Probability at least 3 never respond on a questionnaire When sent a questionnaire, $30\%$ of the recipients respond immediately. Of those who do not respond immediately, $45\%$ respond when sent a follow-up letter. A recipient responding to the questionnaire is independent of other recipients.
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? Options are: A: $0.04$
B: $0.09$
C: $0.14$
D: $0.16$
E: $0.35$
$P(X\geq3) = P(X=3)+P(X=4)$ Probability that $4$ never respond is easy as it is just $${4\choose0}(.3)^0(.7*.55)^4=(.385)^4$$ Now $P(X=3)$ is split between them responding on $1^{st}$ mail and not one the second, or not responding on first and doing so on the second. This is
$${4\choose1}(.385)^3(.3)+{4\choose1}(.385)^3(.45)$$ The final answer is $$(.385)^4+4(.385)^3(.3)+4(.385)^3(.45)=.19317$$ which is not one of the answers.
A: The probability that a particular person does not respond is $0.7\cdot 0.55$.  That's the probability for not responding immediately and not responding to the follow up.
The probability that a particular person does respond is $(1-0.7\cdot 0.55)$.   That's simply the probability for not not responding.   We're not really interested in when they might do so (immediate or follow-up), just whether they do so.
So the probability mass function for exactly $x$ non-responders is simply a binomial distribution pmf.   Specifically: $$\mathsf P(X=x) ~=~\binom 4x (1-0.7\cdot 0.55)^{4-x}(0.7\cdot 0.55)^x\qquad\big[x\in\{0,1,2,3,4\}\big]$$
