SimuLab 5: Random Walk and Pascal's Triangle

Carry out the following steps with the Random Walk program. See Java Applets Page.

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