Stock Market Investor: Probability of Retirement as a Winner

[Lisawmi]

I am struggling with this problem, and overall have found probability to be a very difficult subject. I was hoping someone could explain to me how to work this problem.

A stock market investor owns shares in a stock whose present value is 25. She has decided that she must sell her stock if it either goes down to 10 or up to 40. If each change of price is either up 1 point with probability .55 or down 1 point with probability .45, and the successive changes are independent, what is the probability the investor retired a winner?

Thanks

 

[juvenal]

A few hints:

1. Binomial tree model.

2. You want either a net number (n) of 15 successes(= stock up) or 15 failures (=stock down).

3. Work it out instead with 1 success or failure (should be easy), then with 2 successes or 2 failures, or n=2. Once you figure out n=2, n=15 should be pretty easy.

4. For n>=2, there are only 3 possible types of outcomes:
a. Retired winner in the stock market
b. Retired loser in the stock market
c. An infinite loop where the number of successes is equal to the number of failures.

I think we can ignore case (c) because the weighted probability of such an event is small.