I'd very much appreciate a solution to the following problem. I should point out that I'm not a student and this problem has NOT been presented to me as some form of test, but rather has occurred as a consequence of a much larger system I've been working on.
On with the problem...
At set intervals (for example every minute) a test is performed to see if a patient has deteriorated or stabilised. These two outcomes are mutually exclusive, in that a patient cannot both deteriorate AND stabilise at the same interval.
Once a patient stabilises they can no longer deteriorate, and therefore no more tests will need to be performed.
If a patient deteriorates six times before stabilising then the patient will expire, and no more tests will need to be performed.
The probability of a patient deteriorating at a given interval is 'd', and the probability of a patient stabilising at a given interval is 's'.
The questions I'd very much appreciate answers to are as follows:
1. What is the probability of patient death after 'n' intervals?
2. What is the probability of patient stabilisation (and therefore not death) after 'n' intervals?
3. What is the probability of a patient still being alive after 'n' intervals, having NOT yet stabilised?
1, 2 and 3 should add up to 1.
I have written a computer program to simulate and sample this problem and give increasingly accurate approximations as sample size increases. However I'd very much appreciate an exact algebraic solution. The benefit to having such a program is that any alegraic solution can be compared to the approximate data from the simulation.
Thank you very much for taking the time to read this problem, and I look forward to your responses.