I'm baffled as to how to explicitly solve this problem... I would normally just plug in problem-specific values and use Monte Carlo simulation to solve something complicated like this, but my coursework is asking for a general solution.
A rocket is launched from the Earth. Random forces cause the rocket to move left and right. In an interval of time t, the rocket is moved according to a normal distribution N(t), where it is equally likely to be moved left or right. There is an asteroid belt on the right side x miles away. 1. What is the probability that the rocket will move j miles to the left in a*t (where j is less than x and a<=1)? 2. What is the probability that the rocket will hit the asteroid belt and explode? 3. What is the probability that the rocket will move into the “danger zone” (beginning at y miles, where 0 is less than y is less than x) and then return to safety? 4. What is the probability that the rocket will move into the danger zone and then return to safety z times (without an asteroid belt being present)? With an asteroid belt being present?
I'm thinking that this could be beyond my somewhat obscure graduate textbook's content, so any recommendations to a more comprehensive and clear statistics/probability textbook that would lead to a solution to this kind of problem are welcome. Thanks for your help guys.