Random Walk and Pascal's Triangle
SimuLab
5: Random Walk and Pascal's Triangle
You
need the Random Walk program to complete this SimuLab. You
can download it to your computer or use it online in a Java applet..
Download
Random Walk:
Carry
out the following steps with the Random Walk program.
1. Choose Pascal's Triangle from the New box on the
control panel.
2. Start by doing 100 trials with 10 steps. (If you wish, select
Less Graphics under Options.)
3. Select Tile Windows under Options to place the resulting
bar graph in one corner of the screen (see 3.6).
4. Start the coin-flipping experiment, run 100 trials, then display
the results in a second tiled window to place it next to the Pascal's
Triangle results.
5. Finally bring up the 1D Random Walk program, start it
doing 100 trials, and show the resulting display in a third tiled
window.
Q3.20: Compare the graphs in the three displays. Are they
identical? similar?
Q3.21: If you ran any of the programs twice in a row,
would you get the same result each time? Similar results
each time? If so, what do you mean by "similar?''
Q3.22: Choose your favorite among the three "experiments''
of the Random Walk program (Random Walk
or Pascal's Triangle) and run some trials that
help you answer the following questions: How likely is
it that a walker will be at least four spaces away from
its starting point (right or left) after taking only four
steps? After taking 8 steps? After taking 12 steps? We
need to figure out how this likelihood (probability) changes
as the number of steps changes.
Under the
Options menu there is a command called
Graph Displacement.
For
Random Walk and
Pascal's Triangle this inserts a
small graph in the lower right corner of the screen, which you can
move around like any other window. On this graph, the horizontal axis
shows the number of steps and the vertical axis displays the
square
of the average distance the walker is from the center after that
number of steps. A green straight line shows the average slope of
these data points.
Q3.23: Are the dots more scattered at the beginning of
a run or at the end?
Q3.24: Are the dots more scattered at the end of a 10-trial
run or at the end of a 100-trial run?
Q3.25: If you ran 30,000 trials,
guess what the
value of the slope would be?
Q3.26: What is going on? Usually when you move in a straight
line your distance itself (not its square) increases linearly
with time. Is this case different? If so, how and why?
Previous: 3.4
- Pascal's Triangle
