http://www.ck12.org Chapter 31. Random Walks 1
Question
What is the probability of each of the three possible outcomes?Answer
The probability of landing two steps to the right is the same as that of rolling two heads with the coin above.
This equalsP×P=P^2. Analogously, the probability of landing two steps to the left is( 1 −P)^2. The
probability of landing in the original position is equal to getting one heads and one tails. You might think
this isP( 1 −P), but it’s slightly subtler: since theorderof the heads and tails doesn’t matter, there are two
ways to get this outcome: heads first, tails second, or vice versa (right then left, or left then right). Therefore
we have to add the probability of one such outcome,P( 1 −P), to the probability of the other,( 1 −P)P. So the
result is 2×P( 1 −P).Note thatP^2 + 2 (P)( 1 −P)+( 1 −P)^2 =1. Why is this important and relevant?
Motivation
You might ask at this point: what’s the meaning of this model? Can something so simple actually be useful? And if
the model is useful, what exactly are we trying to find?
These are good questions. First, it turns out that the model is useful; first, because many real world phenomena
— stock prices, gambling wins/losses, certain quantum phenomena — can actually be modeled as one-dimensional
random walks. More importantly, however, many of the one-dimensional results happen to transfer easily to two and
three-dimensional random walks, which actually model a much greater range of phenomena.
Question
What situations can be accurately modeled by two and three-dimensional random walks?