Peoples Physics Book Version-2

31.1. Introduction http://www.ck12.org

1. Consider all the possible outcomes (as we showed above, they will be equally likely)


2. Think which outcomes lead to which end locations — in particular,how manyoutcomes lead to a particular
   end location.


3. Use the fact that

P(End Location) = (Probability of a single outcome)×(Number of outcomes)

The diagram below is analogous to the one for three steps, but now withNsteps. We can divide the possibilities
into 2Nequally likely outcomes, this time each with probability 1/( 2 N). So, we found the first part in the left side of
the equation above.The question is, ’How many outcomes lead to a given end location, say,Dsteps to the right (as
posed above)?’

There is still only one outcome that leads to each of the two ’extreme’ locations, when all steps are taken either right
or left. Their probabilities are 2−N— but what about the other locations?

To find their respective probabilities, we need to remember the fact that end locations depend on thedifference
between the number of steps taken to the left and right (and not their order) to pose the problem in a slightly
different way.

LetLbe the number of steps taken to the left, andRto the right. Since the total number of steps isN,

N=L+R Total of steps adds toN

If the walkers winds upDto the right of the origin, she must have takenDmore steps to the right than to the left:

D=R−L Total distance traveled

Solving these two equations (work through them yourself), we find that:

R= 1 / 2 (D+N) Steps to the right

L=N−R= 1 / 2 (N−D) Steps to the left