Probability that a simple 1d random walk is between [-k,k] in 100 moves What is the probability that a simple 1d random walk is between (-k,k), exclusive, in 100 moves?
My initial though was: $1-\sum_{i=k}^{100}P_i$, where $P_i$ stands for the probability that it reaches $k$ in $i$ number of moves.
 A: You can find a solution by considering two aspects:

*

*The distribution of the one-dimensional random walk on a lattice can be modelled with a binomial distribution.

*If you are not just interested in the position after 100 moves but also about the position of the path in all 99 previous moves, then you can use the reflection principle. This would more or less double the probability of the path extending outside $[-k,k]$ (that would be the case for a continuous distribution with continuous time, for a discrete distribution and discrete time steps it is slightly bit more tricky to work it out and the factor is not exactly 2).

Several questions and answers on this website already have dealt with this and they might help you to get an idea. See: https://stats.stackexchange.com/search?q=%22reflection+principle%22
Another relating post is https://stats.stackexchange.com/a/492091/164061 and you can also compute the probability by considering a Markov chain and compute the distribution step by step, or you can approximate this with an inverse Gaussian distribution.
