One way to picture this is to use one row of a checkerboard or an enlarged photocopy of Figure 2. Do this activity with a partner. The ant steps from one cell to the next, right or left randomly, depending on whether the penny comes up heads or tails, respectively. After ten steps, report the final position of the ant, and whether it is to the right or to the left of the lamp post. Again, the tally keeper puts a big X on a bar graph of the final position on the blackboard, as in Figure 1, above.
Science is about making predictions, often based on measuring some quantities.
You will find that the measure of the mean squared distance is one of the most important and powerful measures of a random walk. In this case symmetry means that moving right is just as likely as moving left. In brief, averaged over many millions of trials, the positive displacements cancel the negative displacements. The average position may be zero, but the spread of positions is not zero. There are various ways to measure this spread.