http://www.ck12.org Chapter 31. Random Walks 1
We have solved for the necessary number of steps left and right in terms of known quantities,NandD. At this point
all that remains is finding how many ways there are to take 1/ 2 (D+N)steps to the right out of a total ofNsteps:
this will give us the number of outcomes that lead to end location ofDsteps to the right.
In the three step case, for instance, ending one space right of the origin required taking two steps right and one step
left; there are three discrete ways to take two achieve this (the left step can be first, second, or third), and so three
outcomes that lead to that location.
For the case ofNtotal steps and 1/ 2 (D+N)steps to the right, the correct result will be given by the ’ways of
choosing’ formula from combinatorics: literally, it is the number of ways to choose 1/ 2 (D+N)positions for the
right steps out of a total ofNpositions. This is written as
(
N
1 / 2 (N+D)
)
where
(
n
k)
=
n!
k!(n−k)!So, according to our earlier result, the probability of finding the walker a distanceDsteps to the right of the origin
is given by the following formula:
P(D) = (^2) ︸︷︷︸−N
P(one outcome)
×
(
N
1 / 2 (N+D)
)
︸ ︷︷ ︸
Number of outcomes that lead to this end positionWe have now answered our original question (finding the probabilities of various end locations) for all unbiased
(P= 1 / 2 )random walks with constant step lengths. Again, we can plot this distribution for several different cases: