Rolling Two Fair Dice

Work Sample & Commentary: Rolling Two Fair Dice

The task
Mohammed and Noreen are studying probability. Their math teacher had a pair of fair but unusual dice. These dice each had six sides but one had a zero (0) on a face instead of a four (4) and the other had a zero on a face instead of a one (1).

a)	Find the probability of rolling a sum of eight with these dice. Show all work.
b)	Which sum(s) would be most likely with this pair of dice? Explain why.

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
with manipulatives	with calculator

Mathematics required by the task
The sample space for this task’s experiment of tossing two special dice can be represented by a 6 by 6 table. (See

in the student work.) Each entry in the table is the sum of the faces of the two dice in one of the 36 possible combinations of one of the 6 faces of one die paired with one of the 6 of the other die. Moreover, each of these 36 entries is equally likely, and has the probability

because:

In tossing one fair die, the 6 possible outcomes are equally likely events with probability

. (Note that this probability is unaffected by the fact that the die faces have non-standard markings.)
The outcome of tossing one die is independent of the outcome of tossing the other. This means the two probabilities for a single toss are multiplied together, giving a probability of

for each entry in the table.

Since the sum “8” occurs 4 times in the table, its probability is (4) (

) =

.

Since the sum “5” and “6” each occurs most often in the table (5 times each), these are the most likely sums, with probability (5) (

) =

This work sample illustrates a standard-setting performance for the following parts of the standards:
h	Statistics and Probability Concepts: Create and use models of probability and understand the role of assumptions.
i	Statistics and Probability Concepts: Use concepts such as equally likely, sample space, outcome, and event in analyzing situations involving chance.
j	Statistics and Probability Concepts: Construct appropriate sample space.
l	Statistics and Probability Concepts: Choose an appropriate probability model and use it to arrive at a theoretical probability for a chance event.
b	Problem Solving and Mathematical Reasoning: Implementation.
c	Mathematical Communication: Organize work and present mathematical procedures and results clearly, systematically, succinctly, and correctly.

What the work shows
h Statistics and Probability Concepts: The student creates and uses models of probability and understands the role of assumptions.

i Statistics and Probability Concepts: The student uses concepts such as equally likely, sample space, outcome, and event in analyzing situations involving chance.

j Statistics and Probability Concepts: The student constructs appropriate sample spaces, and applies the addition and multiplication principles for probabilities.

The student listed the faces of the dice in the experiment and constructed a 6x6 model, which the student referred to as a rug, to generate the sample space, i.e., the 36 possible outcomes for the rolling of the dice.

l Statistics and Probability Concepts: The student chooses an appropriate probability model and uses it to arrive at a theoretical probability for a chance event.

From the model, the student counted the number of ways each possible sum could occur, then listed the probability for each sum.

Next, the student used the probability for rolling a sum of “8” (), to answer part (a) of the task. The explanation is not completely explicit; the student explains the meaning of the 36 in the denominator but not the 4 in the numerator. However, knowledge of the concept is evident in the model of the sample space and the listing of the probabilities for each sum.

The student went on to answer part (b) of the task by comparing the probabilities for each sum and found that the probabilities for rolling a “5” or a “6” were equal () and the chances were greater for rolling those sums than other sums with smaller probability ratios.

b Problem Solving and Mathematical Reasoning: Implementation. The student selects appropriate mathematical concepts and techniques from different areas of mathematics and applies them to the solution of the problem.

The student had to formulate an approach that worked. The student modeled the complete sample
space using a clearly labeled and accurate table, then used the sample space to calculate the probability of each event occurring

c Mathematical Communication: The student organizes work and presents mathematical procedures and results clearly, systematically, succinctly, and correctly.

The student presented an orderly approach to the problem, explained the steps of the solution process clearly and concisely, and arrived at a result that is correct.