Question

In a simple random walk, compute the probability P(−1 ≤ X10 ≤ 1), i.e. compute the probability that in ten steps the system will be in position −1, 0, or 1. Give the answer in terms of p which is the probability that a step of one will be taken to right. Also, compute the answer when p = 1/2.

Answer 1

Note that in 10 steps, the system can be in position 0 when 5 of the 10 steps are towards right and 5 are towards left.

Also in 10 steps, the system cannot be at +1 or -1, this is because 10 is an even number of steps and therefore there could only be even number of positions that the system can take after 10 steps. For example with 4 right steps and 6 left steps, it would be at +4 - 6 = -2 and for 4 left steps and 6 right steps the system would be at +2 right.

Therefore, P( -1 <= X₁₀ <= 1) = P(X₁₀ = 0)

This is computed as the probability of having 5 right and 5 left steps. This is computed using the binomial probability function as:

This is the required probability here.

For p = 0.5, this probability is computed here as:

Therefore 0.2461 is the required probability here.