# Random walk in one dimension multinomial

A particle starting at the origin once every second has a 1/6 probability of jumping two steps forward, a 1/6 probability of jumping one steps forward, a 1/3 probability of jumping one step back, and a 1/3 probability of staying in the same place.

Calculate the probability that at the 36 second mark the particle is more than ten units away from the origen.

I've messed around with this problem and think the expected value is +6 units from the origen and the variance is 41 but I don't know what to do to actually figure out the probabilities here. Thanks.

• Do you know whether this is expected to be done by hand or with a computer? Because by hand the only approach I can see that isn't hopelessly tedious is a normal approximation. For that normal approximation, all you need is the mean and variance that you just calculated.
– Ian
Nov 22, 2019 at 20:50
• Thanks. Definitely by hand. I actually had though of that + that was my response. My TA cautioned me that that wasn't the right approach but it was his first time looking at the problem and I think he was mistaken thinking that multinomial problem solving wouldn't be tedious. Nov 22, 2019 at 21:00