STATISTICS Day 2 AM
Day 2 AM Objectives: To compare probability with statistics and explore the similarities and the differences.
8. TASK: Experimental Pitching (30 minutes)
TEACHER NOTES
Purpose: To surface ideas concerning the
probability vocabulary of experiment, sample
space, event, outcomes.
Core Standards Focus:
7.SP.7: Develop a probability model and use it to
find probabilities of events
S.CP.1: Describe events as subsets of a sample
space.
Launch (Whole Class): Ask what is meant when someone conducts an experiment. Ideas should
include a scientific experiment, or an experiment to see what happens by a random process. Mention
that many things in life can be considered an experiment, and for this task they are to consider the
pitching of a baseball as an experiment.
Explore (Groups): For students struggling to get started, ask them to describe what happens when a
baseball is pitched, what are the variables?
Possible responses could be the types of pitches (fast, curve, etc) or the result of the pitches (ball, hit,
strike, foul)which are the outcomes or possible sample space.
Discuss (Whole Class):
Select students’ responses to share. Make clear the following definitions as part of the students’
responses which should be placed in a journal.
 Within the discipline of probability, an experiment is an activity for which outcomes occur
randomly (i.e., based upon chance). (Depending upon the class’s responses, you may want to
discuss whether the outcomes of baseball pitching would be considered random or not due to
the pitcher’s and the batter’s ability.)
 The set of all outcomes for a given experiment is called the sample space of the experiment.
 An event is a subset of the sample space. That is, an event is a collection of outcomes.
Discuss the place of games in the context of probability as being an experiment. Compare/Contrast
this context with scientific experiments.
Ask whether Dialing your friend’s number on a cell phone is an experiment? (No, because the
number is not random.)
Day 2 AM: 8:30 – 9:00
FACILITATOR NOTES
While we have been talking about statistical
experiments and studies, our focus is
turning to probability experiments.
Discuss whether the data collection in Give
Me a Hand could be considered a
probability experiment or not.
Yes it is a probability experiment if we
consider the activity as a handful and the
randomness as the number of pieces of
cereal a person would pick would change
each time a handful is taken. The event
could be getting more than a certain
number.
It is also a statistical experiment because
we are gathering data to infer a significant
difference between genders.
It is not the activity that counts, but it is the
question you are asking.
Debrief:
How are probability and statistics related?
Discuss the relationship between
probability and statistics.
“Probability is a tool for statistics.” (Gaise
Report p8) Assume a coin is “fair”. If we
toss the coin 5 times, how many head will
we get is a mathematical probability
problem. Pick up a coin. Is this a fair coin?
is a statistics problem that can use the
mathematical probability to seek a solution.
Where would this task be in the learning
cycle?
Where would we go from here?
STATISTICS Day 2 AM
Day 2 AM Objectives: To compare probability with statistics and explore the similarities and the differences.
Is closing your eyes and haphazardly dialing ten digits on a cell phone an experiment? (Yes, because
the number is sort of random.)
Discuss the role of random in those events.
Additional experiments for students to define the outcomes, sample space, and an event of interest:
Taking a swing at a piñata
Playing Pin-the-Tail-on-the-Donkey
STATISTICS Day 2 AM
Day 2 AM Objectives: To compare probability with statistics and explore the similarities and the differences.
9. TASK: Dice Games (60 minutes)
TEACHER NOTES
Day 2 AM: 9:00 – 10:00
FACILITATOR NOTES
Purpose: To use probability to make a decision.
Give a pair of dice to each pair of
participants
Core Standards Focus:
7.SP.5-8: Investigate chance processes and develop, use, and evaluate
probability models.
S.MD:Using Probability to Make Decisions and Evaluate Outcomes of
Decisions
S.CP.1: Describe events as subsets of a sample space as unions,
intersections, or complements of other events.
Related Standards: S.CP.2-5
Materials: A pair of dice for every two students.
Launch (Whole Class): Ask students why they play games? Response should be that they play to
win. Tell them we are going to play games but that they first get to decide which player they want to
be. Read together the rules for the First Game and everyone indicate which player they would want
to be (only give 10 seconds at most so they are going with their gut reaction). Repeat for the Second
game. Have them play the games with their table partner.
Explore Part I (Partners): As the students are playing, watch to see that they are correctly
determining the winner in each roll and keeping a tally. Listen for statements of “Not Fair” to be
discussed as a whole class.
Discuss (Whole Class):
Ask the students who protested the “fairness” to explain what they saw happening. Compile the
number from all the partners of which player won:
Number of Players A Number of Players B
who won
who won
Game One
Game Two
What do our results tell us about fairness?
Ask the class: According to our definition of experiment, is this an experiment? Be sure to define the
outcomes and the sample space. When discussing the outcomes, ask the question, how many ways
Debrief:
Where do the big ideas of probability fit in
the core?
Discuss the place of probability in the core
and in the courses.
Concept of probability is in Math 7 and in
Secondary II the rules of probability are
built.
How can we use dice games to build
understanding of other probability
situations such as intersections or unions?
STATISTICS Day 2 AM
Day 2 AM Objectives: To compare probability with statistics and explore the similarities and the differences.
can we get a particular sum? List all the sums on the board. Discuss whether it is possible to get
certain sums more ways than other sums.
Explore Part II (Partners): Have the partners create a table to show all the different ways to get the
sums. You may need to explain that doubles are only counted once because you cannot tell the
difference. For example you can tell the difference with two colored dice between a green 6 and a red
3 with a green 3 and a red 6. But you cannot tell the difference between a green 3 and a red 3 with a
red 3 and a green 3.
Discuss (Whole Class):
Have someone share their results that was very systematic in thinking about the combinations. Lead
the discussion to going back to the list of sums, and adding the probability for each sum occurring.
Return to the idea of an experiment, the outcomes, and the event. What was the event for First
Game? (If you’re player A, getting an odd sum) What was the event for Second Game? (If you’re
player A, getting a sum between 5 and 9 inclusively)
Direct the partners to consider the two games. What is the probability that each player would win, in
other words, would get their event. After a couple of minutes, discuss how the probability for each
player on the first game is ½, but that on the second game, Player A has a probability of winning of
2/3 while Player B has a probability of winning of 1/3.
Make clear the definition of a game being fair: A game is fair if the each of the players is equally likely
to win.
How did our theoretical probabilities compare to our experimental probability?
Additional problems based on the sum of two dice is Dice Games Part II.
STATISTICS Day 2 AM
Day 2 AM Objectives: To compare probability with statistics and explore the similarities and the differences.
10. TASK: Mad Probabilist (45 minutes)
TEACHER NOTES
Purpose: To use combinations in probability and the use of
random chance in an event.
Core Standards Focus:
S.CP.9: Use permutations and combinations to compute
probabilities of compound events and solve problems.
Related Standards: 7.SP.8
Project a picture of the illustration to aid in discussions.
Launch (Whole Class): Read together the situation by which the
Mad Probabilist would decide his path. Decide as a group a different
path that would get him to the Café. Discuss what the experiment is
(activity is moving along the grids and the outcomes are the
different paths.) Now point out the question “How many different
ways are there?” Ask them to work in their groups to answer that
question.
Explore I (Groups): As students begin exploring ways to get to the
café, they might realize there are a lot of different ways. Allow them
about 5 minutes to begin making a table, creating a list, a tree
diagram, or other method to determine all the ways.
Discuss I (Whole Class): Bring them together to discuss the different methods they used and to
wonder if there’s an easier way. Have them return to the paper (2nd side) and discuss the answer to
the next question “How many moves does it take to get to (5, 3). Then ask where else he might be
after 8 moves to make sure students are getting the idea of the different ending locations for a set
number of moves. Encourage them to use grid or dot paper to answer the rest of the questions and
that we will come back to the question of how many different paths he could take to get to (5, 3)
Explore II (Groups): Watch to make sure students are aware that for a set number of points, there is
a diagonal line of possible locations. As the students get to the second side, watch to see that they are
counting the ways to each set of locations requiring the same number of moves. Watch and listen for
anyone who can see a pattern in the number of paths as the number of moves increase.
Day 2 AM: 10:00 – 10:45
FACILITATOR NOTES
Keep the first Explore to under 10 minutes
so they will have time to work through the
diagram.
Debrief:
How is this task a probability experiment?
Define the sample space, the outcomes, and
the event.
Sample space is all possible routes.
Outcome is a specific route.
The Event is a subset of the sample space
defining a particular point as the target.
Discuss the standards and content topic for
this task.
(This task is an introduction into
combinatorics which is an honor topic in
Sec II. However, this point does not need to
be expressed to participants unless
needed.)
Where in the learning cycle is this task?
STATISTICS Day 2 AM
Day 2 AM Objectives: To compare probability with statistics and explore the similarities and the differences.
Discuss II (Whole Class): Class discussion should center around anyone who has seen the pattern
of adding the previous number of paths coming into a new location. For example the location (2, 2)
would have 3 ways since there were 2 ways to get to (1, 2) and 1 way to get to (2,0). Let students
know this is Pascal’s Triangle and has become a very useful tool in counting problems. By building
the number of paths, we can determine that there are 56 ways to get to the Slow Food Café.
Ask the question, what is the probability that the Mad Probabilist’s random walk will end up at the
Slow Food Café? Review that the event is to end up at (5,3) so the number of paths to (5, 3) is 56 and
the number of paths for 8 moves would be the sum of all the number of ways to get to the locations
after 8 moves (256) the probability is 56/256 or about 22%.
Ask what is the probability he will end up at (7, 1) where the barbershop is. (8/256=3%) Why is this
location a much lower probability? Additional questions using the table created could also be asked
to bring the idea of combinations into probability.
STATISTICS Day 2 AM
Day 2 AM Objectives: To compare probability with statistics and explore the similarities and the differences.
11. TASK: Is the Teacher Really Psychic? (45 minutes)
TEACHER NOTES
Purpose: To use probability to determine the likelihood of a claim.
Core Standards Focus:
7.SP.7: Compare probabilities from a model to observed frequencies.
S.MD.7: Analyze decisions using probability concepts
S.IC.2: Decide if a specified model is consistent with results from a given
data-generating process.
Related Standards: 7.SP.5, 7.SP.6, S.IC.1, S.IC.3
EU6-8: 4. Inferential statistics uses data in a sample selected from a
population to describe features of the population
EU9-12: 3. Hypothesis tests answer the question Do I think this could
have happened by random chance?
Materials needed: Deck of cards for each group.
Ahead of time, seed a display calculator: type 929 (whatever number you choose) press the store key
STO> next to the number 1, then press the Math key arrow over to the PRB menu and select rand
press enter, and enter again. You should see the number 929 on the screen. Select randint(1,4) in the
calculator and when you hit enter 8 times you will get the sequence 3, 1, 2, 4, 4, 4,2,4,1 Memorize the
suits that correspond to these numbers according to the launch: club, heart, diamond, spade, spade,
spade, diamond, spade, heart. You are psychic! Be sure to seed that number right before the activity
without any of the students knowing.
Launch (Whole Class): Tell your students that you are psychic and that you are able to determine
the suit of a card that can be randomly generated from a calculator. Put the following key on the
board: 1 is a heart, a 2 is a diamond, a 3 is a club, and a 4 is spade. Have someone enter the
command randint(1,4) which will generate a number randomly from 1 to 4, but before hitting
enter, you will tell them what the suit will be. Someone will probably suggest that the calculator was
rigged. Ask the students to talk with a table partner about how we could decide whether the teacher
is psychic using cards rather than a calculator. (question #1)
Discuss together the idea of this being an experiment. What is the activity? (Drawing 3 cards with
replacement.) What are the outcomes or sample space? (The four suits three at a time in different
combinations, 64 ways.) What is the event (also known as random variable) we are interested in?
(Getting all 3 correct.) How could this help us decide?
Day 2 AM: 10:45 – 11:30
FACILITATOR NOTES
You will need to seed a display calculator
without anyone seeing.
This task will introduce the use of
simulation. Rather than cards, generate 3
random integers from 1-4 where 1
indicates you guessed correctly, and 2-4
you guessed incorrectly. Have each person
run 5 trials keeping track of how many
times he got 0, 1, 2, or 3 correct. Compile
all the results into one table to determine
class experimental probabilities.
Debrief:
How does a probability simulation allow us
to evaluate a statistical claim?
There is evidence that the performance of
the teacher is successfully identifying 8
cards correct is not random. There is no
evidence that being psychic is the reason
for the non-randomness.
Where would this task be in learning cycle?
Discuss the binomial expansion of
determining probabilities.
STATISTICS Day 2 AM
Day 2 AM Objectives: To compare probability with statistics and explore the similarities and the differences.
Explore (Group): Have students conduct the
experiment 10 times using a deck of cards and
determine the experimental probability of being
psychic.
Have them continue the questions to consider
the theoretical probability.
Discuss (Whole Class):
Have a student explain how to determine
theoretical probability. Compare it to a
combined experimental probability from the
class. Have different students with opposing
beliefs about the psychic ability of the teacher
explain their case. They should use either the
experimental or theoretical probabilities as
evidence of their decision. Discuss what a good
chance is as opposed to a bad chance. Make
clear that we are asking the question of whether
getting three cards correct in a row could
happen by random chance or if there is some
other ability at work to make it happen.
Answers: 4 a) ¼ b) 27/64, 27/64, 9/64, 1/64
c) 15.6%