Abhay,
I have done work based on your original inquiry.
I have found that the probability distribution for your inquiry is closely related to the probability distribution of a random-walk process for n steps.
For example, I studied the following question:
If a given security had a closing price of 10.00 on 'day 1', and a closing price of 10.30 'day 2', what is the probability that 'day 6' would have a higher closing price than 'day 1'?
Relative to day 1, I counted (well, I didn't count it myself, software did) the number downticks over the next 4 days.
Ultimatley, it created a nice combinatorial model which captured all possible outcomes (0 downticks, 1 downtick, 2 downticks, 3 downticks, 4 downticks) over a 4 day period.
The results I came up with were very close to theoretical results (i.e. if you were to just use probability theory to solve this problem).
If this doesn't make any sense, essentially I showed that if a stock price went up every day for four (4) days, there is no statistical advantage to purchasing that stock - it's price might be higer in 5 days in the future, it might be lower in 5 days in the future.
I studied 500 different companies, with all available historical data for each company (or, about 4,000 years of data in total).
Regards,
Brandon