Probability and Statistics
Pre-algebra Probability and Statistics Unit
Linda M. Mrazek
CIED 549: Instructional Design for Educators
University of St. Thomas
East Grand Forks-7 Cohort
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Probability and Statistics
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I ANALYSIS

Rationale/Purpose: The purpose of this unit is to acquaint students with the
probability of independent and dependent events, using theoretical and experimental
probability. Students will also study sample size and population and review graphs
from previous courses in addition to learning about box-and-whiskers and scatter
plots. Probability/data analysis is one of our five math standards, so doing a unit on
this material is critical.

Problem/Needs Assessment: Traditionally, the study of probability and statistics is
tucked into the back of most mathematics texts, so, oftentimes that topic is sacrificed
in order to cover other topics. Unfortunately, our standardized test scores have been
impacted by the omission and/or untimeliness of this unit. Moreover, students
entering the eighth grade have a varying range of knowledge pertaining to probability
and statistics.

Target Group: This unit will be taught to eighth graders enrolled in a pre-algebra
course at South Middle School; all of the math classes are heterogeneous in nature.

Background Content: Since the school year will commence with a portion of this
unit, it is expedient that students' knowledge on tree diagrams, Venn diagrams,
creating and interpreting line graphs, bar graphs, histograms, pie charts and stem plots
be pre-assessed.

Unit Timeline: The first three goals of this unit will be taught within a 10-11 day
time frame, depending on the students' prior knowledge of the material; classes meet
five days a week for 49 minutes each day.

Delivery Options: Lessons from this unit will be delivered through lecture,
demonstration, experiments, cooperative group activities, problem sets from our text,
and various hands-on activities.

Unit Resources: Resources needed for this unit include, but are not limited to,
graphing calculators, rulers, bags and blocks, paper cups, graph paper, two-sided
counters, dice, protractors, compass, and various spinners.

Implementation Instructions: The text will be used for most of the unit, with a few
additional activities included. Students construct their learning through the problem
sets, with the instructor initiating most lessons. Since probability involves
experimentation, there will be hands-on activities through which students will
enhance their learning. Assessments will include daily assignments, exit cards, share
and summarize writing activities, as well as a chapter test and an attitude survey.
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II DESIGN
Goal 1: Students will recognize when previous outcomes influence later ones.
1.1 Given manipulatives (ie.a bag with marbles or blocks, spinners, dice, etc.) students
will answer questions about a single random drawing
1.1.1 Given different scenarios, students will calculate the theoretical
probability of a likely event, a certain event, an impossible event, and
complementary events.
1.2 Given sequential drawings, students will calculate the theoretical probability of
events, both with and without replacement.
1.2.1 Given sequential drawings, students will describe all the elements in the
sample space.
1.2.2 Given different events, students will classify each event as either
dependent or independent.
1.3 Given manipulatives, students will determine experimental probability for various
events.
1.4 Given an experiment, students will compare theoretical and experimental probability.
Goal 2: Students will apply probability to identify whether a game is fair.
2.1 Given various games to analyze, students will determine from the probabilities of the
outcomes and the scoring system whether a game is fair.
2.2 Given various unfair games, students will assign points to game outcomes so that the
games are fair.
2.2.1 Using the probabilities of the outcomes for a particular game, students
will find the least common denominator of the ratios in order to find the
least common multiple of the numerators so as to assign new points to an
unfair game.
2.3 Given an unfair game, students will rewrite the game rules so that the game is fair.
Goal 3: Students will analyze the appropriateness of a sample or a sampling process.
3.1 Given a representative sample, students predict the characteristics of an entire
population.
3.2 Given survey results based on small/restricted samples, students identify inherent
limitations and flaws.
3.2.1 Given a biased survey question, students can identify the bias and rewrite
the survey question in an unbiased way.
3.3 Given different scenarios for surveys, students propose sample
representative populations.
Goal 4: Students will use graphical representations, including scatter plots and boxand-whisker plots, to make inferences and draw conclusions from data sets.
4.1 Given examples of box-and-whiskers plots to analyze, students will correctly identify
range, quartiles, mean, median, maximum, and minimum points.
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4.2 Given sets of data, students will construct a box-and-whisker plot.
4.3 Given displays of scatter plots, students will correctly categorize the graphs by type
of trend depicted (positive, negative, or no trend).
4.4 Given two sets of data, students will construct a scatter plot.
Goal 5: Students will choose appropriate displays for data sets.
5.1 While analyzing any display of a given data set, students will describe the
information that is included and the information that is left out.
5.2 Given a data set, students will construct a display that accurately depicts the data.
Standards:
Grand Forks School District #1
Data Analysis and Probability Standard Grade 8
Students understand and apply concepts of data analysis, probability and statistics.
Benchmark 3.8.1 Formulate questions that can be addressed with data and collect,
organize and display relevant data to answer them.
Critical Knowledge:
 Select, create and use appropriate graphical representations of data, including scatter
plots.
 Formulate questions, design studies and collect data about a characteristic shared by
two populations or different characteristics within one population.
 When using data from other sources, determine which are appropriate, understand
methods of collection, and consider limitations that could affect the interpretation.
Benchmark 3.8.2 Students select and use appropriate statistical methods to analyze data.
Critical Knowledge:
 Use graphical representations, including scatter plots, to make inferences and draw
conclusions from data sets.
Benchmark 3.8.3 Students develop and evaluate inferences and predictions that are
based on data.
Critical Knowledge:
 Make conjectures about possible relationships between two characteristics of a
sample on the basis of scatter plots and approximate lines of fit.
 Use conjectures to formulate new questions and plan new studies to answer them.
Benchmark 3.8.4 Students understand and apply basic concepts of probability.
Critical Knowledge:
 Use methods such as organized lists, tree diagrams, and area models to compare
probabilities for simple and compound events.
 Analyze the concept of fair game and determine from probabilities of outcomes
whether or not a game is fair.
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III DEVELOPMENT
DAYS 1 & 2
Goal 1: Students will recognize when previous outcomes influence later ones.
1.1 Given manipulatives (ie.a bag with blocks, spinners, dice, etc.) students will answer
questions about a single random drawing
1.1.1 Given different scenarios, students will calculate the theoretical probability
of a likely event, a certain event, an impossible event, and complementary
events.
1.2 Given sequential drawings, students will calculate the theoretical probability of
events, both with and without replacement.
1.2.1 Given sequential drawings, students will describe all the elements in the
sample space.
1.2.2 Given different events, students will classify each event as either dependent,
or independent.
DAY 1
Description:
Using a bag containing 6 blue and 4 purple blocks, theoretical probability (favorable
outcomes divided by possible outcomes) will be demonstrated in addition to the
introduction of the terms likely, certain (P(A) = 1), impossible (P(A) = 0), dependent
(P(A, then B) = P(A) x P(B after A)), independent (P(A and B) = P(A) x P(B)), and
complementary events (P(A) + P(~A) = 1). The demonstration will be done with and
without replacement. The same concepts will be reinforced using a single die.
Task 1: Independently, students will complete an investigation, Problem Set A,
page 666, from the text about a single random drawing. A class discussion will ensue
upon completion of this task. (15 minutes)
Task 2: Students will then work in pairs to complete the second investigation
from the text, Problem Set B, page 667; this investigation invokes students to examine
the probabilities when a block is replaced versus not being replaced. Students will create
sample spaces to answer some of the questions. (25 minutes)
DAY 2:
Task 1: Students will engage in a 'share and summarize' writing activity on page
668. This is a formative assessment whereby pairs of students reflect on the results of the
investigations from the day before. (10 minutes)
Task 2*: Students will be given a diagram of a spinner; in pairs, students will find
the probability of complementary events (ie. P(A) and P(~A)).(5 minutes)
Task 3*: Given a spinner coupled with a die and cards with the letters of the
alphabet, students will calculate the probability of independent events. (5 minutes)
Task 4*: Given different events, students will classify each event as either
independent or dependent. (5 minutes)
Task 5: Homework will be given; students will have class time to receive
guidance on this task. (10 minutes)
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Task 6: During the last 5-10 minutes of class, students will respond to three
questions pertaining to the two-day lesson; students will write their responses on an exit
card and present these upon leaving the classroom.
Strategies:
Narrative (introduction of terms), demonstration (bag with blocks and dice), discussion,
guided practice (homework in class), scanning (investigations from text), and reflection
('share and summarize' activity and the exit cards).
Assessments:
Homework: Page 672, #1,2,3, and 6
Exit cards:
1) Give an example of a certain event.
2) Give an example of an impossible event.
3) Given a die, state an event that would have a probability of .5
Reflective writing activity: 'share and summarize,' page 668
* Tasks 2, 3, and 4 can be found on page 14.
DAYS 3 & 4
Goal 1: Students will recognize when previous outcomes influence later ones.
1.3 Given manipulatives, students will determine experimental probability for various
events.
1.4 Given an experiment, students will compare theoretical and experimental probability.
DAY 3
Description:
Task 1: Students will work in pairs, each dropping a paper cup twenty times and
tallying how many times the cup lands rim up, rim down, or on its side. Experimental
probability will then be defined (number of times an event occurs divided by the number
of trials performed). Results will be looked at individually and collectively as a class,
relating the relevance of a large sample space to the validity of experimental
probability.(20 minutes)
Task 2: Students will toss two pennies twenty times, looking for HT, TH, TT, and
HH results. Individual and collective class results will be discussed, comparing
experimental and theoretical probabilities. (20 minutes)
Task 3: Students will begin their homework assignment in class. Students will
complete an exit card before the end of class. (5 minutes)
Strategies:
Narrative (introduction of terms), experimentation (dropping cups, tossing coins),
discussion, guided practice (homework in class), and reflection (exit cards).
Assessment:
Guided practice (homework):
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1) Toss a coin 50 times, recording the results in a table. Find P(H) and P(T).
2) Choose a paragraph from a novel. Find P(a), P(e), P(i), P(o), and P(u). From
the same novel, choose a second paragraph of comparable length and find the
occurrence of vowels again. Compare the two probabilities.
Exit card:
1) Give an example of a pair of complementary events.
2) Why are experimental probability and theoretical probability sometimes
different?
3) If P(red) = 30%, then what is P(~red)?
DAY 4
Description:
Task 1: Students use the graphing calculator to simulate the tossing of a coin(s)
thousands of times. Students will use the results to compare theoretical and experimental
probability, individually and collectively as a class. (20 minutes)
Task 2*: Students complete two worksheets relevant to the lesson.(20 minutes)
Task 3: Students complete an exit card within the last 5 minutes of class.
*Worksheets can be found on pages 15 and 16 of this document.
Strategies:
Narrative (graphing calculator instructions), experimentation (with calculator), discussion
about group results, and guided practice (worksheets started in class).
Assessment:
Guided Practice: worksheets started in class and to be completed as homework
Exit Card: A manufacturer packages 6 mini-boxes of sweetened cereal and 8 mini-boxes
of unsweetened cereal together. You select 2 boxes at random from a package.
Draw a tree diagram that shows the probability of each possible outcome.
DAY 5
Goal 2: Students will apply probability to identify whether a game is fair.
2.1 Given various games to analyze, students will determine from the probabilities of the
outcomes and the scoring system whether a game is fair.
2.2 Given various unfair games, students will assign points to game outcomes so that the
games are fair.
2.2.1 Using the probabilities of the outcomes for a particular game, students will
find the least common denominator of the ratios in order to find the least
common multiple of the numerators so as to assign new points to an unfair
game.
Description:
After introducing what a fair game is (one in which every player has the same chance of
winning), students will begin the class period examining a simple game of tossing 2 fair
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coins. The rules are if the coins show the same faces, you score 1 point; if the coins show
different faces, your opponent scores 2 points.
Task 1: Pairs of students will play 10 rounds of this game for approximately 5
minutes, after which, a discussion will ensue pertaining to the fairness of the game. The
discussion will include the finding of the sample space and subsequent probabilities and
how this affects the allocation of points to make the game fair. (10 minutes)
Task 2: Pairs of students will now investigate a second game with 3 disks, two
that are blue on both sides, and one that is blue on one side and red on the other. After a
flip of all three disks, 1 point is scored if all three are blue, and the opponent scores 2
points if they are not all blue. Fairness and reassigning of points will be discussed. (10
minutes)
Task 3: Students will be in groups of 4, three players and 1 recorder. Groups will
play 9 rounds of the game 1,2,3 Show!, which is depicted on page 677 of the text, and
complete the related problem set (investigation) as a group. (10 minutes)
Task 4: Each group will do the 'share and summarize' reflective writing activity
on page 678. At this point, students will demonstrate whether or not they know how to
reassign points to an unfair game. (10 minutes)
Task 5: Students will complete an exit card reviewing a previous concept. (5
minutes)
Strategies:
Narrative (introduction of terms), experimentation (tossing coins and disks and showing
fingers), discussion, scanning (investigating through problem sets A and B), and
reflection ('share and summarize').
Assessment:
Homework: Students will do #1,2,3,8, and 9 on pages 686-688.
Reflective writing activity: 'share and summarize' on page 678.
Exit card: A rectangular jigsaw puzzle has 200 pieces. The first 100 pieces you take are
not corner pieces. What is the probability that the next piece is a corner piece?
DAYS 6 & 7
Goal 2: Students will apply probability to identify whether a game is fair.
2.3 Given an unfair game, students will rewrite the game rules so that the game is fair.
DAY 6
Description:
Students will spend the entire class period analyzing and consequently rewriting the rules
for the game called What's the Difference? The rules are as follows: two players each roll
a single die; Player A scores 1 point if the difference is 2 or less and Player B scores 1
point if the difference is 3 or more. The first player to reach 5 points wins.
Task 1: Pairs of students will play the game for 5 minutes, recording the results
after each roll. They will then answer the three questions on page 678, which inquire
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about patterns found, both as a pair and collectively as a class, and the fairness of the
game. (10 minutes)
Task 2: Each pair of students will investigate further by describing all the
possible combinations in the sample space. Upon completion of the problem set, students
will be able to state why the game is unfair based on theoretical probability. (10 minutes)
Task 3: For the next 10 minutes, students will complete the third problem set on
pages 679-680. In this third investigation, students will design at least one set of new
rules to make the game fair for 2 players and then rewrite the rules to make the game fair
for 3 players.
Task 4: Students will complete an exit card on a previous concept.
Strategies:
Narrative (introduction of game rules), experimentation (throwing dice), discussion,
scanning (investigating through problem sets C, D, and E).
Assessment:
Homework: Students will do #4 on page 687.
Exit Card: Suppose, when doing a jigsaw puzzle, you like to put together edge and corner
pieces first. You take one piece at a time from the box. If it is a corner or
edge piece, you place it in a pile. If it is neither, you put it back in the box.
Describe the disadvantage of doing a puzzle in this way.
DAY 7
Description:
Task 1: In the same pairings as the day before, students will complete the 'share
and summarize' on page 680. In this reflective exercise, students will create an unfair
two coin toss problem and then rewrite the rules to make the game fair. They must also
justify their solution. (15 minutes)
Task 2: Still in pairs, students will create a fair dart board game on grid paper.
They will assign points and justify why their game is fair. (20 minutes)
Task 3: Students will spend 10 minutes on guided practice.
Task 4: Students complete an exit card on a previous concept.
Strategies:
Reflection ('share and summarize'), demonstration (students will apply what they've
learned about theoretical probability to create a fair game).
Assessment:
Guided practice: Students will do #10 and #12 on pages 688-689 for homework.
Exit Card: 1) You toss 3 coins. Find P( 3 heads ).
2) You roll a die. Find P( not 4 or 5 ).
3) You roll a die and draw an alphabet card. Find P( 6 and A ).
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DAY 8
Goal 3: Students will analyze the appropriateness of a sample or a sampling process.
3.1 Given a representative sample, students predict the characteristics of an entire
population.
Description:
Students will initially engage in two brief activities whereby they will make predictions
about the contents of a bag. In an ensuing discussion, students will share their findings.
Afterwards, the terms sample, population, random sample, sample size, and bias will be
introduced and examples will be given of sampling for medical research and hunting. A
problem set will be completed in addition to a reflective writing assignment and exit card.
Task 1: The first activity involves the drawing of 4 tiles from a bag of 10 tiles,
colored blue, red, green, and yellow; each group will draw three times. The exact
contents are unknown to each pair of students. Each pair will record what they draw (ie.
RRGB) and make an initial prediction, and then refine the prediction after the two
subsequent drawings. A comparison will then be made between the exact contents and
the final prediction. (5 minutes)
Task 2: Given a different bag containing the letters comprising the word
Mississippi, students will do the same process as the first activity. (5 minutes)
Task 3: After the discussion and introduction of terms (10 minutes), students will
complete a problem set on pages 693-694. Students will continue to work with their
partner for this task; they will be answering questions pertaining to two additional
drawings. (15 minutes)
Task 4: Students will complete the 'share and summarize' on page 694. (5
minutes)
Task 5: For the remaining 10 minutes, students will work on guided practice and
complete an exit card on a previously learned concept.
Strategies:
Narrative (introduction of terms and examples shared), reflection ('share and summarize'
and exit card), simulation (drawings and predictions), guided practice (homework
assignment).
Assessment:
Guided practice: Students will do #1-5, 11, and 12, pages 700-704 for homework.
Exit Card: You place 10 cards marked with the digits 0-9 in a box and select one at
random. You put it aside and select another card at random from the box.
a) Find P(3, then 0).
b) Find P(3, then 3).
c) Find P(5, then an even number)
DAY 9
Goal 3: Students will analyze the appropriateness of a sample or a sampling process.
Probability and Statistics 11
3.2 Given survey results based on small/restricted samples, students identify inherent
limitations and flaws.
3.2.1 Given a biased survey question, students can identify the bias and rewrite
the survey question in an unbiased way.
Description:
Students will work in pairs to analyze samples for three different surveys and examine
seven survey questions for bias (attached). After discussing their findings, the class will
then complete a worksheet with the same two main ideas. An exit card will complete the
period's activities.
Task 1: With a partner, students will analyze a bus route survey, a questionnaire
about popular CDs, and a favorite movie survey. Class discussion will ensue. (10
minutes)
Task 2: With a partner, students will examine survey questions about such things
as cars, egg, and friend, for bias. Class discussion will follow. (10 minutes)
Task 3: Students will independently complete, in class, a worksheet that
reinforces the concepts of flawed samples and biased survey questions. (20 minutes)
Task 4: Students will complete an exit card relevant to the objective.
Strategies:
Discussion, scanning (analyzing representative samples and survey questions), reflection
(exit card), guided practice (worksheet in class).
Assessment:
Guided practice: Students will complete a relevant worksheet in class.
Exit Card: Write an example of a biased survey question. Then rewrite the question to
ask for the same information in an unbiased way.
DAY 10
Goal 3: Students will analyze the appropriateness of a sample or a sampling process.
3.3 Given different scenarios for surveys, students propose sample representative
populations.
Description:
Students continue to analyze samples, but now they will be looking at factors other than
size to determine if a sample is truly representative of a population. Students will also
discern whether or not a survey is practical.
Task 1: As a large group, students will brainstorm ideas for collecting a sample
that would help them make a prediction about which of eight activities (playing sports,
playing music, listening to music, watching TV, going to the mall, playing video games,
reading, or drawing/painting) would be the favorites of eighth graders from the four
middle schools in Grand Forks. Students' responses will be kept on the board throughout
the class period. (10 minutes)
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Task 2: With a partner, students will complete the questions in Problem Set C on
page 698; students will be comparing and contrasting 5 strategies for a particular survey.
Class discussion will ensue. (15 minutes)
Task 3: Individually, students will complete Problem Set D on page 699; students
will be making predictions by extracting survey results from a pie chart. Students may
either use a protractor or simply estimate an angle measure to help answer the questions.
(15 minutes)
Task 3: Students will spend the remaining 10 minutes completing the 'share and
summarize' individually. Part of this reflective writing assignment refers to the list of
student responses from the brainstorming activity. For each of the responses, students
need to answer the following questions:
 Is the sample large enough?
 Is the sample representative?
 Is the survey method practical?
Strategies:
Brainstorming, reflection ('share and summarize); tutorial (Problem Sets C and D),
discussion.
Assessment:
Guided practice: Students will do #8-10, 16-18 on pages 703-707 for homework.
DAYS 11 & 12
Description:
Students will complete the chapter review assignment on pages 724-726. A whole class
review will ensue. The test on the first three goals will be given the following day (Day
12)
IV UNIT EVALUATION
Formative Evaluation
Since this unit has not been taught for a couple of years, I will ask our district expert on
probability and statistics to peruse what I've designed and render any constructive
criticism. I will also invite one of the other middle school math teachers to come in and
observe my teaching of this unit, in order to gain feedback and insight. Since my
students work primarily as groups, this affords me the opportunity to observe and
evaluate students on a one-to-one basis. The exit cards that students frequently complete
and the 'share and summarize' reflective writing assignments give day-to-day feedback on
how the students are progressing throughout the unit. On a typical day, this unit will be
taught four times, with one class in the morning and the other three after lunch. What
transpires in the first class sometimes prompts changes in the lesson for the afternoon
classes, so there is ongoing formative evaluation occurring several times a week in
addition to those mentioned above.
Summative Evaluation
Students will be evaluated on their homework performance, which is comparatively a
minor consideration because of students receiving parental assistance as well as help
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from other students outside of class. Quizzes and a final test will be more telling about
what students know, understand, and are able to do regarding probability. Insight will
also be gained from one-to-one evaluations of students that take place in the classroom.
An attitude survey will be given to assess how students feel about what they learned in
the unit. Lastly, during and after each unit taught, copious notes will be taken pertaining
to changes needed to be made to the unit prior to being utilized again next year.
Probability and Statistics 14
ATTACHMENTS
DAY 2
Tasks 2, 3, and 4:
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Day 4 Worksheets
Probability and Statistics 16
Day 4 Worksheets (Cont.)
Probability and Statistics 17
DAY 9 (Class Problem Sets)
Task 1:
A. You think that your school bus stops are too far apart. You want to see if riders on all
the buses agree. Tell whether each survey plan describes a good sample.
a. Interview 50 students who enter the school building.
b. Interview several friends on your bus.
c. Pick four buses at random. Interview every fifth rider as he or she gets off the
bus.
B. A rock band wants to know which songs on their new CD are most popular. Choose
the best survey plan.
a. Interview every sixth person who visits a CD store.
b. Interview 100 people who listened to the rock band's new CD.
c. Interview 50 people who work in the recording business.
C. Suppose you want to find out how often teenagers in town go to movies. For each of
the following, tell whether each survey plan describes a good sample. Justify your
answer.
a. You select a theater at random. One night, you interview all the teenagers who
attend.
b. You select teenagers at random from a list of town residents. Then you interview
them.
c. You interview every student in your English class.
d. You go to your favorite movie theater. You ask people near you how often they
attend.
Task 2: Explain why each of the following questions is biased.
1. Do you like the luxurious car A, or the cheaper model, car B?
Probability and Statistics 18
2. How do you like your eggs cooked?
3. Which shoes would you buy: the high-priced brand-name shoes or the well-made
discount pair?
4. What kind of dog do you prefer to have as a pet?
5. Do you prefer fast food for lunch or a healthy meal from the cafeteria?
6. Is your brother one of your closest friends?
7. Do you think that warm, cuddly, little kittens are nicer than drooling puppies?
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End of Unit Assessment:
Chapter 10 Test-Probability
Name__________________
1. There are eight cards in a bag: one jack, two kings, one queen, and four aces. Sara
reaches into the bag without looking and picks a card.
a. What is the probability that Sara will pick a queen?
b. Is Sara more likely to pick a jack or a king? Explain.
c. The first card that Sara picked was an ace. Which will give her a better chance to
pick a queen on her next pick: putting the ace back in the bag first or leaving it
out? Support your answer with actual probabilities.
2. You have a package of seeds that has 50 pumpkin seeds, 25 squash seeds, 100 carrot
seeds, and 75 tomato seeds. Find the following probabilities:
a. P(pumpkin seed)
b. P(~pumpkin seed)
c. P(squash or carrot seed)
d. P(cucumber seed)
3. Oscar and Chloe are playing a game with two dice, each with sides numbered
1,2,3,4,5, and 6. They roll the dice and add the points. If the sum is less than or
equal to 6, Chloe scores 1 point. If the sum of the numbers is greater than 6, Oscar
scores 2 points.
a. Create the sample space for this game.
b. Is this a fair game? Why or why not?
4. Three coins are tossed. Player A gets 12 points if exactly 3 heads show. Player B
gets 4 points if exactly 2 heads show. Player C gets 3 points if either 0 or 1 head
shows. Is this a fair game? Why or why not?
Probability and Statistics 21
5. Students and teachers at Horace Mann Middle School formed a committee to evaluate
the school lunch service. One thing they wanted to know was the average amount of
money each student spends on lunch in the cafeteria. Various people offered
suggestions for how to gather the information. For each, decide how representative
and how practical the suggestion is. Justify your answers.
a. Hand out a survey to all students.
b. Hand out a survey to students eating lunch in the cafeteria on a particular
Tuesday.
c. Publish a survey in the paper that students can put in a box at the end of a school
day.
d. Have a computer randomly pick 50 students from all the students in the school,
and ask these
students to fill out the survey.
e.
For a week, have the cashiers at the checkout registers record the amount paid by
each student eating lunch.
6. A bag has 3 green markers, 3 blue markers, and 2 yellow markers. You randomly
choose one marker and then replace it. Then you choose a second marker. Find each
probability.
a. P(green and yellow) b. P(green and blue)
c. P(both yellow)
7. A drawer has 3 green socks, 4 blue socks, and 2 black socks. You pick one sock at a
time and don't replace it. Find each probability.
a. P(blue, then black)
b. P(green, then blue)
c. P(black, then green)
Probability and Statistics 22
8. Suppose Jacob tossed 2 coins 10,000 times. How do you think P(2 heads) would
differ from his results using only 20 tosses?
9. A paper cup was tossed twenty times. It landed on its top 5 times. If the 20 tosses is
a representative sampling, in 400 tosses of the cup, how many times would you
expect it to land on its top?
10. Do you prefer Dazzle toothpaste or no toothpaste?
Is this question biased or unbiased?
Explain your answer.
11. Are the events dependent or independent?
a. ______________Flipping a coin twice.
b. ______________Choosing a hammer and a paint color at Menard's.
c. ______________Picking a board from a pile, nailing it on a fence, then picking
another board from the pile.