When sent a questionnaire, the probability is $.5$ that any particular individual to whom it is sent will respond immediately to that questionnaire. For an individual who did not respond immediately, there is a probability of $.4$ that the individual will respond when sent a follow-up letter. If the questionnaire is sent to $4$ persons and a follow-up letter is sent to any of the $4$ who do not respond immediately, what is the probability that at least $3$ never respond?
I can sort of get the solution heuristically because it is a nice "textbook" problem. But this is not good, because I don't think I understand exactly what I'm doing so I can never be $100 \%$ sure of my answer on this type of question (even though I know my answer is right on this one).
I would like to see a more formal solution which defines the experiment(s), defines the events, and states what underlying assumptions we're making (e.g. I'm sure in my solution I assumed some independence somewhere, but I'm not sure where).
Here is my heuristic solution: The probability that one person doesn't respong in both attempts is $(.5)(.6)=.3$, so the probability that four people never respond is $.3^4$.
The probability that $3$ never respond is $4(.3)^3 (.7)$. Therefore the probability that at least $3$ don't respond is $.3^4+4(.3)^3 (.7)$.