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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 $.3^4+4(.3)^3 (.7)$.

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Your heuristic solution confuses two concepts: the percentage of recipients who respond at a given stage and the probability of a given recipient responding at a given stage.

The easy answer to your question is:

The questionnaire is sent to 4 people, of which 2 (50%) respond immediately. This means that, regardless of how many people respond after the follow up letter (40% of 2 people is 0.8 people, so it's not clear how many responses we should expect), no more than 2 people will never respond. Therefore, the probability that at least 3 people never respond is 0.

The heuristic you present is the solution for the word problem:

When sent a questionnaire, all recipients are 50% likely to respond immediately. Those who do not respond immediately receive a follow-up letter. All recipients of a follow-up letter are 40% likely to respond to the questionnaire. A person's decision to respond to the questionnaire upon receipt and after the follow-up letter is independent of the decision made by other people. If the questionnaire is sent to 4 people and a follow-up letter is sent to any of the 4 people who do not respond immediately, what is the probability that at least 3 never respond?

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  • $\begingroup$ I am so sorry about the misunderstanding; the problem did indeed look like your restatement. I double checked and copied the problem word for word (my problem doesn't mention independence, but I guess it's assumed). I have updated the OP accordingly. $\endgroup$ – Ovi Aug 31 '17 at 13:21
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perhaps you can define you outcome to be a vector of four paired elements, {{R_1^1,R_2^1},{R_1^2,R_2^2},{R_1^3,R_2^3},{R_1^4,R_2^4}}, one pair for each person, representing the first and follow up response. The value is 1 if a person responds, and 0 otherwise.

The assumption could be each person is independent of each other, and their choice of follow up is also independent of the first attempt; and once a person responds, then the second follow up outcome is 'NA'

Then the event is summation of those R's to be smaller and equal to 1, you list all possible cases and compute the probability

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