Exploring Probability with Fathom
Some questions to consider:
Suppose you are playing Monopoly and have found yourself in jail. You are debating on
paying $50 or rolling for doubles. You begin to wonder how many rolls, on average, will
you have to do to get doubles. How many rolls do you think it would take?
Suppose you are flipping a coin ten times. You flip heads on the first two flips and then
ask yourself how many times you think heads will come up in this set of ten flips. How
many do you expect?
Probability Simulations
Rolling a die:
Open a new document window in Fathom. Drag a case table into the window and
create two columns: trial and roll. Under the Collection menu select add cases and
enter 10. Edit the formula for the Trial column (Edit menu, Edit Formula) and
type in CaseIndex for the formula. This will simply copy the case table row
number into the column titled trials. Next, edit the formula for Roll by entering
RandomInteger(1,6).
Construct a graph of the rolls. Add 10 more cases to the table. From the
Collection menu select Rerandomize and observe the changes in the graph.
Change the graph to a histogram and add 80 cases. Then add 500 cases.
Rerandomize to observe any patterns.
Select the graph (click on it) and under the Graph menu select Scale and Relative
frequency. What is the histogram now displaying?
Suppose you want to plot a function that shows where the tops of the histogram
bars should be theoretically. What function should you plot?
Once you’ve plotted the function (use Plot Function on the Graph menu), consider
the area formed by the blue line and the vertical lines that begin and end the bars
of the histogram. What is the area of this rectangle?
Counting 6’s: How often did you “roll” a 6?
Make a new column titled Roll_of_6. Enter the formula
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are needed to s ee this pic ture.
Make another new column titled Count_of_6. Enter the formula Roll_of_6 +

Prev(Count_of_6)
Now look at the number of 6’s that were rolled after 6 rolls (when Trial=6) and
consider what you expected. Does your experimental data match the theoretical
outcome? Continue to look at the number of 6’s after 12 rolls, 18 rolls, 24 rolls,
etc. and note any patterns.
Make another new column titled Freq_of_6. Enter the formula Count_of_6 ÷
Trial.
Drag a new graph into the document window. Place Trial on the x-axis and
Freq_of_6 on the y-axis. Select Rerandomize and observe any patterns. Add 1200
additional cases to bring the total trials to 1800.
As you select Rerandomize, observe the histogram and the number of cases in
each bin as well as the trend in the frequency of 6’s.
What kinds of observations and class discussion might these graphs prompt?
The Distribution of the Sum of Two Dice
Open a new Fathom document and drag a case table onto the document window.
Build a table with columns for the roll of each of two dice and the sum of those
rolls as shown below. Put 72 trials in your table to begin. (New Cases under the
collection menu)
QuickTime™ and a
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Construct a histogram of the sums. Use the Rerandomize tool several times and
try to determine what the most frequently occurring sum is. Select the graph and
under the Graph menu select Scale and Relative frequency. Calculate the area of
each rectangle. What do the areas represent? What is the sum of the areas?
Examining the distribution of sums:
Drag a new graph to the document window. Drag Roll_1 to the x-axis while
holding down on the shift key and Roll_2 to the y-axis while holding down on the
shift key. Change the type of graph from bar graph to breakdown plot.
What does each dot represent in the breakdown plot? (Hint: Consider how many
dots there are)
How does this plot help explain the distribution of the sums that were displayed in
the histogram?
Click on one bar after another in the histogram. You can see that one diagonal
lights up in the breakdown plot for each bar in the histogram. Each cell in the
breakdown plot represents an equally likely roll of the two dice. The height of
each bar in the histogram represents the number of cells in the breakdown plot
that correspond to it.
Compare the theoretical and empirical probabilities displayed in the graphs.
Select Rerandomize several times and examine the patterns.
Extension: Counting Doubles
Create a new column titled Doubles. Edit the formula so it identifies whether
Roll_1 and Roll_2 are equal and assigns a 1 or a 0 depending if they are equal or
not as shown below.
Make another new column titled Count_Doubles. Enter the formula Doubles +
Prev(Count_Doubles). Similar to what we did with the 6’s, look at the number of
doubles that were rolled after 6 rolls (scroll down in the table to when Trial=6)
and consider what you expected. Does your experimental data match the
theoretical outcome? Continue to look at the table and the number of doubles
after 12 rolls, 18 rolls, 24 rolls, etc. and note any patterns.
To observe the long-term trend make another new column titled Freq_Doubles.
Enter the formula Count_Doubles ÷ Trial.
Drag a new graph into the document window. Place Trial on the x-axis and
Freq_Doubles on the y-axis. Add 1118 additional cases to bring the total trials to
1200 and then select Rerandomize observe any patterns
The following are a portion of the NYS Core Curriculum
Standards from the Statistics and Probability Strand in
Algebra that were addressed in this activity.
Algebra
Probability:
A.S.18 Know the definition of conditional probability and use it to solve for
probabilities in finite sample spaces
A.S.19 Determine the number of elements in a sample space and the number of
favorable events
A.S.20 Calculate the probability of an event and its complement
A.S.21 Determine empirical probabilities based on specific sample data
A.S.22 Determine, based on calculated probability of a set of events, if:
 some or all are equally likely to occur
 one is more likely to occur than another
 whether or not an event is certain to happen or not to
happen
Analysis of Data:
A.S.9 Analyze and interpret a frequency distribution table or histogram, a
cumulative frequency distribution table or histogram, or a box-and-whisker plot