worksheet.
Explore:
Students may then complete Blood Type and Flower Problems in small groups. Observe student work.
Share:
Have groups volunteer to share answers on the smart board for Blood Type and the Flower Problems.
Discuss any questions or disagreements students may have.
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Contingency Table Example
An anthropologist working with a horticultural group notices that when the community shares the meat
of a large animal, the hunter gives as a present the head of the animal to somebody of his choice. Both
males and females of high and low status receive the present. The anthropologist records the following
data set.
Female
Male
High
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20
Low
18
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a) What is the total number of heads given away in this data set?
b) What is the proportion of female community members that receive a head?
c) Given that a community member has low status, what proportion of females receive a
head?
d) What proportion of community members that receive a head are considered as having high
status?
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Blood Type
In a survey of 500 randomly selected individuals, a phlebotomist obtained the following data contained
in a contingency table.
Blood Type
Rh-Level
A
B
AB
O
Rh+
176
28
22
198
Rh—
30
12
4
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1. What proportion of individuals have B blood type?
2. What proportion of individuals have an Rh+ Rh-level?
3. If an individual from this group of 500 is randomly selected, what is the probability they have
blood type O?
4. If an individual from this group of 500 is randomly selected, what is the probability they are B—?
5. What proportion of individuals that have an Rh-level of Rh— have blood type AB?
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Flower Problems
A bag of tulip bulbs was found in the garage. The tulips come in four colors: red, purple, yellow, and
orange. They also come in three types based on how quickly they will mature: early, normal, and late.
The bulbs look alike and you cannot tell them apart. The bulb information is contained in the following
contingency table.
Color
Type
Red
Purple
Yellow
Orange
Early
12
6
2
8
Normal
14
10
4
10
Late
8
5
12
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1. How many bulbs are in the bag?
2. What is the probability that if you select one bulb from the bag it will grow to be a yellow tulip?
3. What is the probability that if you select one bulb from the bag it will mature late?
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4. What percentage of bulbs are red or purple?
5. What percentage of bulbs mature early and are orange?
6. What is the probability that if you select one bulb from the bag it will mature late and it will be
purple?
7.
If you select one bulb from the bag and plant it and it starts growing early, what is the
probability that it will be red?
8. If I come to your house and see a purple tulip on your table in a vase, what is the probability that
it started growing late?
9. Write out the distribution of tulip bulbs based on color. In other words, how many are there of
each color?
10. If a tulip bulb is selected from the bag, which color tulip is most likely to be selected?
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Probability Test
Name___________________________
1. Use the maze to find the probability of landing on section A and section B. Use a geometric
model and tree diagram to show how you arrived at your answer.
2.
Find the probability of landing in the purple shaded region.
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3. Four friends-Bob, Bill, Ruth, and Raquel-are among the last to board a jet for a long flight home.
Before they board, the flight attendant announces that because of seat limitations, only two of
them may sit together. List the sample space for the seating arrangement.
4. A fair coin was flipped 3 times and landed on heads all three times. Is it more likely that the next
flip will be a tail? Explain your answer.
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5. You are going to play a game of roller derby. How will you arrange your markers? Use a sample
space to justify your answer.
6. Suppose that 100 individuals are randomly sampled from a very large population and part of a
study of gender differences in handedness. For males, 43 were right-handed and 9 were lefthanded. For females, 44 were right-handed and 4 were left-handed. Organize the data into a
matrix and answer the following questions.
A) How many people were left-handed?
B) What is the probability of selecting a right-handed male?
C) If only females are taken into account, what is the probability of selecting a lefthanded female?
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MCA-II and PASS Sample Questions
11. The probability of 2 people in any given group of 13 people having the same
birthday is about 15%. Based on this information, what is the probability of 2 people in
any given group of 13 not having the same birthday?
A. 2%
B. 13%
C. 50%
D. 85%
13. The odds of picking a green marble from a bag that contains only blue, white, and
green marbles are 4 to 9. What is the probability of picking a green marble from the
bag?
31. Matthew had exactly 20 pennies in his pocket. Five were Canadian pennies and the
rest were U.S. pennies. If he took 1 penny out of his pocket at random, what are the
odds that the penny would be Canadian?
A. 1 to 3
B. 1 to 4
C. 1 to 5
D. 1 to 20
A bag contains 4 yellow tiles and 2 green tiles. A tile will be taken from the bag at
random and set aside. A second tile will then be taken from the bag at random.
What is the probability that both tiles will be green?
A. 31
B. 91
C. 151
D. 181