I was working with someone on computing certain probabilities related to random walks. The original problem was the following:
> An investment with initial value zero can go up or down $10,000 per
> month with equal likelihood. (a) What is probability that the
> investment will reach $1,000,000 without reaching –$250,000?
We worked out the probability adapting the problem to the gambler's ruin structure, essentially just re-scaling the numbers so that the wins are $\pm 1 $ the starting wealth is 25 and the goal is to reach 125.
We asked ourselves this question: given that we reached 125 before going to zero, what is the probability that now we reach, say 150 before going to zero. Again, by conditioning not a big problem.
But then one asked .. what is the probability the investment will reach \$1,000,000 twice without reaching –$250,000? Here my knowledge started to falter. I get that the probability is quite high because we are starting, in our rendition of the problem, at 125 and we want to reach 125 again before reaching 0 and we have a lot of coins to spend, so to speak before getting ruined. The probability to re-hit 125 in 2, 4, 6, etc. steps can be computed easily, but it is the part "before going to zero" that I was not able to handle...
I would like to inquire about the existence of some kind of formula for computing this type of probabilities.
Thank you