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The task
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 pay-off 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.

Circumstances of performance
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 standard-setting
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.

What the work shows
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 well-explained, 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.