Students were given the following task:
The parent/teacher association is organizing the annual fundraising carnival.
The eighth grade class will be allocated one booth in which to conduct
a fundraising activity. About 500 people usually attend this event.
Your task is to design an activity that uses multiple probability events
and that would attract lots of players. In your plan, please explain why
you think your activity would be fun to play, how much you would charge
to play, how much you would pay a winning contestant, and how much profit
you would expect to make from your activity. A team of students and parents
will select the proposal they think would be the most successful.
The task called for students to design an “unfair” game
in which the organizers could be expected to make money. The game
requires multiple probability events, which could be either independent
or dependent events. The task includes a subjective criterion in
requiring students to predict how many times the anticipated 500
attendees will play the game. Upon that assumption, students then
needed to determine the expected payoff of the proposed game of
chance.
The task is clear in stating the information required if the student
understands “multiple probability events,” yet it also
leaves much freedom for the student to develop the problem, both
in concocting the game and in predicting how many players the game
will draw.


A good extension for this task would be to make conjectures about the
numbers of game players when adjusting the values of either the prize
money or the chances of winning. In this case, an optimization problem,
based on the hypothesized numbers of players, manifests itself. The students’
carnival choice could then be the one that would seem to yield maximum
profit.
This sample of student work was produced
under the following conditions: 
 alone 
in a group 
 in class 
as homework 
with teacher feedback 
with peer feedback 
 timed 
opportunity for revision 
This is an unrevised draft of a timed assignment completed in 40 minutes.
This work sample illustrates a standardsetting
performance for the following parts of the standards:

h 
Statistics and Probability
Concepts: Represent and determine probability, recognize equally likely
outcomes, and construct sample spaces. 
i 
Statistics and Probability
Concepts: Make predictions based on experimental or theoretical probabilities. 
j 
Statistics and Probability
Concepts: Predict the result of a series of trials once the probability
for one trial is known. 
a 
Problem Solving
and Mathematical Reasoning: Formulation. 
b 
Mathematical Communication:
Organize work, explain a solution orally and in writing, and use other
techniques to make meaning clear to the audience. 
d 
Mathematical Communication:
Exhibit developing reasoning abilities by justifying statements and
defending work. 
The work provides strong evidence for parts of
. It provides evidence for determining probabilities of events, for example,
the probabilities of coin tossing and dice throwing and of the multiple
event created by combining the independent events into one game. The work
shows the student computing the profit he can expect at the booth, based
on his assumptions about how many people will play. The student works
within the constraints of the task to formulate his game of chance and
predict the profits to be made. Thus, his work provides evidence for part
of
. The solution is wellexplained, providing evidence for
.
h
Statistics and Probability Concepts: The student
represents and determines probability as a fraction of a set of equally
likely outcomes; recognizes equally likely outcomes and constructs sample
spaces….
The student
was careful to clarify and justify the claim that P[rolling 11 or 12]
= ,
by alluding to the sample space of 36 equally likely outcomes when tossing
two dice.
i
Statistics and Probability Concepts: The student
makes predictions based on experimental or theoretical probabilities.
j
Statistics and Probability Concepts: The student
predicts the result of a series of trials once the probability for one
trial is known.
The student
displayed understanding of expected value, realizing that his
probability implied that he could expect one winner per 24 games played.
a
Problem Solving and Mathematical Reasoning:
Formulation. The student formulates and solves a variety of meaningful
problems….
In
the first two paragraphs, the student formulated the problem needing
analysis by deciding on a game, a fee, and a prize, and by making
a guess as to the number of people who will play. One could challenge
the assumption of 400 game players. Would that many people really
pay $1 for a chance to win $10 when the odds are so slim (1 in 24)?
The student
noted correctly that his expected profit is $14 for each group of
24 players. Since a total of 400 players contains between 16 and 17
groups of 24, his expected profit should be between 16 x $14 = $224
and 17 x $14 = $238, so the range $224 to $238 should replace the
student’s response of $384 to $408 (which are 16 x $24 and 17
x $24). This error does not detract from the evidence of understanding
in the rest of the work. 

b
Mathematical Communication: The student organizes
work, explains facets of a solution orally and in writing, labels drawings,
and uses other techniques to make meaning clear to the audience.
d
Mathematical Communication: The student exhibits
developing reasoning abilities by justifying statements and defending
work.
The student
was careful to clarify and justify the claim that P[rolling 11 or 12]
= , by alluding
to the sample space of 36 equally likely outcomes when tossing two dice.
