http://www.ck12.org Chapter 31. Random Walks 1
The first case is equivalent to flipping three heads in a row and therefore taking three right step, the second to flipping
the sequence heads, tails, heads, and so on. Since in this case the coin is fair, the eight outcomes (step combinations)
shown above are equally likely to occur: they each have a probability of( 1 / 2 )^3 = 1 /8.
These outcomes, however, do not all result in different end locations (the four low arrows) for the walker: this is
determined by the difference between the number of steps taken to the right and the number taken to the left. So
while only one outcome corresponds to an end location of three steps to the right or three to the left, three outcomes
correspond to an end location of one step to the right or one to the left, analogously to our calculations in the
bookkeeping section above. So the eight equally likely outcomes result in four possible end locations that are clearly
not equally likely. This is even clearer when we look at a graph of the possible walks, where we can trace the paths
that lead to the same end locations:
Question
Show each of the eight possible step combinations illustrated earlier on the figure above.As we noted earlier, the probability of stopping at a particular end location (such as one step to the right of the
starting point) occurs will equal to the sum of the probabilities of the outcomes that lead to it. This is important, and