I am writing a paper for psychology in school on bipolar disorder and one of my hypothesis was that most people don't know anyone with this disease, but the data from my survey indicates that actually 51% of people know someone with this disorder.
I would like to back this up with a calculation of actual probability that a random person knows someone with that disease. I have tried to look for some studies on the internet but wasn't able to find anything, if you find anything please let me know. Otherwise, I would appreciate if you can do some probabilistic estimate, which I don't know how to, since there are too many variables. I would also like to mention that in my study not knowing someone means knowing 0 people with it and knowing someone with it means at least 1 person with it. The probability that a random person has this disorder is 4%, according to my sources.
Thank you,
Matthew
Best Answer
Let's say you know $n$ people. If the first person has the disease, you already know at least one person with the disease. The probability of that happening is $4$%. If the first person does not have the disease but the second person does, you're done as well. The probability is $96$% times $4$% (i.e. $0.96\cdot 0.04$), and so on. All in all, the probability that you know at least one person with the disease will be $0.04\sum_{i=0}^{n-1} 0.96^n = 0.04\frac{1-0.96^{n}}{0.04} = 1-0.96^{n}$.